├── .github └── workflows │ └── main.yml ├── .gitignore ├── LICENSE ├── Makefile ├── README.rst ├── docs ├── Makefile └── source │ ├── conf.py │ └── index.rst ├── flake.lock ├── flake.nix ├── papis_zotero ├── __init__.py ├── bibtex.py ├── server.py ├── sql.py └── utils.py ├── pyproject.toml ├── tests ├── .gitkeep ├── __init__.py ├── resources │ ├── bibtex │ │ ├── files │ │ │ ├── 8 │ │ │ │ └── Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf │ │ │ ├── 10 │ │ │ │ └── De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf │ │ │ ├── 12 │ │ │ │ └── Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf │ │ │ └── 15 │ │ │ │ └── Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf │ │ └── zotero-library.bib │ ├── bibtex_out.yaml │ ├── sql │ │ ├── storage │ │ │ ├── 5KW7TMDH │ │ │ │ ├── .zotero-ft-cache │ │ │ │ ├── .zotero-ft-info │ │ │ │ ├── .zotero-pdf-state │ │ │ │ └── Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf │ │ │ ├── J8FIHBUY │ │ │ │ ├── .zotero-ft-cache │ │ │ │ ├── .zotero-ft-info │ │ │ │ ├── .zotero-pdf-state │ │ │ │ └── Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf │ │ │ ├── PIMHYJGK │ │ │ │ ├── .zotero-ft-cache │ │ │ │ ├── .zotero-ft-info │ │ │ │ ├── .zotero-pdf-state │ │ │ │ └── De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf │ │ │ └── WN7WJBGS │ │ │ │ ├── .zotero-ft-cache │ │ │ │ ├── .zotero-ft-info │ │ │ │ ├── .zotero-pdf-state │ │ │ │ └── Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf │ │ └── zotero.sqlite │ └── sql_out.yaml ├── test_bibtex.py └── test_sql.py └── tools └── update-pypi.sh /.github/workflows/main.yml: -------------------------------------------------------------------------------- 1 | name: CI 2 | 3 | on: 4 | workflow_dispatch: 5 | push: 6 | branches: [ "main" ] 7 | tags: [ 'v**' ] 8 | pull_request: 9 | branches: [ "main" ] 10 | schedule: 11 | # 17:00 on Friday (UTC) 12 | - cron: "00 17 * * 5" 13 | 14 | concurrency: 15 | group: ${{ github.workflow }}-${{ github.event.pull_request.number || github.ref }} 16 | cancel-in-progress: true 17 | 18 | jobs: 19 | build: 20 | runs-on: ${{ matrix.os }} 21 | strategy: 22 | matrix: 23 | os: [ubuntu-latest, macos-latest, windows-latest] 24 | python-version: ["3.8", "3.9", "3.10", "3.11", "3.12", "3.13"] 25 | fail-fast: False 26 | 27 | steps: 28 | - uses: actions/checkout@v3 29 | - name: Set up Python ${{ matrix.python-version }} 30 | uses: actions/setup-python@v4 31 | with: 32 | python-version: ${{ matrix.python-version }} 33 | 34 | - name: Install dependencies 35 | run: | 36 | make ci-install 37 | 38 | - name: Lint with flake8 39 | run: | 40 | make flake8 41 | 42 | - name: Lint with mypy 43 | run: | 44 | make mypy 45 | 46 | - name: Test with pytest 47 | run: | 48 | make pytest 49 | 50 | pypi-release: 51 | needs: [build] 52 | name: PyPI Release 53 | environment: pypi 54 | permissions: 55 | contents: write 56 | id-token: write 57 | runs-on: ubuntu-latest 58 | steps: 59 | - uses: actions/checkout@v4 60 | with: 61 | fetch-depth: 0 62 | 63 | - uses: actions/setup-python@v5 64 | with: 65 | python-version: '3.12' 66 | 67 | - id: dist 68 | run: | 69 | make ci-install-build-system 70 | python -m build . 71 | 72 | - if: github.event_name == 'push' && startsWith(github.ref, 'refs/tags') 73 | name: Publish Package to PyPI 74 | uses: pypa/gh-action-pypi-publish@release/v1 75 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | doc/build/ 2 | #Vim swap files 3 | *.swp 4 | *.swo 5 | 6 | # Byte-compiled / optimized / DLL files 7 | tags 8 | __pycache__/ 9 | *.py[cod] 10 | *$py.class 11 | 12 | # C extensions 13 | *.so 14 | 15 | # Distribution / packaging 16 | .Python 17 | env/ 18 | build/ 19 | develop-eggs/ 20 | dist/ 21 | downloads/ 22 | eggs/ 23 | .eggs/ 24 | lib/ 25 | lib64/ 26 | parts/ 27 | sdist/ 28 | var/ 29 | *.egg-info/ 30 | .installed.cfg 31 | *.egg 32 | 33 | # PyInstaller 34 | # Usually these files are written by a python script from a template 35 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 36 | *.manifest 37 | *.spec 38 | 39 | # Installer logs 40 | pip-log.txt 41 | pip-delete-this-directory.txt 42 | 43 | # Unit test / coverage reports 44 | htmlcov/ 45 | .tox/ 46 | .coverage 47 | .coverage.* 48 | .cache 49 | nosetests.xml 50 | coverage.xml 51 | *,cover 52 | .hypothesis/ 53 | 54 | # Translations 55 | *.mo 56 | *.pot 57 | 58 | # Django stuff: 59 | *.log 60 | local_settings.py 61 | 62 | # Flask stuff: 63 | instance/ 64 | .webassets-cache 65 | 66 | # Scrapy stuff: 67 | .scrapy 68 | 69 | # Sphinx documentation 70 | docs/_build/ 71 | 72 | # PyBuilder 73 | target/ 74 | 75 | # IPython Notebook 76 | .ipynb_checkpoints 77 | 78 | # pyenv 79 | .python-version 80 | 81 | # celery beat schedule file 82 | celerybeat-schedule 83 | 84 | # dotenv 85 | .env 86 | 87 | # virtualenv 88 | venv/ 89 | ENV/ 90 | 91 | # Spyder project settings 92 | .spyderproject 93 | 94 | # Rope project settings 95 | .ropeproject 96 | changes 97 | *~ 98 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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You may not convey a covered 525 | work if you are a party to an arrangement with a third party that is 526 | in the business of distributing software, under which you make payment 527 | to the third party based on the extent of your activity of conveying 528 | the work, and under which the third party grants, to any of the 529 | parties who would receive the covered work from you, a discriminatory 530 | patent license (a) in connection with copies of the covered work 531 | conveyed by you (or copies made from those copies), or (b) primarily 532 | for and in connection with specific products or compilations that 533 | contain the covered work, unless you entered into that arrangement, 534 | or that patent license was granted, prior to 28 March 2007. 535 | 536 | Nothing in this License shall be construed as excluding or limiting 537 | any implied license or other defenses to infringement that may 538 | otherwise be available to you under applicable patent law. 539 | 540 | 12. No Surrender of Others' Freedom. 541 | 542 | If conditions are imposed on you (whether by court order, agreement or 543 | otherwise) that contradict the conditions of this License, they do not 544 | excuse you from the conditions of this License. If you cannot convey a 545 | covered work so as to satisfy simultaneously your obligations under this 546 | License and any other pertinent obligations, then as a consequence you may 547 | not convey it at all. For example, if you agree to terms that obligate you 548 | to collect a royalty for further conveying from those to whom you convey 549 | the Program, the only way you could satisfy both those terms and this 550 | License would be to refrain entirely from conveying the Program. 551 | 552 | 13. Use with the GNU Affero General Public License. 553 | 554 | Notwithstanding any other provision of this License, you have 555 | permission to link or combine any covered work with a work licensed 556 | under version 3 of the GNU Affero General Public License into a single 557 | combined work, and to convey the resulting work. The terms of this 558 | License will continue to apply to the part which is the covered work, 559 | but the special requirements of the GNU Affero General Public License, 560 | section 13, concerning interaction through a network will apply to the 561 | combination as such. 562 | 563 | 14. Revised Versions of this License. 564 | 565 | The Free Software Foundation may publish revised and/or new versions of 566 | the GNU General Public License from time to time. Such new versions will 567 | be similar in spirit to the present version, but may differ in detail to 568 | address new problems or concerns. 569 | 570 | Each version is given a distinguishing version number. If the 571 | Program specifies that a certain numbered version of the GNU General 572 | Public License "or any later version" applies to it, you have the 573 | option of following the terms and conditions either of that numbered 574 | version or of any later version published by the Free Software 575 | Foundation. If the Program does not specify a version number of the 576 | GNU General Public License, you may choose any version ever published 577 | by the Free Software Foundation. 578 | 579 | If the Program specifies that a proxy can decide which future 580 | versions of the GNU General Public License can be used, that proxy's 581 | public statement of acceptance of a version permanently authorizes you 582 | to choose that version for the Program. 583 | 584 | Later license versions may give you additional or different 585 | permissions. However, no additional obligations are imposed on any 586 | author or copyright holder as a result of your choosing to follow a 587 | later version. 588 | 589 | 15. Disclaimer of Warranty. 590 | 591 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 592 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 593 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 594 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 595 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 596 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 597 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 598 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 599 | 600 | 16. Limitation of Liability. 601 | 602 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 603 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 604 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 605 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 606 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 607 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 608 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 609 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 610 | SUCH DAMAGES. 611 | 612 | 17. Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | all: help 2 | 3 | help: ## Show this help 4 | @echo -e "Specify a command. The choices are:\n" 5 | @grep -E '^[0-9a-zA-Z_-]+:.*?## .*$$' $(MAKEFILE_LIST) | awk 'BEGIN {FS = ":.*?## "}; {printf " \033[0;36m%-18s\033[m %s\n", $$1, $$2}' 6 | @echo "" 7 | .PHONY: help 8 | 9 | tags: ## Generate ctags for main codebase 10 | ctags -f tags \ 11 | --recurse=yes \ 12 | --tag-relative=yes \ 13 | --fields=+l \ 14 | --kinds-python=-i \ 15 | --language-force=python \ 16 | papis_zotero 17 | .PHONY: tags 18 | 19 | flake8: ## Run flake8 checks 20 | python -m flake8 papis_zotero tests 21 | .PHONY: flake8 22 | 23 | mypy: ## Run (strict) mypy checks 24 | python -m mypy papis_zotero tests 25 | .PHONY: mypy 26 | 27 | pytest: ## Run pytest test and doctests 28 | python -m pytest -rswx -v -s tests 29 | .PHONY: pytest 30 | 31 | ci-install-build-system: 32 | python -m pip install --upgrade pip hatchling wheel build 33 | .PHONY: ci-install-build-system 34 | 35 | ci-install: ci-install-build-system ## Run pip and install dependencies on CI 36 | python -m pip install -e '.[develop]' 37 | .PHONY: ci-install 38 | -------------------------------------------------------------------------------- /README.rst: -------------------------------------------------------------------------------- 1 | |pypi| |ci| 2 | 3 | Zotero compatibility for Papis 4 | ============================== 5 | 6 | Installation 7 | ------------ 8 | 9 | Pip 10 | ^^^ 11 | 12 | To install the latest release from PyPI 13 | 14 | .. code:: bash 15 | 16 | python -m pip install papis-zotero 17 | 18 | To install the latest development version 19 | 20 | .. code:: bash 21 | 22 | python -m pip install papis-zotero@https://github.com/papis/papis-zotero.git#egg=papis-zotero 23 | 24 | Nix 25 | ^^^ 26 | 27 | For Nix and NixOS users, a Nix flake is included in this repository and can be 28 | used to install the package. There are many ways of doing so, for instance like so: 29 | 30 | .. code:: nix 31 | 32 | { 33 | pkgs, 34 | inputs, 35 | ... 36 | }: { 37 | home.packages = with pkgs; [ 38 | ( 39 | python3.withPackages 40 | ( 41 | ps: [ 42 | inputs.papis.packages.${system}.default 43 | inputs.papis-zotero.packages.${system}.default 44 | # you can add other packages you might want to make available for papis 45 | # ps.jinja2 46 | ] 47 | ) 48 | ) 49 | # Here you can list other packages, such as 50 | # typst 51 | # hayagriva 52 | # zotero_7 53 | ]; 54 | } 55 | 56 | Arch 57 | ^^^^ 58 | 59 | Arch users can use the AUR to install `the package 60 | `__. 61 | 62 | Importing from Zotero SQLite (preferred) 63 | ---------------------------------------- 64 | 65 | Zotero also maintains a database of all its files and collections under a 66 | ``zotero.sqlite`` file. You can check where this file is located by going to 67 | ``Edit > Preferences > Advanced > Data Directory Location`` (may vary depending 68 | on the Zotero version). The Zotero data directory should contain the ``zotero.sqlite`` 69 | file and a ``storage`` directory with the files for each document. 70 | 71 | The SQLite database maintained by Zotero can be imported directly (without 72 | using a BibTeX export) by ``papis-zotero``. This can be done with: 73 | 74 | .. code:: bash 75 | 76 | papis zotero import --from-sql-folder 77 | 78 | Here, ``ZOTERO_DATA_DIRECTORY`` is the folder containing the ``zotero.sqlite`` 79 | file. By default, ``papis-zotero`` will add the imported documents to your 80 | current library directory, but it can be customized using the 81 | ``--outfolder`` argument. 82 | 83 | Importing from BibTeX (alternative) 84 | ----------------------------------- 85 | 86 | Zotero can export different variants of BibTeX or BibLaTeX files 87 | (from ``Files > Export Library``). You could import the resulting ``.bib`` file 88 | directly with Papis (with the ``papis bibtex`` command), but ``papis-zotero`` 89 | provides a specialised command. This command has better support for special Zotero 90 | fields. To import a given exported library run: 91 | 92 | .. code:: bash 93 | 94 | papis zotero import --from-bibtex library.bib 95 | 96 | BibTeX files exported by Zotero can include attached files as shown in the below 97 | example: 98 | 99 | .. code:: bibtex 100 | 101 | @article{Einstein1905Photon, 102 | author = { A. Einstein }, 103 | doi = { 10.1002/andp.19053220607 }, 104 | journal = { Ann. Phys. }, 105 | pages = { 132--148 }, 106 | title = { Über einen die Erzeugung und Verwandlung des Lichtes 107 | betreffenden heuristischen Gesichtspunkt }, 108 | file = { Full Text:path/to/some/relative/file.pdf }, 109 | volume = { 322 }, 110 | year = { 1905 }, 111 | } 112 | 113 | Given this, ``papis-zotero`` will interpret the path of the ``file`` entry 114 | as a relative path to the ``library.bib`` passed to the import command using 115 | ``--from-bibtex``. The files are skipped if they do not exist at the expected 116 | location. 117 | 118 | By default, ``papis-zotero`` will add the documents to your current library. 119 | When initially importing a big library, it is recommended to always import it 120 | into a scratch folder, so that you can verify the import. This can be easily done 121 | using: 122 | 123 | .. code:: bash 124 | 125 | papis zotero import --from-bibtex library.bib --outfolder some/folder/lib 126 | 127 | When you are ready you can move this folder to a final Papis library. 128 | 129 | Using Zotero connectors 130 | ----------------------- 131 | 132 | This plugin can also connect to a Zotero connector browser plugin. First, such 133 | a plugin should be installed from the 134 | `Zotero website `__. Then, make sure that 135 | Zotero itself is not running (and connected to the connector) and run: 136 | 137 | .. code:: bash 138 | 139 | papis zotero serve 140 | 141 | Papis now starts listening to your browser for incoming data. Whenever you click the 142 | Zotero button to add a paper, ``papis-zotero`` will add this paper to the Papis 143 | library instead. 144 | 145 | Development 146 | ----------- 147 | 148 | This project uses ``pyproject.toml`` and ``hatchling`` for its build system. 149 | To develop the code, it is recommended to start up a 150 | `virtual environment `__ and 151 | install the project in editable mode using, e.g.:: 152 | 153 | python -m pip install -e '.[develop]' 154 | 155 | After installation, always check that the command is correctly recognized, e.g. 156 | by looking at the help output 157 | 158 | .. code:: bash 159 | 160 | papis zotero --help 161 | 162 | If you use the Nix flake, you can also use the included ``devShell`` with 163 | ``nix develop``. 164 | 165 | 166 | .. |pypi| image:: https://badge.fury.io/py/papis-zotero.svg 167 | :target: https://badge.fury.io/py/papis-zotero 168 | .. |ci| image:: https://github.com/papis/papis-zotero/workflows/CI/badge.svg 169 | :target: https://github.com/papis/papis-zotero/actions?query=branch%3Amain+workflow%3ACI 170 | -------------------------------------------------------------------------------- /docs/Makefile: -------------------------------------------------------------------------------- 1 | # Minimal makefile for Sphinx documentation 2 | # 3 | 4 | # You can set these variables from the command line. 5 | SPHINXOPTS = 6 | SPHINXBUILD = sphinx-build 7 | SPHINXPROJ = PapisZotero 8 | SOURCEDIR = source 9 | BUILDDIR = build 10 | 11 | # Put it first so that "make" without argument is like "make help". 12 | help: 13 | @$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) 14 | 15 | .PHONY: help Makefile 16 | 17 | # Catch-all target: route all unknown targets to Sphinx using the new 18 | # "make mode" option. $(O) is meant as a shortcut for $(SPHINXOPTS). 19 | %: Makefile 20 | @$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) -------------------------------------------------------------------------------- /docs/source/conf.py: -------------------------------------------------------------------------------- 1 | # 2 | # Configuration file for the Sphinx documentation builder. 3 | # 4 | # This file does only contain a selection of the most common options. For a 5 | # full list see the documentation: 6 | # http://www.sphinx-doc.org/en/stable/config 7 | 8 | # -- Path setup -------------------------------------------------------------- 9 | 10 | # If extensions (or modules to document with autodoc) are in another directory, 11 | # add these directories to sys.path here. If the directory is relative to the 12 | # documentation root, use os.path.abspath to make it absolute, like shown here. 13 | # 14 | # import os 15 | # import sys 16 | # sys.path.insert(0, os.path.abspath('.')) 17 | 18 | # -- Project information ----------------------------------------------------- 19 | 20 | project = "Papis Zotero" 21 | copyright = "2019, Papis community" 22 | author = "Papis community" 23 | 24 | # The short X.Y version 25 | version = "" 26 | # The full version, including alpha/beta/rc tags 27 | release = "" 28 | 29 | # -- General configuration --------------------------------------------------- 30 | 31 | # If your documentation needs a minimal Sphinx version, state it here. 32 | # 33 | # needs_sphinx = '1.0' 34 | 35 | # Add any Sphinx extension module names here, as strings. They can be 36 | # extensions coming with Sphinx (named 'sphinx.ext.*') or your custom 37 | # ones. 38 | extensions = [ 39 | "sphinx.ext.autodoc", 40 | "sphinx.ext.doctest", 41 | "sphinx.ext.todo", 42 | "sphinx.ext.coverage", 43 | "sphinx.ext.ifconfig", 44 | "sphinx.ext.viewcode", 45 | "sphinx_click.ext", 46 | ] 47 | 48 | # Add any paths that contain templates here, relative to this directory. 49 | templates_path = ["_templates"] 50 | 51 | # The suffix(es) of source filenames. 52 | # You can specify multiple suffix as a list of string: 53 | # 54 | # source_suffix = ['.rst', '.md'] 55 | source_suffix = ".rst" 56 | 57 | # The master toctree document. 58 | master_doc = "index" 59 | 60 | # The language for content autogenerated by Sphinx. Refer to documentation 61 | # for a list of supported languages. 62 | # 63 | # This is also used if you do content translation via gettext catalogs. 64 | # Usually you set "language" from the command line for these cases. 65 | # language = None 66 | 67 | # List of patterns, relative to source directory, that match files and 68 | # directories to ignore when looking for source files. 69 | # This pattern also affects html_static_path and html_extra_path . 70 | exclude_patterns = [] 71 | 72 | # The name of the Pygments (syntax highlighting) style to use. 73 | pygments_style = "sphinx" 74 | 75 | # -- Options for HTML output ------------------------------------------------- 76 | 77 | # The theme to use for HTML and HTML Help pages. See the documentation for 78 | # a list of builtin themes. 79 | # 80 | html_theme = "sphinx_rtd_theme" 81 | 82 | # Theme options are theme-specific and customize the look and feel of a theme 83 | # further. For a list of options available for each theme, see the 84 | # documentation. 85 | # 86 | # html_theme_options = {} 87 | 88 | # Add any paths that contain custom static files (such as style sheets) here, 89 | # relative to this directory. They are copied after the builtin static files, 90 | # so a file named "default.css" will overwrite the builtin "default.css". 91 | # html_static_path = ["_static"] 92 | 93 | # Custom sidebar templates, must be a dictionary that maps document names 94 | # to template names. 95 | # 96 | # The default sidebars (for documents that don't match any pattern) are 97 | # defined by theme itself. Builtin themes are using these templates by 98 | # default: ``['localtoc.html', 'relations.html', 'sourcelink.html', 99 | # 'searchbox.html']``. 100 | # 101 | # html_sidebars = {} 102 | 103 | # -- Options for HTMLHelp output --------------------------------------------- 104 | 105 | # Output file base name for HTML help builder. 106 | htmlhelp_basename = "papiszoterodoc" 107 | 108 | # -- Options for LaTeX output ------------------------------------------------ 109 | 110 | latex_elements = { 111 | # The paper size ('letterpaper' or 'a4paper'). 112 | # 113 | # 'papersize': 'letterpaper', 114 | 115 | # The font size ('10pt', '11pt' or '12pt'). 116 | # 117 | # 'pointsize': '10pt', 118 | 119 | # Additional stuff for the LaTeX preamble. 120 | # 121 | # 'preamble': '', 122 | 123 | # Latex figure (float) alignment 124 | # 125 | # 'figure_align': 'htbp', 126 | } 127 | 128 | # Grouping the document tree into LaTeX files. List of tuples 129 | # (source start file, target name, title, 130 | # author, documentclass [howto, manual, or own class]). 131 | latex_documents = [ 132 | (master_doc, "PapisZotero.tex", "Papis Zotero Documentation", 133 | "Papis community", "manual"), 134 | ] 135 | 136 | # -- Options for manual page output ------------------------------------------ 137 | 138 | # One entry per manual page. List of tuples 139 | # (source start file, name, description, authors, manual section). 140 | man_pages = [ 141 | (master_doc, "papiszotero", "Papis Zotero Documentation", 142 | [author], 1) 143 | ] 144 | 145 | # -- Options for Texinfo output ---------------------------------------------- 146 | 147 | # Grouping the document tree into Texinfo files. List of tuples 148 | # (source start file, target name, title, author, 149 | # dir menu entry, description, category) 150 | texinfo_documents = [ 151 | (master_doc, "PapisZotero", "Papis Zotero Documentation", 152 | author, "PapisZotero", "One line description of project.", 153 | "Miscellaneous"), 154 | ] 155 | 156 | # -- Extension configuration ------------------------------------------------- 157 | 158 | # -- Options for todo extension ---------------------------------------------- 159 | 160 | # If true, `todo` and `todoList` produce output, else they produce nothing. 161 | todo_include_todos = True 162 | -------------------------------------------------------------------------------- /docs/source/index.rst: -------------------------------------------------------------------------------- 1 | Welcome to Papis Zotero's documentation! 2 | ======================================== 3 | 4 | .. toctree:: 5 | :maxdepth: 2 6 | :caption: Contents: 7 | 8 | Command-line interface 9 | ---------------------- 10 | 11 | .. click:: papis_zotero:main 12 | :prog: papis zotero 13 | 14 | .. click:: papis_zotero:serve 15 | :prog: papis zotero serve 16 | 17 | .. click:: papis_zotero:do_importer 18 | :prog: papis zotero import 19 | 20 | Indices and tables 21 | ================== 22 | 23 | * :ref:`genindex` 24 | * :ref:`modindex` 25 | * :ref:`search` 26 | -------------------------------------------------------------------------------- /flake.lock: -------------------------------------------------------------------------------- 1 | { 2 | "nodes": { 3 | "flake-utils": { 4 | "inputs": { 5 | "systems": "systems" 6 | }, 7 | "locked": { 8 | "lastModified": 1731533236, 9 | "narHash": "sha256-l0KFg5HjrsfsO/JpG+r7fRrqm12kzFHyUHqHCVpMMbI=", 10 | "owner": "numtide", 11 | "repo": "flake-utils", 12 | "rev": "11707dc2f618dd54ca8739b309ec4fc024de578b", 13 | "type": "github" 14 | }, 15 | "original": { 16 | "owner": "numtide", 17 | "repo": "flake-utils", 18 | "type": "github" 19 | } 20 | }, 21 | "flake-utils_2": { 22 | "inputs": { 23 | "systems": "systems_2" 24 | }, 25 | "locked": { 26 | "lastModified": 1731533236, 27 | "narHash": "sha256-l0KFg5HjrsfsO/JpG+r7fRrqm12kzFHyUHqHCVpMMbI=", 28 | "owner": "numtide", 29 | "repo": "flake-utils", 30 | "rev": "11707dc2f618dd54ca8739b309ec4fc024de578b", 31 | "type": "github" 32 | }, 33 | "original": { 34 | "owner": "numtide", 35 | "repo": "flake-utils", 36 | "type": "github" 37 | } 38 | }, 39 | "nixpkgs": { 40 | "locked": { 41 | "lastModified": 1744463964, 42 | "narHash": "sha256-LWqduOgLHCFxiTNYi3Uj5Lgz0SR+Xhw3kr/3Xd0GPTM=", 43 | "owner": "nixos", 44 | "repo": "nixpkgs", 45 | "rev": "2631b0b7abcea6e640ce31cd78ea58910d31e650", 46 | "type": "github" 47 | }, 48 | "original": { 49 | "owner": "nixos", 50 | "ref": "nixos-unstable", 51 | "repo": "nixpkgs", 52 | "type": "github" 53 | } 54 | }, 55 | "papis": { 56 | "inputs": { 57 | "flake-utils": "flake-utils_2", 58 | "nixpkgs": [ 59 | "nixpkgs" 60 | ], 61 | "pyproject-nix": "pyproject-nix" 62 | }, 63 | "locked": { 64 | "lastModified": 1743954027, 65 | "narHash": "sha256-s0MtTSuVP4MBpLO2U9TlGuEI3R8goCjlKnmqo9wIPRM=", 66 | "owner": "papis", 67 | "repo": "papis", 68 | "rev": "9cffc6e4eb616402d8e28689a633764088850e26", 69 | "type": "github" 70 | }, 71 | "original": { 72 | "owner": "papis", 73 | "repo": "papis", 74 | "type": "github" 75 | } 76 | }, 77 | "pyproject-nix": { 78 | "inputs": { 79 | "nixpkgs": [ 80 | "papis", 81 | "nixpkgs" 82 | ] 83 | }, 84 | "locked": { 85 | "lastModified": 1741648141, 86 | "narHash": "sha256-jQEZCSCgm60NGmBg3JPu290DDhNVI1GVVEd0P8VCnME=", 87 | "owner": "nix-community", 88 | "repo": "pyproject.nix", 89 | "rev": "7747e5a058245c7abe033a798f818f0572d8e155", 90 | "type": "github" 91 | }, 92 | "original": { 93 | "owner": "nix-community", 94 | "repo": "pyproject.nix", 95 | "type": "github" 96 | } 97 | }, 98 | "pyproject-nix_2": { 99 | "inputs": { 100 | "nixpkgs": [ 101 | "nixpkgs" 102 | ] 103 | }, 104 | "locked": { 105 | "lastModified": 1743438845, 106 | "narHash": "sha256-1GSaoubGtvsLRwoYwHjeKYq40tLwvuFFVhGrG8J9Oek=", 107 | "owner": "nix-community", 108 | "repo": "pyproject.nix", 109 | "rev": "8063ec98edc459571d042a640b1c5e334ecfca1e", 110 | "type": "github" 111 | }, 112 | "original": { 113 | "owner": "nix-community", 114 | "repo": "pyproject.nix", 115 | "type": "github" 116 | } 117 | }, 118 | "root": { 119 | "inputs": { 120 | "flake-utils": "flake-utils", 121 | "nixpkgs": "nixpkgs", 122 | "papis": "papis", 123 | "pyproject-nix": "pyproject-nix_2" 124 | } 125 | }, 126 | "systems": { 127 | "locked": { 128 | "lastModified": 1681028828, 129 | "narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=", 130 | "owner": "nix-systems", 131 | "repo": "default", 132 | "rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e", 133 | "type": "github" 134 | }, 135 | "original": { 136 | "owner": "nix-systems", 137 | "repo": "default", 138 | "type": "github" 139 | } 140 | }, 141 | "systems_2": { 142 | "locked": { 143 | "lastModified": 1681028828, 144 | "narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=", 145 | "owner": "nix-systems", 146 | "repo": "default", 147 | "rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e", 148 | "type": "github" 149 | }, 150 | "original": { 151 | "owner": "nix-systems", 152 | "repo": "default", 153 | "type": "github" 154 | } 155 | } 156 | }, 157 | "root": "root", 158 | "version": 7 159 | } 160 | -------------------------------------------------------------------------------- /flake.nix: -------------------------------------------------------------------------------- 1 | { 2 | description = "Zotero compatibility layer for Papis"; 3 | 4 | inputs = { 5 | nixpkgs.url = "github:nixos/nixpkgs?ref=nixos-unstable"; 6 | papis = { 7 | url = "github:papis/papis"; 8 | inputs.nixpkgs.follows = "nixpkgs"; 9 | }; 10 | pyproject-nix = { 11 | url = "github:nix-community/pyproject.nix"; 12 | inputs.nixpkgs.follows = "nixpkgs"; 13 | }; 14 | flake-utils.url = "github:numtide/flake-utils"; 15 | }; 16 | 17 | outputs = { 18 | self, 19 | nixpkgs, 20 | flake-utils, 21 | pyproject-nix, 22 | papis, 23 | }: 24 | flake-utils.lib.eachDefaultSystem ( 25 | system: let 26 | pypkgs = pkgs.python3Packages; 27 | pkgs = nixpkgs.legacyPackages.${system}; 28 | python = pkgs.python3.override { 29 | packageOverrides = self: super: { 30 | papis = papis.packages.${system}.default; 31 | flake8-quotes = flake8-quotes; 32 | flake8-pyproject = flake8-pyproject; 33 | python-coveralls = python-coveralls; 34 | }; 35 | }; 36 | project = pyproject-nix.lib.project.loadPyproject {projectRoot = ./.;}; 37 | 38 | flake8-quotes = python.pkgs.buildPythonPackage rec { 39 | pname = "flake8-quotes"; 40 | version = "3.4.0"; 41 | 42 | src = python.pkgs.fetchPypi { 43 | inherit pname version; 44 | sha256 = "sha256-qthJL7cQotPqvmjF+GoUKN5lDISEEn4UxD0FBLowJ2w="; 45 | }; 46 | 47 | doCheck = false; 48 | checkInputs = []; 49 | 50 | meta = with pkgs.lib; { 51 | homepage = "http://github.com/zheller/flake8-quotes"; 52 | description = "Flake8 lint for quotes."; 53 | license = licenses.mit; 54 | }; 55 | }; 56 | 57 | flake8-pyproject = python.pkgs.buildPythonPackage { 58 | pname = "flake8-pyproject"; 59 | version = "1.2.3"; 60 | pyproject = true; 61 | 62 | src = pkgs.fetchFromGitHub { 63 | owner = "john-hen"; 64 | repo = "Flake8-pyproject"; 65 | rev = "30b8444781d16edd54c11df08210a7c8fb79258d"; 66 | hash = "sha256-bPRIj7tYmm6I9eo1ZjiibmpVmGcHctZSuTvnKX+raPg="; 67 | }; 68 | 69 | doCheck = false; 70 | checkInputs = []; 71 | propagatedBuildInputs = [pypkgs.flit-core pypkgs.flake8]; 72 | 73 | meta = with pkgs.lib; { 74 | homepage = "https://github.com/john-hen/Flake8-pyproject"; 75 | description = "Flake8 plug-in loading the configuration from pyproject.toml"; 76 | license = licenses.mit; 77 | }; 78 | }; 79 | 80 | python-coveralls = python.pkgs.buildPythonPackage rec { 81 | pname = "python-coveralls"; 82 | version = "2.9.3"; 83 | 84 | src = python.pkgs.fetchPypi { 85 | inherit pname version; 86 | sha256 = "sha256-v694EefcVijoO2sWKWKk4khdv/GEsw5J84A3TtG87lU="; 87 | }; 88 | 89 | doCheck = false; 90 | checkInputs = []; 91 | 92 | meta = with pkgs.lib; { 93 | homepage = "http://github.com/z4r/python-coveralls"; 94 | description = "Python interface to coveralls.io API "; 95 | license = licenses.asl20; 96 | }; 97 | }; 98 | in { 99 | packages = { 100 | papis-zotero = let 101 | attrs = project.renderers.buildPythonPackage { 102 | inherit python; 103 | }; 104 | in 105 | python.pkgs.buildPythonPackage (attrs 106 | // { 107 | version = 108 | if (self ? rev) 109 | then self.shortRev 110 | else self.dirtyShortRev; 111 | propagatedBuildInputs = [ 112 | papis.packages.${system}.default 113 | ]; 114 | }); 115 | default = self.packages.${system}.papis-zotero; 116 | }; 117 | 118 | devShells = { 119 | default = let 120 | arg = project.renderers.withPackages { 121 | inherit python; 122 | extras = ["develop"]; 123 | }; 124 | pythonEnv = python.withPackages arg; 125 | in 126 | pkgs.mkShell { 127 | packages = [ 128 | pythonEnv 129 | self.packages.${system}.papis-zotero 130 | ]; 131 | shellHook = '' 132 | export PYTHONPATH="$(pwd):$PYTHONPATH" 133 | ''; 134 | }; 135 | }; 136 | } 137 | ); 138 | } 139 | -------------------------------------------------------------------------------- /papis_zotero/__init__.py: -------------------------------------------------------------------------------- 1 | from functools import partial 2 | import os 3 | import http.server 4 | from typing import List, Optional, Tuple 5 | 6 | import click 7 | 8 | import papis.config 9 | import papis.logging 10 | import papis_zotero.server 11 | 12 | logger = papis.logging.get_logger(__name__) 13 | 14 | 15 | @click.group("zotero") 16 | @click.help_option("-h", "--help") 17 | def main() -> None: 18 | """Zotero interface for papis.""" 19 | 20 | 21 | @main.command("serve") 22 | @click.help_option("-h", "--help") 23 | @click.option("--port", 24 | help="Port to listen to", 25 | default=papis_zotero.server.ZOTERO_PORT, 26 | type=int) 27 | @click.option("--address", help="Address to bind", default="localhost") 28 | @click.option( 29 | "-s", "--set", "set_list", 30 | help="Set imported document metadata as . Can be used multiple times.", 31 | multiple=True, 32 | type=(str, str)) 33 | def serve(address: str, port: int, set_list: List[Tuple[str, str]],) -> None: 34 | """Start a ``zotero-connector`` server.""" 35 | 36 | logger.warning("The 'zotero-connector' server is experimental. " 37 | "Please report bugs and improvements at " 38 | "https://github.com/papis/papis-zotero/issues.") 39 | 40 | server_address = (address, port) 41 | request_handler = partial(papis_zotero.server.PapisRequestHandler, set_list) 42 | try: 43 | httpd = http.server.HTTPServer(server_address, request_handler) 44 | except OSError: 45 | logger.error( 46 | "Address '%s:%s' is already in use. This may be because you " 47 | "have the Zotero application open.", address, port) 48 | logger.error("papis zotero serve requires to be the only one " 49 | "listening on that port. Zotero must quit before this " 50 | "command can be used!") 51 | return 52 | 53 | logger.info("Starting server in address https://%s:%s.", address, port) 54 | logger.info("Press Ctrl-C to exit.") 55 | 56 | httpd.serve_forever() 57 | 58 | 59 | @main.command("import") 60 | @click.help_option("-h", "--help") 61 | @click.option( 62 | "-f", 63 | "--from-bibtex", 64 | "from_bibtex", 65 | help="Import Zotero library from a BibTeX dump, the files fields in " 66 | "the BibTeX files should point to valid paths", 67 | default=None, 68 | type=click.Path(exists=True)) 69 | @click.option("-s", 70 | "--from-sql", 71 | "--from-sql-folder", 72 | "from_sql", 73 | help="Path to the FOLDER where the 'zotero.sqlite' file resides", 74 | default=None, 75 | type=click.Path(exists=True)) 76 | @click.option("-o", 77 | "--outfolder", 78 | help="Folder to save the imported library", 79 | default=None, 80 | type=str) 81 | @click.option("--link", 82 | help="Whether to link the pdf files or copy them", 83 | is_flag=True, 84 | default=False) 85 | def do_importer(from_bibtex: Optional[str], from_sql: Optional[str], 86 | outfolder: Optional[str], link: bool) -> None: 87 | """Import zotero libraries into papis libraries.""" 88 | import papis_zotero.bibtex 89 | import papis_zotero.sql 90 | 91 | if outfolder is None: 92 | outfolder = papis.config.get_lib_dirs()[0] 93 | 94 | if not os.path.exists(outfolder): 95 | os.makedirs(outfolder) 96 | 97 | if from_bibtex is not None: 98 | import papis_zotero.bibtex 99 | papis_zotero.bibtex.add_from_bibtex(from_bibtex, outfolder, link) 100 | elif from_sql is not None: 101 | import papis_zotero.sql 102 | try: 103 | papis_zotero.sql.add_from_sql(from_sql, outfolder, link) 104 | except Exception as exc: 105 | logger.error("Failed to import from file: %s", 106 | from_sql, 107 | exc_info=exc) 108 | else: 109 | logger.error("Either '--from-bibtex' or '--from-sql-folder' should be " 110 | "passed to import from Zotero.") 111 | -------------------------------------------------------------------------------- /papis_zotero/bibtex.py: -------------------------------------------------------------------------------- 1 | import os 2 | import re 3 | from typing import Any, Dict, Optional 4 | 5 | import papis.bibtex 6 | import papis.commands.add 7 | import papis.config 8 | import papis.logging 9 | 10 | logger = papis.logging.get_logger(__name__) 11 | 12 | RE_SEPARATOR = re.compile(r"\s*,\s*") 13 | 14 | 15 | def add_from_bibtex(bib_file: str, 16 | out_folder: Optional[str] = None, 17 | link: bool = False) -> None: 18 | if out_folder is not None: 19 | papis.config.set_lib_from_name(out_folder) 20 | 21 | entries = papis.bibtex.bibtex_to_dict(bib_file) 22 | nentries = len(entries) 23 | for i, entry in enumerate(entries): 24 | result: Dict[str, Any] = entry.copy() 25 | 26 | # cleanup date 27 | if "date" in result: 28 | date = str(result.pop("date")).split("-") 29 | result["year"] = int(date[0]) 30 | result["month"] = int(date[1]) 31 | 32 | # cleanup tags 33 | if "keywords" in result: 34 | result["tags"] = RE_SEPARATOR.split(result.pop("keywords")) 35 | 36 | if "ref" in result: 37 | result["ref"] = papis.bibtex.ref_cleanup(result["ref"]) 38 | else: 39 | result["ref"] = papis.bibtex.create_reference(result) 40 | 41 | # get file 42 | pdf_file = result.pop("file", None) 43 | if pdf_file is not None: 44 | pdf_file = pdf_file.split(":")[1] 45 | pdf_file = os.path.join(*pdf_file.split("/")) 46 | pdf_file = os.path.join(os.path.dirname(bib_file), pdf_file) 47 | 48 | if os.path.exists(pdf_file): 49 | logger.info("Document file found: '%s'.", pdf_file) 50 | else: 51 | logger.warning("Document file not found: '%s'.", pdf_file) 52 | pdf_file = None 53 | 54 | # add to library 55 | logger.info("[%4d/%-4d] Exporting item with ref '%s'.", 56 | i, nentries, result["ref"]) 57 | 58 | papis.commands.add.run([pdf_file] if pdf_file is not None else [], 59 | data=result, 60 | link=link) 61 | -------------------------------------------------------------------------------- /papis_zotero/server.py: -------------------------------------------------------------------------------- 1 | """Start a web server listening on port 23119. This server is 2 | compatible with the `zotero connector`. This means that if zotero is 3 | *not* running, you can have items from your web browser added directly 4 | into papis. 5 | 6 | """ 7 | 8 | import json 9 | import http.server 10 | from typing import Any, Dict, List, Tuple 11 | 12 | import papis.api 13 | import papis.crossref 14 | import papis.document 15 | import papis.commands.add 16 | import papis.logging 17 | 18 | import papis_zotero.utils 19 | 20 | logger = papis.logging.get_logger(__name__) 21 | 22 | # NOTE: 5.0.75 was released in October 8, 2019 at the same time with Python 3.8 23 | ZOTERO_CONNECTOR_API_VERSION = 2 24 | ZOTERO_VERSION = "5.0.75" 25 | ZOTERO_PORT = 23119 26 | 27 | _k = papis.document.KeyConversionPair 28 | 29 | ZOTERO_TO_PAPIS_CONVERSIONS = [ 30 | _k("creators", [{ 31 | "key": "author_list", 32 | "action": lambda a: zotero_authors(a) 33 | }]), 34 | _k("tags", [{ 35 | "key": "tags", 36 | "action": lambda t: [tag["tag"] for tag in t] 37 | }]), 38 | _k("date", [ 39 | {"key": "year", "action": lambda d: int(d.split("-")[0])}, 40 | {"key": "month", "action": lambda d: int(d.split("-")[1])}, 41 | ]), 42 | _k("archiveID", [ 43 | {"key": "eprint", "action": lambda a: a.split(":")[-1]} 44 | ]), 45 | _k("type", [ 46 | {"key": "type", "action": papis_zotero.utils.ZOTERO_TO_PAPIS_TYPES.get} 47 | ]), 48 | ] 49 | 50 | 51 | def zotero_authors(creators: List[Dict[str, str]]) -> List[Dict[str, str]]: 52 | authors = [] 53 | for creator in creators: 54 | if creator["creatorType"] != "author": 55 | continue 56 | 57 | authors.append({ 58 | "given": creator["firstName"], 59 | "family": creator["lastName"], 60 | }) 61 | 62 | return authors 63 | 64 | 65 | def zotero_data_to_papis_data(item: Dict[str, Any]) -> Dict[str, Any]: 66 | item.pop("id", None) 67 | item.pop("attachments", None) 68 | item.pop("html", None) 69 | item.pop("detailedCookies", None) 70 | item.pop("uri", None) 71 | item.pop("sessionID", None) 72 | 73 | if item.get("referrer") == "": 74 | item.pop("referrer", None) 75 | 76 | for foreign_key, key in papis_zotero.utils.ZOTERO_TO_PAPIS_FIELDS.items(): 77 | if foreign_key in item: 78 | item[key] = item.pop(foreign_key) 79 | 80 | item = papis.document.keyconversion_to_data(ZOTERO_TO_PAPIS_CONVERSIONS, 81 | item, 82 | keep_unknown_keys=True) 83 | for key in papis_zotero.utils.ZOTERO_EXCLUDED_FIELDS: 84 | if key in item: 85 | del item[key] 86 | 87 | # try to get information from Crossref as well 88 | if "doi" in item: 89 | try: 90 | crossref_data = papis.crossref.doi_to_data(item["doi"]) 91 | crossref_data.pop("title", None) 92 | logger.info("Updating document with data from Crossref.") 93 | except ValueError: 94 | crossref_data = {} 95 | 96 | item.update(crossref_data) 97 | 98 | logger.info("Document metadata: %s", item) 99 | return item 100 | 101 | 102 | def download_zotero_attachments(attachments: List[Dict[str, str]]) -> List[str]: 103 | files = [] 104 | 105 | for attachment in attachments: 106 | logger.info("Checking attachment: %s", attachment) 107 | 108 | mime = str(attachment.get("mimeType")) 109 | if mime not in papis_zotero.utils.ZOTERO_SUPPORTED_MIMETYPES_TO_EXTENSION: 110 | continue 111 | 112 | url = attachment["url"] 113 | extension = papis_zotero.utils.ZOTERO_SUPPORTED_MIMETYPES_TO_EXTENSION[mime] 114 | logger.info("Downloading file (%s): '%s'.", mime, url) 115 | 116 | filename = papis_zotero.utils.download_document( 117 | url, expected_document_extension=extension) 118 | if filename is not None: 119 | files.append(filename) 120 | 121 | return files 122 | 123 | 124 | class PapisRequestHandler(http.server.BaseHTTPRequestHandler): 125 | def __init__(self, set_list: List[Tuple[str, str]], request: Any, 126 | client_address: Any, server: Any) -> None: 127 | self.set_list = set_list 128 | super().__init__(request, client_address, server) 129 | 130 | def log_message(self, fmt: str, *args: Any) -> None: 131 | logger.info(fmt, *args) 132 | 133 | def set_zotero_headers(self) -> None: 134 | self.send_header("X-Zotero-Version", ZOTERO_VERSION) 135 | self.send_header("X-Zotero-Connector-API-Version", 136 | str(ZOTERO_CONNECTOR_API_VERSION)) 137 | self.end_headers() 138 | 139 | def read_input(self) -> bytes: 140 | length = int(self.headers["content-length"]) 141 | return self.rfile.read(length) 142 | 143 | def do_GET(self) -> None: # noqa: N802 144 | logger.info("Received GET request at '%s'", self.path) 145 | if self.path == "/connector/ping": 146 | self.handle_get_ping() 147 | 148 | def handle_get_ping(self) -> None: 149 | self.send_response(200) 150 | self.send_header("Content-Type", "text/html") 151 | self.set_zotero_headers() 152 | response = """\ 153 | 154 | 155 | 156 | Zotero Connector Server is Available 157 | 158 | 159 | Zotero Connector Server is Available 160 | 161 | 162 | """ 163 | 164 | self.wfile.write(response.encode("utf-8")) 165 | 166 | def do_POST(self) -> None: # noqa: N802 167 | logger.info("Received POST request at '%s'", self.path) 168 | if self.path == "/connector/ping": 169 | self.handle_post_ping() 170 | elif self.path == "/connector/getSelectedCollection": 171 | self.handle_post_collection() 172 | elif self.path == "/connector/saveSnapshot": 173 | self.handle_post_snapshot() 174 | elif self.path == "/connector/saveItems": 175 | self.handle_post_add() 176 | 177 | def handle_post_ping(self) -> None: 178 | self.send_response(200) 179 | self.send_header("Content-Type", "application/json") 180 | self.set_zotero_headers() 181 | response = json.dumps({"prefs": {"automaticSnapshots": True}}) 182 | 183 | self.wfile.write(response.encode("utf-8")) 184 | 185 | def handle_post_collection(self) -> None: 186 | self.send_response(200) 187 | self.send_header("Content-Type", "application/json") 188 | self.set_zotero_headers() 189 | papis_library = papis.api.get_lib_name() 190 | 191 | response = json.dumps({ 192 | "libraryID": 1, 193 | "libraryName": papis_library, 194 | "libraryEditable": True, 195 | "editable": True, 196 | "id": None, 197 | "name": papis_library 198 | }) 199 | 200 | self.wfile.write(response.encode("utf-8")) 201 | 202 | def handle_post_add(self) -> None: 203 | logger.info("Adding paper from the Zotero Connector.") 204 | rawinput = self.read_input() 205 | data = json.loads(rawinput.decode("utf-8")) 206 | 207 | logger.info("Response: %s", data) 208 | for item in data["items"]: 209 | attachments = item.get("attachments", []) 210 | if attachments: 211 | files = download_zotero_attachments(attachments) 212 | else: 213 | logger.info("Document has no attachments.") 214 | files = [] 215 | 216 | papis_item = zotero_data_to_papis_data(item) 217 | if self.set_list: 218 | papis_item.update(self.set_list) 219 | 220 | logger.info("Adding paper to papis.") 221 | papis.commands.add.run(files, data=papis_item) 222 | 223 | self.send_response(201) 224 | self.set_zotero_headers() 225 | 226 | self.wfile.write(rawinput) 227 | 228 | def handle_post_snapshot(self) -> None: 229 | import tempfile 230 | import datetime 231 | import urllib.parse 232 | 233 | rawinput = self.read_input() 234 | try: 235 | data = json.loads(rawinput.decode("utf-8")) 236 | except json.JSONDecodeError as e: 237 | logger.error("Failed to decode data from the Zotero connector.", exc_info=e) 238 | 239 | html_template = """ 240 | 241 | 242 | {html} 243 | 244 | """ 245 | full_html = html_template.lstrip().format(**data) 246 | temp_html = tempfile.mktemp(suffix=".html") 247 | logger.debug("Writing temp html to '%s'", temp_html) 248 | with open(temp_html, mode="w") as f: 249 | f.write(full_html) 250 | 251 | current_date = datetime.datetime.now() 252 | data["date"] = data.get("date", current_date.isoformat()) 253 | 254 | url = urllib.parse.urlparse(data["url"]) 255 | data["author"] = url.hostname 256 | 257 | papis_item = zotero_data_to_papis_data(data) 258 | 259 | logger.info("Adding snapshot to papis.") 260 | papis.commands.add.run([temp_html], data=papis_item, 261 | folder_name=papis.config.getstring("add-folder-name") 262 | ) 263 | 264 | self.send_response(201) 265 | self.set_zotero_headers() 266 | -------------------------------------------------------------------------------- /papis_zotero/sql.py: -------------------------------------------------------------------------------- 1 | import os 2 | import re 3 | import sqlite3 4 | from datetime import datetime 5 | from typing import Any, Dict, List, Optional 6 | 7 | import papis.config 8 | import papis.bibtex 9 | import papis.strings 10 | import papis.document 11 | import papis.logging 12 | import papis.commands.add 13 | 14 | import papis_zotero.utils 15 | 16 | logger = papis.logging.get_logger(__name__) 17 | 18 | # fuzzy date matching 19 | ISO_DATE_RE = re.compile(r"(?P\d{4})-?(?P\d{2})?-?(?P\d{2})?") 20 | 21 | 22 | ZOTERO_QUERY_ITEM_FIELD = """ 23 | SELECT 24 | fields.fieldName, 25 | itemDataValues.value 26 | FROM 27 | fields, 28 | itemData, 29 | itemDataValues 30 | WHERE 31 | itemData.itemID = ? AND 32 | fields.fieldID = itemData.fieldID AND 33 | itemDataValues.valueID = itemData.valueID 34 | """ 35 | 36 | 37 | def get_fields(connection: sqlite3.Connection, item_id: str) -> Dict[str, str]: 38 | """ 39 | :arg item_id: an identifier for the item to query. 40 | :returns: a dictionary mapping fields to their values, e.g. ``"doi"``. 41 | """ 42 | cursor = connection.cursor() 43 | cursor.execute(ZOTERO_QUERY_ITEM_FIELD, (item_id,)) 44 | 45 | # get fields 46 | fields = {} 47 | for name, value in cursor: 48 | if name in papis_zotero.utils.ZOTERO_EXCLUDED_FIELDS: 49 | continue 50 | 51 | name = papis_zotero.utils.ZOTERO_TO_PAPIS_FIELDS.get(name, name) 52 | fields[name] = value 53 | 54 | # get year and month from date if available 55 | date = fields.pop("date", None) 56 | if date is not None: 57 | m = ISO_DATE_RE.match(date) 58 | if m: 59 | if m.group("year"): 60 | fields["year"] = int(m.group("year")) 61 | if m.group("month"): 62 | fields["month"] = int(m.group("month")) 63 | else: 64 | # NOTE: didn't manage to match, so just save the whole date 65 | fields["date"] = date 66 | 67 | return fields 68 | 69 | 70 | ZOTERO_QUERY_ITEM_CREATORS = """ 71 | SELECT 72 | creatorTypes.creatorType, 73 | creators.firstName, 74 | creators.lastName 75 | FROM 76 | creatorTypes, 77 | creators, 78 | itemCreators 79 | WHERE 80 | itemCreators.itemID = ? AND 81 | creatorTypes.creatorTypeID = itemCreators.creatorTypeID AND 82 | creators.creatorID = itemCreators.creatorID 83 | ORDER BY 84 | creatorTypes.creatorType, 85 | itemCreators.orderIndex 86 | """ 87 | 88 | 89 | def get_creators(connection: sqlite3.Connection, 90 | item_id: str) -> Dict[str, List[str]]: 91 | cursor = connection.cursor() 92 | cursor.execute(ZOTERO_QUERY_ITEM_CREATORS, (item_id,)) 93 | 94 | # gather creators 95 | creators_by_type: Dict[str, List[Dict[str, str]]] = {} 96 | for ctype, given_name, family_name in cursor: 97 | creators_by_type.setdefault(ctype.lower(), []).append({ 98 | "given": given_name, 99 | "family": family_name, 100 | }) 101 | 102 | # convert to papis format 103 | result: Dict[str, Any] = {} 104 | for ctype, creators in creators_by_type.items(): 105 | result[ctype] = papis.document.author_list_to_author({"author_list": creators}) 106 | result[f"{ctype}_list"] = creators 107 | 108 | return result 109 | 110 | 111 | ZOTERO_QUERY_ITEM_ATTACHMENTS = """ 112 | SELECT 113 | items.key, 114 | itemAttachments.path, 115 | itemAttachments.contentType 116 | FROM 117 | itemAttachments, 118 | items 119 | WHERE 120 | itemAttachments.parentItemID = ? AND 121 | itemAttachments.contentType IN ({}) AND 122 | items.itemID = itemAttachments.itemID 123 | """.format(",".join(["?"] * len( 124 | papis_zotero.utils.ZOTERO_SUPPORTED_MIMETYPES_TO_EXTENSION))) 125 | 126 | 127 | def get_files(connection: sqlite3.Connection, item_id: str, item_key: str, 128 | input_path: str, out_folder: str) -> List[str]: 129 | cursor = connection.cursor() 130 | cursor.execute( 131 | ZOTERO_QUERY_ITEM_ATTACHMENTS, 132 | (item_id,) + tuple(papis_zotero.utils.ZOTERO_SUPPORTED_MIMETYPES_TO_EXTENSION)) 133 | 134 | files = [] 135 | for key, path, mime_type in cursor: 136 | if path is None: 137 | logger.warning("Attachment %s (with type %s) skipped. Path not specified.", 138 | key, mime_type) 139 | continue 140 | 141 | if match := re.match("storage:(.*)", path): 142 | file_name = match.group(1) 143 | files.append(os.path.join(input_path, "storage", key, file_name)) 144 | elif os.path.exists(path): 145 | # NOTE: this is likely a symlink to some other on-disk location 146 | files.append(path) 147 | else: 148 | logger.error("Failed to export attachment %s (with type %s) from path '%s'", 149 | key, mime_type, path) 150 | 151 | return files 152 | 153 | 154 | ZOTERO_QUERY_ITEM_TAGS = """ 155 | SELECT 156 | tags.name 157 | FROM 158 | tags, 159 | itemTags 160 | WHERE 161 | itemTags.itemID = ? AND 162 | tags.tagID = itemTags.tagID 163 | """ 164 | 165 | 166 | def get_tags(connection: sqlite3.Connection, item_id: str) -> Dict[str, List[str]]: 167 | cursor = connection.cursor() 168 | cursor.execute(ZOTERO_QUERY_ITEM_TAGS, (item_id,)) 169 | 170 | tags = [str(row[0]) for row in cursor] 171 | return {"tags": tags} if tags else {} 172 | 173 | 174 | ZOTERO_QUERY_ITEM_COLLECTIONS = """ 175 | SELECT 176 | collections.collectionName 177 | FROM 178 | collections, 179 | collectionItems 180 | WHERE 181 | collectionItems.itemID = ? AND 182 | collections.collectionID = collectionItems.collectionID 183 | """ 184 | 185 | 186 | def get_collections(connection: sqlite3.Connection, 187 | item_id: str) -> Dict[str, List[str]]: 188 | cursor = connection.cursor() 189 | cursor.execute(ZOTERO_QUERY_ITEM_COLLECTIONS, (item_id,)) 190 | 191 | collections = [name for name, in cursor] 192 | return {"collections": collections} if collections else {} 193 | 194 | 195 | ZOTERO_QUERY_ITEM_COUNT = """ 196 | SELECT 197 | COUNT(item.itemID) 198 | FROM 199 | items item, 200 | itemTypes itemType 201 | WHERE 202 | itemType.itemTypeID = item.itemTypeID AND 203 | itemType.typeName NOT IN ({}) 204 | ORDER BY 205 | item.itemID 206 | """.format(",".join(["?"] * len(papis_zotero.utils.ZOTERO_EXCLUDED_ITEM_TYPES))) 207 | 208 | ZOTERO_QUERY_ITEMS = """ 209 | SELECT 210 | item.itemID, 211 | itemType.typeName, 212 | key, 213 | dateAdded 214 | FROM 215 | items item, 216 | itemTypes itemType 217 | WHERE 218 | itemType.itemTypeID = item.itemTypeID AND 219 | itemType.typeName NOT IN ({}) 220 | ORDER BY 221 | item.itemID 222 | """.format(",".join(["?"] * len(papis_zotero.utils.ZOTERO_EXCLUDED_ITEM_TYPES))) 223 | 224 | 225 | def add_from_sql(input_path: str, 226 | out_folder: Optional[str] = None, 227 | link: bool = False) -> None: 228 | """ 229 | :param inpath: path to zotero SQLite database "zoter.sqlite" and 230 | "storage" to be imported 231 | :param outpath: path where all items will be exported to created if not 232 | existing 233 | """ 234 | 235 | if out_folder is None: 236 | out_folder = papis.config.get_lib_dirs()[0] 237 | 238 | if not os.path.exists(input_path): 239 | raise FileNotFoundError( 240 | "[Errno 2] No such file or directory: '{}'".format(input_path)) 241 | 242 | if not os.path.exists(out_folder): 243 | raise FileNotFoundError( 244 | "[Errno 2] No such file or directory: '{}'".format(out_folder)) 245 | 246 | zotero_sqlite_file = os.path.join(input_path, "zotero.sqlite") 247 | if not os.path.exists(zotero_sqlite_file): 248 | raise FileNotFoundError( 249 | "No 'zotero.sqlite' file found in '{}'".format(input_path)) 250 | 251 | connection = sqlite3.connect(zotero_sqlite_file) 252 | cursor = connection.cursor() 253 | 254 | cursor.execute(ZOTERO_QUERY_ITEM_COUNT, 255 | papis_zotero.utils.ZOTERO_EXCLUDED_ITEM_TYPES) 256 | for row in cursor: 257 | items_count = row[0] 258 | 259 | cursor.execute(ZOTERO_QUERY_ITEMS, 260 | papis_zotero.utils.ZOTERO_EXCLUDED_ITEM_TYPES) 261 | if out_folder is not None: 262 | papis.config.set_lib_from_name(out_folder) 263 | 264 | folder_name = papis.config.getstring("add-folder-name") 265 | for i, (item_id, item_type, item_key, date_added) in enumerate(cursor, start=1): 266 | # convert fields 267 | date_added = ( 268 | datetime.strptime(date_added, "%Y-%m-%d %H:%M:%S") 269 | .strftime(papis.strings.time_format)) 270 | item_type = papis_zotero.utils.ZOTERO_TO_PAPIS_TYPES.get(item_type, item_type) 271 | 272 | # get Zotero metadata 273 | fields = get_fields(connection, item_id) 274 | files = get_files(connection, 275 | item_id, 276 | item_key, 277 | input_path=input_path, 278 | out_folder=out_folder) 279 | 280 | item = {"type": item_type, "time-added": date_added, "files": files} 281 | item.update(fields) 282 | item.update(get_creators(connection, item_id)) 283 | item.update(get_tags(connection, item_id)) 284 | item.update(get_collections(connection, item_id)) 285 | 286 | logger.info("[%4d/%-4d] Exporting item '%s' to library '%s'.", 287 | i, items_count, item_key, out_folder) 288 | 289 | papis.commands.add.run(paths=files, data=item, link=link, 290 | folder_name=folder_name 291 | ) 292 | 293 | logger.info("Finished exporting from '%s'.", input_path) 294 | logger.info("Exported files can be found at '%s'.", out_folder) 295 | -------------------------------------------------------------------------------- /papis_zotero/utils.py: -------------------------------------------------------------------------------- 1 | import tempfile 2 | from typing import Any, Dict, Optional 3 | 4 | import papis.utils 5 | import papis.logging 6 | 7 | logger = papis.logging.get_logger(__name__) 8 | 9 | # Zotero item types to be excluded when converting to papis 10 | ZOTERO_EXCLUDED_ITEM_TYPES = ("attachment", "note") 11 | 12 | # Zotero excluded fields 13 | ZOTERO_EXCLUDED_FIELDS = frozenset({ 14 | "accessDate", 15 | "id", 16 | "shortTitle", 17 | "attachments", 18 | }) 19 | 20 | # dictionary of Zotero attachments mimetypes to be included 21 | # NOTE: mapped onto their respective extension to be used in papis 22 | ZOTERO_SUPPORTED_MIMETYPES_TO_EXTENSION = { 23 | "application/vnd.ms-htmlhelp": "chm", 24 | "image/vnd.djvu": "djvu", 25 | "application/msword": "doc", 26 | "application/vnd.openxmlformats-officedocument.wordprocessingml.document": 27 | "docx", 28 | "application/epub+zip": "epub", 29 | "application/octet-stream": "fb2", 30 | "application/x-mobipocket-ebook": "mobi", 31 | "application/pdf": "pdf", 32 | "text/rtf": "rtf", 33 | "application/zip": "zip", 34 | } 35 | 36 | # dictionary translating from zotero to papis field names 37 | ZOTERO_TO_PAPIS_FIELDS = { 38 | "abstractNote": "abstract", 39 | "publicationTitle": "journal", 40 | "DOI": "doi", 41 | "itemType": "type", 42 | "ISBN": "isbn", 43 | "ISSN": "issn", 44 | } 45 | 46 | # TODO: This mapping is copied from 'papis.bibtex.bibtex_type_converter' with 47 | # no changes. It will be available in papis>0.13, so it should be deleted and 48 | # replaced when we can depend on a newer version 49 | 50 | ZOTERO_TO_PAPIS_TYPES: Dict[str, str] = { 51 | # Zotero 52 | "annotation": "misc", 53 | "attachment": "misc", 54 | "audioRecording": "audio", 55 | "bill": "legislation", 56 | "blogPost": "online", 57 | "bookSection": "inbook", 58 | "case": "jurisdiction", 59 | "computerProgram": "software", 60 | "conferencePaper": "inproceedings", 61 | "dictionaryEntry": "misc", 62 | "document": "article", 63 | "email": "online", 64 | "encyclopediaArticle": "article", 65 | "film": "video", 66 | "forumPost": "online", 67 | "hearing": "jurisdiction", 68 | "instantMessage": "online", 69 | "interview": "article", 70 | "journalArticle": "article", 71 | "magazineArticle": "article", 72 | "manuscript": "unpublished", 73 | "map": "misc", 74 | "newspaperArticle": "article", 75 | "note": "misc", 76 | "podcast": "audio", 77 | "preprint": "unpublished", 78 | "presentation": "misc", 79 | "radioBroadcast": "audio", 80 | "statute": "jurisdiction", 81 | "tvBroadcast": "video", 82 | "videoRecording": "video", 83 | "webpage": "online", 84 | # Others 85 | "journal": "article", 86 | "monograph": "book", 87 | } 88 | 89 | 90 | # TODO: this function is copied from `papis.downloaders.__init__` with no 91 | # changes. It will be available in papis>0.13, so it should be deleted when 92 | # we can depend on a newer version 93 | 94 | def download_document( 95 | url: str, 96 | expected_document_extension: Optional[str] = None, 97 | cookies: Optional[Dict[str, Any]] = None, 98 | ) -> Optional[str]: 99 | """Download a document from *url* and store it in a local file. 100 | 101 | :param url: the URL of a remote file. 102 | :param expected_document_extension: an expected file type. If *None*, then 103 | an extension is guessed from the file contents, but this can also fail. 104 | :returns: a path to a local file containing the data from *url*. 105 | """ 106 | if cookies is None: 107 | cookies = {} 108 | 109 | try: 110 | with papis.utils.get_session() as session: 111 | response = session.get(url, cookies=cookies, allow_redirects=True) 112 | except Exception as exc: 113 | logger.error("Failed to fetch '%s'.", url, exc_info=exc) 114 | return None 115 | 116 | if not response.ok: 117 | logger.error("Could not download document '%s'. (HTTP status: %s %d).", 118 | url, response.reason, response.status_code) 119 | return None 120 | 121 | ext = expected_document_extension 122 | if ext is None: 123 | from papis.filetype import guess_content_extension 124 | ext = guess_content_extension(response.content) 125 | if not ext: 126 | logger.warning("Downloaded document does not have a " 127 | "recognizable (binary) mimetype: '%s'.", 128 | response.headers["Content-Type"]) 129 | 130 | ext = ".{}".format(ext) if ext else "" 131 | with tempfile.NamedTemporaryFile( 132 | mode="wb+", 133 | suffix=ext, 134 | delete=False) as f: 135 | f.write(response.content) 136 | 137 | return f.name 138 | -------------------------------------------------------------------------------- /pyproject.toml: -------------------------------------------------------------------------------- 1 | [build-system] 2 | build-backend = "hatchling.build" 3 | requires = [ "hatchling>=1.10" ] 4 | 5 | [project] 6 | name = "papis-zotero" 7 | version = "0.2" 8 | description = "Interact with Zotero using papis" 9 | readme = "README.rst" 10 | keywords = [ "bibtex", "biliography", "cli", "management", "papis", "zotero" ] 11 | license = { text = "GPL-3.0-or-later" } 12 | maintainers = [ { name = "Alejandro Gallo", email = "aamsgallo@gmail.com" } ] 13 | authors = [ { name = "Alejandro Gallo", email = "aamsgallo@gmail.com" } ] 14 | requires-python = ">=3.8" 15 | classifiers = [ 16 | "Environment :: Console", 17 | "Environment :: Console :: Curses", 18 | "Intended Audience :: Developers", 19 | "Intended Audience :: Education", 20 | "Intended Audience :: Science/Research", 21 | "Intended Audience :: System Administrators", 22 | "License :: OSI Approved :: GNU General Public License v3 (GPLv3)", 23 | "Operating System :: MacOS", 24 | "Operating System :: Microsoft", 25 | "Operating System :: OS Independent", 26 | "Operating System :: POSIX", 27 | "Operating System :: Unix", 28 | "Programming Language :: Python :: 3 :: Only", 29 | "Programming Language :: Python :: 3.8", 30 | "Programming Language :: Python :: 3.9", 31 | "Programming Language :: Python :: 3.10", 32 | "Programming Language :: Python :: 3.11", 33 | "Programming Language :: Python :: 3.12", 34 | "Programming Language :: Python :: 3.13", 35 | "Topic :: Utilities", 36 | ] 37 | dependencies = [ "papis>=0.14,<0.15" ] 38 | 39 | [project.optional-dependencies] 40 | develop = [ 41 | "flake8", 42 | "flake8-bugbear", 43 | "flake8-pyproject", 44 | "flake8-quotes", 45 | "mypy>=0.7", 46 | "pep8-naming", 47 | "pytest", 48 | "pytest-cov", 49 | "python-coveralls", 50 | "types-pyyaml", 51 | ] 52 | 53 | [project.urls] 54 | Repository = "https://github.com/papis/papis-zotero" 55 | 56 | [project.entry-points."papis.command"] 57 | zotero = "papis_zotero:main" 58 | 59 | [tool.flake8] 60 | select = [ "B", "D", "E", "F", "N", "Q", "W" ] 61 | extend-ignore = [ "B019", "E123", "N818", "W503" ] 62 | max-line-length = 88 63 | inline-quotes = "double" 64 | multiline-quotes = "double" 65 | 66 | [tool.pytest.ini_options] 67 | addopts = [ 68 | "--doctest-modules", 69 | "--cov=papis_zotero", 70 | ] 71 | norecursedirs = ".git doc build dist" 72 | python_files = "*.py" 73 | markers = [ 74 | "config_setup: setup for tmp_config", 75 | "library_setup: setup for tmp_library", 76 | ] 77 | 78 | [tool.mypy] 79 | strict = true 80 | show_column_numbers = true 81 | hide_error_codes = false 82 | pretty = true 83 | warn_unused_ignores = true 84 | -------------------------------------------------------------------------------- /tests/.gitkeep: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/.gitkeep -------------------------------------------------------------------------------- /tests/__init__.py: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/__init__.py -------------------------------------------------------------------------------- /tests/resources/bibtex/files/10/De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/bibtex/files/10/De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf -------------------------------------------------------------------------------- /tests/resources/bibtex/files/12/Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/bibtex/files/12/Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf -------------------------------------------------------------------------------- /tests/resources/bibtex/files/15/Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/bibtex/files/15/Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf -------------------------------------------------------------------------------- /tests/resources/bibtex/files/8/Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/bibtex/files/8/Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf -------------------------------------------------------------------------------- /tests/resources/bibtex/zotero-library.bib: -------------------------------------------------------------------------------- 1 | 2 | @article{schaeffer_efficient_2013, 3 | title = {Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations}, 4 | volume = {14}, 5 | issn = {15252027}, 6 | url = {http://doi.wiley.com/10.1002/ggge.20071}, 7 | doi = {10.1002/ggge.20071}, 8 | shorttitle = {Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations}, 9 | pages = {751--758}, 10 | number = {3}, 11 | journaltitle = {Geochem. Geophys. Geosyst.}, 12 | author = {Schaeffer, Nathanaël}, 13 | urldate = {2023-02-26}, 14 | date = {2013-03}, 15 | langid = {english} 16 | file = {Full Text:files/8/Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf:application/pdf}, 17 | } 18 | 19 | @article{de_lellis_euler_2009, 20 | title = {The Euler equations as a differential inclusion}, 21 | volume = {170}, 22 | issn = {0003-486X}, 23 | url = {http://annals.math.princeton.edu/2009/170-3/p09}, 24 | doi = {10.4007/annals.2009.170.1417}, 25 | pages = {1417--1436}, 26 | number = {3}, 27 | journaltitle = {Ann. Math.}, 28 | author = {De Lellis, Camillo and Székelyhidi, László}, 29 | urldate = {2023-02-26}, 30 | date = {2009-11-01}, 31 | langid = {english}, 32 | file = {Submitted Version:files/10/De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf:application/pdf}, 33 | } 34 | 35 | @article{grubb_fractional_2015, 36 | title = {Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators}, 37 | volume = {268}, 38 | issn = {00018708}, 39 | url = {https://linkinghub.elsevier.com/retrieve/pii/S0001870814003302}, 40 | doi = {10.1016/j.aim.2014.09.018}, 41 | pages = {478--528}, 42 | journaltitle = {Advances in Mathematics}, 43 | author = {Grubb, Gerd}, 44 | urldate = {2023-02-26}, 45 | date = {2015-01}, 46 | langid = {english}, 47 | file = {Full Text:files/12/Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf:application/pdf}, 48 | } 49 | 50 | @article{morinishi_fully_1998, 51 | title = {Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow}, 52 | volume = {143}, 53 | issn = {00219991}, 54 | url = {https://linkinghub.elsevier.com/retrieve/pii/S0021999198959629}, 55 | doi = {10.1006/jcph.1998.5962}, 56 | pages = {90--124}, 57 | number = {1}, 58 | journaltitle = {Journal of Computational Physics}, 59 | author = {Morinishi, Y. and Lund, T.S. and Vasilyev, O.V. and Moin, P.}, 60 | urldate = {2023-02-26}, 61 | date = {1998-06}, 62 | langid = {english}, 63 | } 64 | 65 | @article{svard_review_2014, 66 | title = {Review of summation-by-parts schemes for initial–boundary-value problems}, 67 | volume = {268}, 68 | issn = {00219991}, 69 | url = {https://linkinghub.elsevier.com/retrieve/pii/S002199911400151X}, 70 | doi = {10.1016/j.jcp.2014.02.031}, 71 | pages = {17--38}, 72 | journaltitle = {Journal of Computational Physics}, 73 | author = {Svärd, Magnus and Nordström, Jan}, 74 | urldate = {2023-02-26}, 75 | date = {2014-07}, 76 | langid = {english}, 77 | keywords = {sbp}, 78 | file = {Submitted Version:files/15/Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf:application/pdf}, 79 | } 80 | -------------------------------------------------------------------------------- /tests/resources/bibtex_out.yaml: -------------------------------------------------------------------------------- 1 | author: Svärd, Magnus and Nordström, Jan 2 | author_list: 3 | - family: Svärd 4 | given: Magnus 5 | - family: Nordström 6 | given: Jan 7 | doi: 10.1016/j.jcp.2014.02.031 8 | files: 9 | - svard-and-nordstrom-2014-review-of-summation-by-parts-schemes-for-initial-b.pdf 10 | issn: 00219991 11 | journaltitle: Journal of Computational Physics 12 | langid: english 13 | month: 7 14 | pages: 17--38 15 | ref: svard_review_2014 16 | tags: 17 | - sbp 18 | title: Review of summation-by-parts schemes for initial–boundary-value problems 19 | type: article 20 | url: https://linkinghub.elsevier.com/retrieve/pii/S002199911400151X 21 | urldate: '2023-02-26' 22 | volume: '268' 23 | year: 2014 24 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/5KW7TMDH/.zotero-ft-cache: -------------------------------------------------------------------------------- 1 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2 | 3 | Article Volume 14, Number 3 4 | 6 March 2013 doi:10.1002/ggge.20071 5 | ISSN: 1525-2027 6 | 7 | Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations 8 | Nathanaël Schaeffer 9 | ISTerre, Université de Grenoble 1, CNRS, F-38041 Grenoble, France (nathanael.schaeffer@ujf-grenoble.fr) 10 | [1] In this paper, we report on very efficient algorithms for spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library, which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre-associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Although the complexity of the algorithms implemented in SHTns are in OðN 3Þ (where N is the maximum harmonic degree of the transform), they perform much better than any third-party implementation, including lower-complexity algorithms, even for truncations as high as N = 1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software. 11 | Components: 3,700 words, 5 figures. 12 | Keywords: spherical harmonics; performance; mathematical software. 13 | Index Terms: 1932 Informatics: High-performance computing; 1976 Informatics: Software tools and services; 3255 Mathematical Geophysics (0500, 4307, 4314, 4400, 7833): Spectral analysis (3205, 3280, 4319); 1906 Informatics: Computational models, algorithms; 1510 Geomagnetism And Paleomagnetism: Dynamo: theories and simulations. 14 | Received 11 December 2012; Revised 15 January 2013; Accepted 15 January 2013; Published 6 March 2013. 15 | Schaeffer, N. (2013), Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations, Geochem. Geophys. Geosyst., 14, 751–758, doi:10.1002/ggge.20071. 16 | 17 | 1. Introduction 18 | [2] Spherical harmonics are the eigenfunctions of the Laplace operator on the 2-sphere. They form a basis and are useful and convenient to describe data on a sphere in a consistent way in spectral space. Spherical harmonic transforms (SHT) are the spherical counterpart of the Fourier transform, casting spatial data to the spectral domain and vice versa. They are commonly used in various pseudospectral direct numerical simulations in spherical geometry, 19 | 20 | for simulating the Sun or the liquid core of the Earth among others [Glatzmaier, 1984; Sakuraba, 1999; Christensen et al., 2001; Brun & Rempel, 2009; Wicht & Tilgner, 2010]. 21 | [3] All numerical simulations that take advantage of spherical harmonics use the classical Gauss-Legendre algorithm (see section 2) with complexity OðN 3Þ for a truncation at spherical harmonic degree N. As a consequence of this high computational cost when N increases, high-resolution spherical codes currently spend most of their time performing SHT. A few years 22 | 23 | ©2013. American Geophysical Union. All Rights Reserved. 24 | 25 | 751 26 | 27 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 28 | 29 | Geochemistry 30 | 3 Geophysics 31 | G Geosystems 32 | 33 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 34 | 35 | 10.1002/ggge.20071 36 | 37 | ago, state-of-the-art numerical simulations used N = 255 [Sakuraba & Roberts, 2009]. 38 | [4] However, there exist several asymptotically fast algorithms [Driscoll & Healy, 1994; Potts et al., 1998; Mohlenkamp, 1999; Suda & Takami, 2002; Healy et al., 2003; Tygert, 2008], but the overhead for these fast algorithms is such that they do not claim to be effectively faster for N < 512. In addition, some of them lack stability (the error becomes too large even for moderate N) and flexibility (e.g., N + 1 must be a power of 2). 39 | [5] Among the asymptotically fast algorithms, only two have open-source implementations, and the only one that seems to perform reasonably well is SpharmonicKit, based on the algorithms described by Healy et al. [Healy et al., 2003]. Its main drawback is the need of a latitudinal grid of size 2(N + 1), while the Gauss-Legendre quadrature allows the use of only N + 1 collocation points. Thus, even if it were as fast as the GaussLegendre approach for the same truncation N, the overall numerical simulation would be slower because it would operate on twice as many points. These facts explain why the Gauss-Legendre algorithm is still the most efficient solution for numerical simulations. 40 | [6] A recent paper [Dickson et al., 2011] reports that a carefully tuned software could finally run nine times faster on the same CPU than the initial nonoptimized version, and insists on the importance of vectorization and careful optimization of the code. As the goal of this work is to speed up numerical simulations, we have written a highly optimized and explicitly vectorized version of the Gauss-Legendre SHT algorithm. The next section recalls the basics of spherical harmonic transforms. We then describe the optimizations we used and compare the performance of our transform to other SHT implementations. We conclude this paper by a short summary and perspectives for future developments. 41 | 42 | 2. Spherical Harmonic Transform 43 | 2.1. Definitions and Properties [7] The orthonormalized spherical harmonics of degree n and order À n ≤ m ≤ n are functions defined on the sphere as: 44 | 45 | Ynmðθ; ’Þ ¼ PnmðcosθÞexpðim’Þ 46 | 47 | (1) 48 | 49 | where θ is the colatitude, ’ is the longitude, and Pnm are the associated Legendre polynomials normal- 50 | 51 | ized for spherical harmonics 52 | 53 | rffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 54 | 55 | PnmðxÞ ¼ ðÀ1Þm 56 | 57 | 2n þ 1 4p 58 | 59 | ðn ðn 60 | 61 | À þ 62 | 63 | jmjÞ!À jmjÞ! 1 64 | 65 | À 66 | 67 | x2Ájmj=2 68 | 69 | djmj dxjmj 70 | 71 | Pn 72 | 73 | ðxÞ 74 | 75 | (2) 76 | 77 | which involve derivatives of Legendre polynomials Pn(x) defined by the following recurrence: 78 | P0ðxÞ ¼ 1 P1ðxÞ ¼ x nPnðxÞ ¼ ð2n À 1ÞxPnÀ1ðxÞ À ðn À 1ÞPnÀ2ðxÞ 79 | The spherical harmonics Ynmðθ; ’Þ form an orthonormal basis for functions defined on the sphere: 80 | 81 | Z 2p Z p 82 | 83 | Ynmðθ; ’ÞYlkðθ; ’Þsinθdθd’ ¼ dnldmk 84 | 85 | (3) 86 | 87 | 00 88 | 89 | with dij the Kronecker symbol. By construction, they are eigenfunctions of the Laplace operator on 90 | the unit sphere: 91 | 92 | ΔYnm ¼ Ànðn þ 1ÞYnm 93 | 94 | (4) 95 | 96 | This property is very appealing for solving many physical problems in spherical geometry involving the Laplace operator. 97 | 98 | 2.2. Synthesis or Inverse Transform 99 | [8] The spherical harmonic synthesis is the evaluation of the sum 100 | 101 | XN Xn 102 | 103 | f ðθ; ’Þ ¼ 104 | 105 | fnmYnmðθ; ’Þ 106 | 107 | (5) 108 | 109 | n¼0 m¼Àn 110 | 111 | up to degree n = N, given the complex coefficients 112 | 113 | Àfnfmnm. ÁIÃf, 114 | 115 | f(θ,’) is a where z* 116 | 117 | real-valued function, then fnÀm ¼ stands for the complex conjugate 118 | 119 | of z. 120 | 121 | [9] The sums can be exchanged; and using the ex- 122 | 123 | pression of Ynm we can write 124 | 125 | 0 126 | 127 | 1 128 | 129 | XN XN 130 | 131 | f ðθ; ’Þ ¼ 132 | 133 | @ fnmPnmðcosθÞAeim’ 134 | 135 | (6) 136 | 137 | m¼ÀN n¼jmj 138 | 139 | From this last expression, it appears that the summation over m is a regular Fourier transform. Hence, the remaining task is to evaluate 140 | 141 | XN 142 | 143 | fmðθÞ ¼ 144 | 145 | fnmPnmðcosθÞ 146 | 147 | (7) 148 | 149 | n¼jmj 150 | 151 | or its discrete version at given collocation points θj. 152 | 153 | 752 154 | 155 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 156 | 157 | Geochemistry 158 | 3 Geophysics 159 | G Geosystems 160 | 161 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 162 | 163 | 10.1002/ggge.20071 164 | 165 | 2.3. Analysis or Forward Transform 166 | 167 | [10] The analysis step of the SHT consists in computing the coefficients 168 | 169 | Z 2p Z p 170 | 171 | fnm ¼ 172 | 173 | f ðθ; ’ÞYnmðθ; ’Þsinθdθd’ 174 | 175 | (8) 176 | 177 | 00 178 | 179 | The integral over ’ is obtained using the Fourier 180 | 181 | transform: 182 | 183 | Z 2p 184 | 185 | fmðθÞ ¼ f ðθ; ’Þeim’d’ 186 | 187 | (9) 188 | 189 | 0 190 | 191 | so the remaining Legendre transform reads 192 | 193 | Zp 194 | 195 | fnm ¼ fmðθÞPnmðcosθÞsinθdθ 196 | 197 | (10) 198 | 199 | 0 200 | 201 | The discrete problem reduces to the appropriate 202 | quadrature rule to evaluate the integral (10) knowing only the values fm(θj). In particular, the use of the Gauss-Legendre quadrature replaces the inte- 203 | gral of expression (10) by the sum 204 | 205 | X Nθ À Á À Á 206 | 207 | fnm ¼ fm θj Pnm cosθj wj 208 | 209 | (11) 210 | 211 | j¼1 212 | 213 | where θj and wj are, respectively, the Gauss nodes and weights [Temme, 2011]. Note that the sum equals the integral if fmðθÞPnmðcosθÞ is a polynomial in cosθ of order 2Nθ À 1 or less. If fm(θ) is given by expression (7), then fmðθÞPnmðcosθÞ is always a polynomial in cosθ, of degree at most 2N. Hence, the GaussLegendre quadrature is exact for Nθ ≥ N + 1. 214 | [11] A discrete spherical harmonic transform using Gauss nodes as latitudinal grid points and a Gauss-Legendre quadrature for the analysis step is referred to as a Gauss-Legendre algorithm. 215 | 216 | 3. Optimization of the Gauss-Legendre Algorithm 217 | 3.1. Standard Optimizations [12] Let us first recall some standard optimizations found in almost every serious implementation of the Gauss-Legendre algorithm. All the following optimizations are used in the SHTns library. 218 | 3.1.1. Use the Fast-Fourier Transform 219 | [13] The expressions in section 2 show that part of the SHT is in fact a Fourier transform. The fast Fourier transform (FFT) should be used for this part, as it improves accuracy and speed. SHTns uses the FFTW 220 | 221 | library [Frigo & Johnson, 2005], a portable, flexible, and highly efficient FFT implementation. 222 | 223 | 3.1.2. Take Advantage of Hermitian Symmetry for Real Data 224 | 225 | [14] tral 226 | 227 | When dealing with coefficients fulfill 228 | 229 | frnÀeaml-¼vaÀlufnemdÁÃd,atsao, 230 | 231 | the we 232 | 233 | speconly 234 | 235 | need to store them for m ≥ 0. This also allows the 236 | 237 | use of faster real-valued FFTs. 238 | 239 | 3.1.3. Take Advantage of Mirror Symmetry 240 | [15] Due to the defined symmetry of spherical harmonics with respect to a reflection about the equator 241 | Pnmðcosðp À θÞÞ ¼ ðÀ1ÞnþmPnmðcosθÞ 242 | one can reduce by a factor of 2 the operation count of both forward and inverse transforms. 243 | 244 | 3.1.4. Precompute Values of Pnm ÀÁ 245 | [16] The coefficients Pnm cosθj appear in both synthesis and analysis expressions (7 and 10), and can be precomputed and stored for all (n,m,j). When performing multiple transforms, it avoids computing the Legendre polynomial recursion at every transform and saves some computing power, at the expense of memory bandwidth. This may or may not be efficient, as we will discuss later. 246 | 247 | 3.1.5. Polar Optimization 248 | 249 | [17] High-order spherical harmonics have their magnitude decrease exponentially when approaching the poles as shown in Figure 1. Hence, the integral of expression (10) can be reduced to 250 | 251 | Z pÀθm0 n 252 | 253 | fnm ¼ 254 | 255 | fmðθÞPnmðcosθÞsinθdθ 256 | 257 | (12) 258 | 259 | θm0 n 260 | 261 | where θm0 n≥0 is a threshold below which Pnm is consid- 262 | 263 | ered to be zero. Similarly, the synthesis of fm(θ) (equa- 264 | 265 | tion uses 266 | 267 | (7)) is only a threshold 268 | 269 | needed θm0 n that 270 | 271 | for θm0 n≤θ≤p À θm0 n does not depend on 272 | 273 | . SHTns n, which 274 | 275 | leads to around 5% to 20% speed increase, depending 276 | 277 | on the desired accuracy and the truncation N. 278 | 279 | 3.2. On-the-Fly Algorithms and Vectorization [18] It can be shown that PnmðxÞ can be computed recursively by 280 | 753 281 | 282 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 283 | 284 | Geochemistry 285 | 3 Geophysics 286 | G Geosystems 287 | 288 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 289 | 290 | 10.1002/ggge.20071 291 | 292 | 1.5 293 | 294 | 1 295 | 296 | 0.5 297 | 298 | 0 299 | 300 | -0.5 301 | 302 | -1 303 | 304 | -1.5 305 | 306 | 0 307 | 308 | 0.5 309 | 310 | 1 311 | 312 | 1.5 313 | 314 | 2 315 | 316 | 2.5 317 | 318 | 3 319 | 320 | θ 321 | 322 | Figure 1. Two associated Legendre polynomials of degree n = 40 and order m = 33 (blue) and m = 36 (red), showing the localization near the equator. 323 | 324 | PmmðxÞ 325 | 326 | ¼ 327 | 328 | amm 329 | 330 | À 1 331 | 332 | À 333 | 334 | x2Ájmj=2 335 | 336 | (13) 337 | 338 | Pmmþ1ðxÞ ¼ ammþ1xPmmðxÞ 339 | 340 | (14) 341 | 342 | PnmðxÞ ¼ amn xPnmÀ1ðxÞ þ bmn PnmÀ2ðxÞ 343 | 344 | (15) 345 | 346 | with 347 | 348 | amm 349 | 350 | ¼ 351 | 352 | v u u tffiffi1ffiffiffiffiffiffiY ffijffimffiffijffiffiffi2ffiffiffikffiffiffiþffiffiffiffiffi1ffiffi 4p k¼1 2k 353 | 354 | (16) 355 | 356 | rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 357 | 358 | amn ¼ 359 | 360 | 4n2 À 1 n2 À m2 361 | 362 | (17) 363 | 364 | sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 365 | 366 | bmn ¼ À 367 | 368 | 2n þ 1 ðn À 1Þ2 À m2 2n À 3 n2 À m2 369 | 370 | (18) 371 | 372 | The coefficients amn and bmn do not depend on x, and 373 | 374 | can be easily precomputed and stored into an array 375 | 376 | of (N order 377 | 378 | + 1)2 values. N3 values of 379 | 380 | ThisÀ hÁas Pnm xj , 381 | 382 | to be which 383 | 384 | compared to the are usually pre- 385 | 386 | computed and stored in the spherical harmonic 387 | 388 | transforms implemented in numerical simulations. 389 | 390 | TheÀ aÁmount of memory required to Pnm xj in double-precision is at least 391 | 392 | store all 2(N + 1)3 393 | 394 | bytes, which gives 2 Gb for N = 1023. Our on-the- 395 | 396 | fly algorithm only needs about 8(N + 1)2 bytes of 397 | 398 | storage (same size as a spectral representation fnm), 399 | 400 | that is, 8 Mb for N = 1023. When N becomeÀs Ávery large, it is no longer possible to store Pnm xj in 401 | 402 | memory (for N ≳1024 nowadays) and on-the-fly ÀÁ 403 | algorithms (which recompute Pnm xj from the recurrence relation when needed) are then the only 404 | 405 | possibility. 406 | 407 | [19] We would like to stress that even far from that storage limit, on-the-fly algorithm can be significantly faster thanks to vector capabilities of modern 408 | 409 | processors. Most desktop and laptop computers, as well as many high-performance computing clusters, have support for single-instruction, multiple-data (SIMD) operations in double precision. The SSE2 instruction set is available since year 2000 and currently supported by almost every PC, allowing the performance of the same double-precision arithmetic operations on a vector of two doubleprecision numbers, effectively doubling the computing power. The recently introduced AVX instruction set increases the vector size to four double-precision numbers. This means that PnmðxÞ can be computed from the recursion relation (15) (which requires three multiplications and one addition) for two or four values of x simultaneously, which may be faster than loading precomputed values from memory. Hence, as already pointed out by Dickson et al. [Dickson et al., 2011], it is therefore very important to use the vector capabilities of modern processors to address their full computing power. Furthermore, when running multiple transforms on the different cores of a computer, the performance of on-the-fly transforms (which use less memory bandwidth) scales much better than algorithms with precomputed matrices, because the memory bandwidth is shared between cores. Superscalar architectures that do not have double-precision SIMD instructions but have many computation units per core (like the POWER7 or SPARC64) could also benefit from on-the-fly transforms by saturating the many computation units with independent computations (at different x). 410 | [20] Figure 2 shows the benefit of explicit vectorization of on-the-fly algorithms on an Intel Xeon E5-2680 (Sandy Bridge architecture with AVX instruction set running at 2.7 GHz) and compares on-the-fly algorithms with algorithms based on precomputed matrices. With the four vectors of AVX, the fastest algorithm is always on the fly, while for two vectors, the fastest algorithm uses precomputed matrices for N ≲200 . In the forthcoming years, wider vector architecture is expected to become widely available, and the benefits of on-the-fly vectorized transforms will become even more important. 411 | 3.2.1. Runtime Tuning 412 | [21] We have now two different available algorithms: one uses precomputed values for PnmðxÞ and the other one computes them on the fly at each transform. The SHTns library compares the time taken by those algorithms (and variants) at startup and chooses the fastest, similarly to what the FFTW 413 | 754 414 | 415 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 416 | 417 | Geochemistry 418 | 3 Geophysics 419 | G Geosystems 420 | 421 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 422 | 423 | 10.1002/ggge.20071 424 | 425 | Figure 2. Efficiency (N + 1)3/(2tf ) of various algorithms, where t is the execution time and f the frequency of the Xeon E5-2680 CPU (2.7 GHz). On-the-fly algorithms with two different vector sizes are compared with the algorithm using precomputed matrices. Note the influence of hardware vector size for on-the-fly algorithms (AVX vectors pack four double-precision floating point numbers where SSE3 vectors pack only two). The efficiency of the algorithm based on precomputed matrices drops above N = 127 probably due to cache size limitations. 426 | library [Frigo & Johnson, 2005] does. The time overhead required by runtime tuning can be several orders of magnitude larger than that of a single transform. The observed performance gain varies between 10% and 30%. This is significant for numerical simulations, but runtime tuning can be entirely skipped for applications performing only a few transforms, in which case there is no noticeable overhead. 427 | 3.3. Multithreaded Transform 428 | [22] Modern computers have several computing cores. We use OpenMP to implement a multithreaded algorithm for the Legendre transform including the above optimizations and the on-the-fly approach. The lower memory bandwidth requirements for the on-the-fly approach is an asset for a multithreaded transform because if each thread would read a different portion of a large matrix, it can saturate the memory bus very quickly. The multithreaded Fourier transform is left to the FFTW library. 429 | [23] We need to decide how to share the work between different threads. Because we compute the Pnm on the fly using the recurrence relation (15), we are left with each thread computing different θ, or different m. As the analysis step involves a sum over θ, we choose the latter option. 430 | [24] From equation (7), we see that the number of terms involved in the sum depends on m, so that the computing cost will also depend on m. To 431 | 432 | achieve the best workload balance between a team of p threads, the thread number i (0 ≤ i < p) handles m = i + kp ≤ N, with integer k from 0 to (N + 1)p. [25] For different thread number b, we have measured the time Ts(p) and Ta(p) needed for a scalar spherical harmonic synthesis and analysis, respectively (including the FFT). [26] Figure 3 shows the speedup T(1)/T(p), where T(p) is the largest of Ts(p) and Ta(p), and T(1) is the time of the fastest single threaded transform. It shows that there is no point in doing a parallel transform with N below 128. The speedup is good for N ¼ 255 or above, and excellent up to eight threads for N ≥ 511 or up to 16 threads for very large transform (N ≥ 2047). 433 | 3.4. Performance Comparisons [27] Table 1 reports the timing measurements of two SHT libraries, compared to the optimized GaussLegendre implementation found in the SHTnslibrary (this work). We compare with the Gauss-Legendre implementation of libpsht [Reinecke, 2011] a parallel spherical harmonic transform library targeting very large N, and with SpharmonicKit 2.7 (DH) which implements one of the Driscoll-Healy fast algorithms [Healy et al., 2003]. All the timings are for a complete SHT, which includes the fast Fourier transform. Note that the Gauss-Legendre algorithm is by far (a factor of order 2) the fastest algorithm of the libpsht library. Note also that SpharmonicKit is limited to N + 1 being a power of two, requires 2 (N + 1) latitudinal colocation points, and crashed for N = 2047. The software library implementing the fast 434 | Figure 3. Speedup obtained with multiple threads using OpenMP (gcc 4.6.3) on a 16-core Intel Xeon E5-2680 (Sandy Bridge architecture with AVX instruction set running at 2.7 GHz). 435 | 755 436 | 437 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 438 | 439 | Geochemistry 440 | 3 Geophysics 441 | G Geosystems 442 | 443 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 444 | 445 | 10.1002/ggge.20071 446 | 447 | Table 1. Comparison of Execution Time for Different SHT Implementations 448 | 449 | N 450 | 451 | 63 452 | 453 | 127 454 | 455 | 255 456 | 457 | 511 458 | 459 | 1023 460 | 461 | 2047 462 | 463 | 4095 464 | 465 | libpsht (1 thread) DH (fast) SHTns (1 thread) 466 | 467 | 1.05 ms 1.1 ms 0.09 ms 468 | 469 | 4.7 ms 5.5 ms 0.60 ms 470 | 471 | 27 ms 21 ms 472 | 4.2 ms 473 | 474 | 162 ms 110 ms 475 | 28 ms 476 | 477 | 850 ms 600 ms 216 ms 478 | 479 | 4.4 s NA 480 | 1.6 s 481 | 482 | 30.5 s NA 11.8 s 483 | 484 | The numbers correspond to the average execution time for forward and backward scalar transform (including the FFT) on an Intel Xeon X5650 (2.67GHz) with 12 cores. The programs were compiled with gcc 4.4.5 and -O3 -march=native -ffast-math compilation options. 485 | 486 | Legendre transform described by Mohlenkamp [Mohlenkamp, 1999], libftsh, has also been tested and found to be of comparable performance to that of SpharmonicKit, although the comparison is not straightforward because libftsh did not include the Fourier transform. Again, that fast library could not operate at N = 2047 because of memory limitations. Note finally that these measurements were performed on a machine that did not support the new AVX instruction set. [28] To ease the comparison, we define the efficiency of the SHT by (N + 1)3/(2Tf ), where T is the execution time (reported in Table 1) and f the frequency of the CPU. Note that (N + 1)3/2 reflects the number of computation elements of a Gauss-Legendre algorithm [the number of modes (N + 1)(N + 2)/2 times the number of latitudinal points N + 1]. An efficiency that does not depend on N corresponds to an algorithm with an execution time proportional to N3. [29] The efficiency of the tested algorithms is displayed in Figure 4. Not surprisingly, the Driscoll-Healy implementation has the largest slope, which means that its efficiency grows fastest with N, as expected for a fast algorithm. It also performs slightly better than libpsht for N ≥ 511. However, even for N = 1023 (the largest size that it can compute), it is still 2.8 times slower than the Gauss-Legendre algorithm implemented in 487 | Figure 4. Efficiency (N + 1)3/(2Tf ) of the implementations from Table 1, where T is the execution time and f the frequency of the Xeon X5650 CPU (2.67 GHz) with 12 cores. 488 | 489 | SHTns. It is remarkable that SHTns achieves an efficiency very close to 1, meaning that almost one element per clock cycle is computed for N = 511 and N = 1023. Overall, SHTns is between 2 and 10 times faster than the best alternative. 490 | 491 | 3.5. Accuracy 492 | 493 | [30] One cannot write about an SHT implementation without addressing its accuracy. The Gauss-Legendre quadrature ensures very good accuracy, at least on par with other high-quality implementations. 494 | 495 | [31] The recurrence relation we use (see section 3.2) is numerically stable, but for N ≳1500 , the value PmmðxÞ can become so small that it cannot be represented by a double-precision number anymore. To avoid this underflow problem, the code dynamically rescales the values of PnmðxÞ during the recursion, when they reach a given threshold. 496 | The number of rescalings is stored in an integer, 497 | which acts as an enhanced exponent. Our 498 | implementation of the rescaling does not impact 499 | performance negatively, as it is compensated by 500 | dynamic polar optimization: these very small values 501 | are treated as zero in the transform (equations (7) 502 | and (11)), but not in the recurrence. This technique 503 | ensures good accuracy up to N = 8191 at least, but 504 | partial transforms have been performed successfully 505 | up to N = 43,600. 506 | 507 | [32] To quantify the error, we start with random 508 | spherical harmonic coefficients Qmn with each real part and imaginary part between -1 and + 1. After 509 | 510 | a backward and forward transform (with orthonor- 511 | 512 | mal spherical harmonics), we compare the resulting coefficients Rmn with the originals Qmn . We use two different error measurements: the maximum error 513 | 514 | is defined as 515 | 516 | emax 517 | 518 | ¼ 519 | 520 | max 521 | n;m 522 | 523 | Rmn 524 | 525 | À 526 | 527 | Qmn  528 | 529 | while the root mean square (rms) error is defined as 530 | erms ¼ sffiðffiNffiffiffiffiffiþffiffiffiffiffi1ffiffiÞffi2ffiðffiffiNffiffiffiffiffiþffiffiffiffiffi2ffiffiÞffiffiffiffiX ffinffiffi;ffimffiffiffiffiffiRffiffiffimnffiffiffiffiÀffiffiffiffiffiQffiffiffimnffiffiffiffi2ffiffi 531 | 532 | 756 533 | 534 | 15252027, 2013, 3, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1002/ggge.20071 by Cochrane Romania, Wiley Online Library on [26/02/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 535 | 536 | Geochemistry 537 | 3 Geophysics 538 | G Geosystems 539 | 540 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 541 | 542 | 10.1002/ggge.20071 543 | 544 | vectorized on-the-fly implementations, we should be able to run spectral geodynamo simulations at N = 1023 in the next few years. Such high-resolution simulations will operate in a regime much closer to the dynamics of the Earth’s core. 545 | 546 | Acknowledgments 547 | 548 | Figure 5. Accuracy of the on-the-fly Gauss-Legendre algorithm with the default polar optimization. 549 | 550 | [36] The author thanks Alexandre Fournier and Daniel Lemire 551 | for their comments that helped to improve the paper. Some computations have been carried out at the Service Commun de Calcul Intensif de l’Observatoire de Grenoble (SCCI) and other were run on the PRACE Research Infrastructure Curie at the TGCC (grant PA1039). 552 | References 553 | 554 | The error measurements for our on-the-fly Gauss-Legendre implementation with the default polar optimization and for various truncation degrees N are shown in Figure 5. The errors steadily increase with N and are comparable to other implementations. For N < 2048, we have emax < 10À 11, which is negligible compared to other sources of errors in most numerical simulations. 555 | 4. Conclusion and Perspectives 556 | [33] Despite the many fast spherical harmonic transform algorithms published, the few with a publicly available implementation are far from the performance of a carefully written Gauss-Legendre algorithm, as implemented in the SHTns library, even for quite large truncation (N = 1023). 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See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 577 | 578 | Geochemistry 579 | 3 Geophysics 580 | G Geosystems 581 | 582 | SCHAEFFER: EFFICIENT SPHERICAL HARMONIC TRANSFORM 583 | 584 | 10.1002/ggge.20071 585 | 586 | Schaeffer, N., D., Jault, P., Cardin, and M. Drouard (2012), On the reflection of alfvén waves and its implication for earth’s core modelling, Geophys. J. Int., 191(2), 508–516, doi:10.1111/j.1365-246X.2012.05611.x. 587 | Suda, R., and M. Takami (2002), A fast spherical harmonics transform algorithm, Math. Comput., 71(238), 703–715, doi:10.1090/S0025-5718-01-01386-2. 588 | Temme, N. M. (2011), Gauss quadrature, in Digital 589 | Library of Mathematical Functions (DLMF), chap. 3.5 590 | 591 | (v), National Institute of Standards and Technology (NIST). Tygert, M. (2008), Fast algorithms for spherical harmonic expansions, II, J. 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Rev., 152(1), 501–542, doi:10.1007/s11214-010-9638-y. 592 | 593 | 758 594 | 595 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/5KW7TMDH/.zotero-ft-info: -------------------------------------------------------------------------------- 1 | Title: ggge20071 751..758 2 | Subject: Geochem Geophys Geosyst 2013.14:751-758 3 | Creator: Arbortext Advanced Print Publisher 9.0.114/W Unicode 4 | Producer: PDFlib PLOP 2.0.0p6 (SunOS)/Acrobat Distiller 9.0.0 (Windows); modified using iText 4.2.0 by 1T3XT 5 | CreationDate: Fri Apr 12 05:07:27 2013 6 | ModDate: Sun Feb 26 00:48:57 2023 7 | Tagged: no 8 | Form: none 9 | Pages: 8 10 | Encrypted: no 11 | Page size: 612 x 792 pts (letter) (rotated 0 degrees) 12 | File size: 463114 bytes 13 | Optimized: no 14 | PDF version: 1.4 15 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/5KW7TMDH/.zotero-pdf-state: -------------------------------------------------------------------------------- 1 | {"pageIndex":0,"scale":"page-width","top":792,"left":-6,"scrollMode":0,"spreadMode":0} -------------------------------------------------------------------------------- /tests/resources/sql/storage/5KW7TMDH/Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/storage/5KW7TMDH/Schaeffer - 2013 - Efficient spherical harmonic transforms aimed at p.pdf -------------------------------------------------------------------------------- /tests/resources/sql/storage/J8FIHBUY/.zotero-ft-info: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/storage/J8FIHBUY/.zotero-ft-info -------------------------------------------------------------------------------- /tests/resources/sql/storage/J8FIHBUY/.zotero-pdf-state: -------------------------------------------------------------------------------- 1 | {"pageIndex":0,"scale":"page-width","top":845,"left":-6,"scrollMode":0,"spreadMode":0} -------------------------------------------------------------------------------- /tests/resources/sql/storage/J8FIHBUY/Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/storage/J8FIHBUY/Grubb - 2015 - Fractional Laplacians on domains, a development of.pdf -------------------------------------------------------------------------------- /tests/resources/sql/storage/PIMHYJGK/.zotero-ft-cache: -------------------------------------------------------------------------------- 1 | arXiv:math/0702079v3 [math.AP] 27 Nov 2007 2 | 3 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 4 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´KELYHIDI JR. 5 | Abstract. In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in Rn with n ≥ 2. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy–decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure. 6 | 7 | 1. Introduction 8 | 9 | Consider the Euler equations in n space dimensions, describing the motion of an ideal incompressible fluid, 10 | 11 | ∂tv + div (v ⊗ v) + ∇p − f = 0 div v = 0 . 12 | 13 | (1) 14 | 15 | Classical (i.e. sufficiently smooth) solutions of the Cauchy problem exist 16 | locally in time for sufficiently regular initial data and driving forces (see 17 | Chapter 3.2 in [16]). In two dimensions such existence results are available 18 | also for global solutions (e.g. Chapters 3.3 and 8.2 in [16] and the references 19 | therein). Classical solutions of Euler’s equations with f = 0 conserve the energy, that is t → |v(x, t)|2 dx is a constant function. Hence the energy space for (1) is L∞ t (L2x). 20 | A recurrent issue in the modern theory of PDEs is that one needs to go 21 | beyond classical solutions, in particular down to the energy space (see for instance [6, 8, 16, 25]). A divergence–free vector field v ∈ L2loc is a weak solution of (1) if 22 | 23 | v∂tϕ + v ⊗ v, ∇ϕ + ϕ · f dx dt = 0 24 | 25 | (2) 26 | 27 | for every test function ϕ ∈ Cc∞(Rnx ×Rt, Rn) with div ϕ = 0. It is well–known that then the pressure is determined up to a function depending only on time (see [28]). In the case of Euler strong motivation for considering weak solutions comes also from mathematical physics, especially the theory of turbulence laid down by Kolmogorov in 1941 [3, 11]. A celebrated criterion of Onsager related to Kolmogorov’s theory says, roughly speaking, that dissipative weak solutions cannot have a H¨older exponent greater than 1/3 28 | 1 29 | 30 | 2 31 | 32 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 33 | 34 | (see [4, 9, 10, 19]). It is therefore of interest to construct weak solutions with limited regularity. 35 | Weak solutions are not unique. In a well–known paper [21] Scheffer constructed a surprising example of a weak solution to (1) with compact support in space and time when f = 0 and n = 2. Scheffer’s proof is very long and complicated and a simpler construction was later given by Shnirelman in [22]. However, Shnirelman’s proof is still quite difficult. In this paper we obtain a short and elementary proof of the following theorem. 36 | 37 | Theorem 1.1. Let f = 0. There exists v ∈ L∞(Rnx × Rt; Rn) and p ∈ L∞(Rnx × Rt) solving (1) in the sense of distributions, such that v is not identically zero, and supp v and supp p are compact in space-time Rnx × Rt. 38 | In mathematical physics weak solutions to the Euler equations that dissipate energy underlie the Kolmogorov theory of turbulence. In another groundbreaking paper [23] Shnirelman proved the existence of L2 distributional solutions with f = 0 and energy which decreases in time. His methods are completely unrelated to those in [21] and [22]. In contrast, the following extension of his existence theorem is a simple corollary of our construction. 39 | 40 | Theorem 1.2. There exists (v, p) as in Theorem 1.1 such that, in addition: • |v(x, t)|2 dx = 1 for almost every t ∈] − 1, 1[, • v(x, t) = 0 for |t| > 1. 41 | 42 | Our method has several interesting features. First of all, our approach fits 43 | 44 | nicely in the well–known framework of L. Tartar for the analysis of oscil- 45 | 46 | lations in linear partial differential systems coupled with nonlinear point- 47 | 48 | wise constraints [7, 15, 26, 27]. Roughly speaking, Tartar’s framework 49 | 50 | amounts to a plane–wave analysis localized in physical space, in contrast 51 | 52 | with Shnirelman’s method in [22], which is based rather on a wave analy- 53 | 54 | sis in Fourier space. In combination with Gromov’s convex integration or 55 | 56 | with Baire category arguments, Tartar’s approach leads to a well under- 57 | 58 | stood mechanism for generating irregular oscillatory solutions to differential 59 | 60 | inclusions (see [14, 15, 17]). 61 | 62 | Secondly, the velocity field we construct belongs to the energy space 63 | 64 | L∞ t (L2x). This was not the case for the solutions in [21, 22], and it was a natural question whether weak solutions in the energy space were unique. 65 | 66 | Our first theorem shows that even higher summability assumptions of v do 67 | 68 | not rule out such pathologies. The pressure in [21, 22] is only a distribution 69 | 70 | solving (1). In our construction p is actually the potential–theoretic solution 71 | 72 | of 73 | 74 | − ∆p = ∂x2ixj (vivj ) − ∂xi fi . 75 | 76 | (3) 77 | 78 | However, being bounded, it has slightly better regularity than the BM O 79 | 80 | given by the classical estimates for (3). 81 | 82 | Next, our point of view reveals connections between the apparently un- 83 | 84 | related constructions of Scheffer and Shnirelman. Shnirelman considers se- 85 | 86 | quences of driving forces fk converging to 0 in some negative Sobolev space. 87 | 88 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 89 | 90 | 3 91 | 92 | In particular he shows that for a suitable choice of fk the corresponding solutions of (1) converge in L2 to a nonzero solution of (1) with f = 0. Scheffer builds his solution by iterating a certain piecewise constant construction at small scales. On the one hand both our proof and Scheffer’s proof are based on oscillations localized in physical space. On the other hand, our proof gives as an easy byproduct the following approximation result in Shnirelman’s spirit. 93 | Theorem 1.3. All the solutions (v, p) constructed in the proofs of Theorem 1.1 and in Theorem 1.2 have the following property. There exist three sequences {vk}, {fk}, {pk} ⊂ Cc∞ solving (1) such that 94 | • fk converges to 0 in H−1, • vk ∞ + pk ∞ is uniformly bounded, • (vk, pk) → (v, p) in Lq for every q < ∞. 95 | Our results give interesting information on which kind of additional (entropy) condition could restore uniqueness of solutions. As already remarked, belonging to the energy space is not sufficient. In fact, in view of our method of construction, there is strong evidence that neither energy–decreasing nor energy–preserving solutions are unique. In a forthcoming paper we plan to investigate this issue, and also the class of initial data for which our method yields energy–decreasing solutions. 96 | The rest of the paper is organized as follows. In Section 2 we carry out the plane wave analysis of the Euler equations in the spirit of Tartar, and we formulate the core of our construction (Proposition 2.2). In Section 3 we prove Proposition 2.2. In Section 4 we show how our main results follow from the Proposition. We emphasize that the concluding argument in Section 4 appeals to the – by now standard – methods for solving differential inclusions, either by appealing to the Baire category theorem [1, 2, 5, 13], or by the more explicit convex integration method [12, 17, 18]. In our opinion, the Baire category argument developed in [14] and used in Section 4 is, for the purposes of this paper, the most efficient and elegant tool. However, we include in Section 5 an alternative proof which follows the convex integration approach, as it makes easier to ”visualize” the solutions constructed in this paper. 97 | In fact we believe that for n ≥ 3 a suitable modification of the original approach of Gromov (see [12]) would also work, yielding solutions which are even continuous (work in progress). 98 | 99 | 2. Plane wave analysis of Euler’s equations 100 | 101 | We start by briefly explaining Tartar’s framework [26]. One considers 102 | 103 | nonlinear PDEs that can be expressed as a system of linear PDEs (conser- 104 | 105 | vation laws) 106 | 107 | m 108 | 109 | Ai∂iz = 0 110 | 111 | (4) 112 | 113 | i=1 114 | 115 | 4 116 | 117 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 118 | 119 | coupled with a pointwise nonlinear constraint (constitutive relations) 120 | 121 | z(x) ∈ K ⊂ Rd a.e., 122 | 123 | (5) 124 | 125 | where z : Ω ⊂ Rm → Rd is the unknown state variable. The idea is then to consider plane wave solutions to (4), that is, solutions of the form 126 | 127 | z(x) = ah(x · ξ), 128 | 129 | (6) 130 | 131 | where h : R → R. The wave cone Λ is given by the states a ∈ Rd such that for any choice of the profile h the function (6) solves (4), that is, 132 | 133 | m 134 | 135 | Λ := a ∈ Rd : ∃ξ ∈ Rm \ {0} with 136 | 137 | ξiAia = 0 . 138 | 139 | (7) 140 | 141 | i=1 142 | 143 | The oscillatory behavior of solutions to the nonlinear problem is then deter- 144 | 145 | mined by the compatibility of the set K with the cone Λ. 146 | 147 | The Euler equations can be naturally rewritten in this framework. The 148 | 149 | domain is Rm = Rn+1, and the state variable z is defined as z = (v, u, q), 150 | 151 | where 152 | 153 | q 154 | 155 | =p+ 156 | 157 | 1 n 158 | 159 | |v|2 160 | 161 | , 162 | 163 | and 164 | 165 | u=v⊗v− 166 | 167 | 1 n 168 | 169 | |v|2 170 | 171 | In, 172 | 173 | so that u is a symmetric n × n matrix with vanishing trace and In denotes the n × n identity matrix. From now on the linear space of symmetric n × n 174 | 175 | matrices will be denoted by Sn and the subspace of trace–free symmetric 176 | 177 | matrices by S0n. The following lemma is straightforward. 178 | 179 | Lemma 2.1. Suppose v ∈ L∞(Rnx × Rt; Rn), u ∈ L∞(Rnx × Rt; S0n), and q ∈ L∞(Rnx × Rt) solve 180 | 181 | ∂tv + div u + ∇q = 0, div v = 0, 182 | 183 | (8) 184 | 185 | in the sense of distributions. If in addition 186 | 187 | u 188 | 189 | = 190 | 191 | v 192 | 193 | ⊗ 194 | 195 | v 196 | 197 | − 198 | 199 | 1 n 200 | 201 | |v|2 202 | 203 | In 204 | 205 | a.e. in Rnx × Rt, 206 | 207 | (9) 208 | 209 | then 210 | 211 | v 212 | 213 | and 214 | 215 | p := 216 | 217 | q− 218 | 219 | 1 n 220 | 221 | |v|2 222 | 223 | are 224 | 225 | a 226 | 227 | solution 228 | 229 | to 230 | 231 | (1) 232 | 233 | with 234 | 235 | f 236 | 237 | ≡ 0. 238 | 239 | Conversely, 240 | 241 | if 242 | 243 | v 244 | 245 | and 246 | 247 | p 248 | 249 | solve 250 | 251 | (1) 252 | 253 | distributionally, 254 | 255 | then 256 | 257 | v, 258 | 259 | u 260 | 261 | := 262 | 263 | v⊗v− 264 | 265 | 1 n 266 | 267 | |v|2In 268 | 269 | and 270 | 271 | q 272 | 273 | := 274 | 275 | p+ 276 | 277 | 1 n 278 | 279 | |v|2 280 | 281 | solve (8) and (9). 282 | 283 | Consider the (n + 1) × (n + 1) symmetric matrix in block form 284 | 285 | U= 286 | 287 | u + qIn v 288 | 289 | v 0 290 | 291 | , 292 | 293 | (10) 294 | 295 | where In is the n × n identity matrix. Notice that by introducing new coordinates y = (x, t) ∈ Rn+1 the equation (8) becomes simply 296 | 297 | divyU = 0. 298 | 299 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 300 | 301 | 5 302 | 303 | Here, as usual, a divergence–free matrix field is a matrix of functions with rows that are divergence–free vectors. Therefore the wave cone corresponding to (8) is given by 304 | 305 | Λ= 306 | 307 | (v, u, q) ∈ Rn × S0n × R : det 308 | 309 | u + qIn v 310 | 311 | v 0 312 | 313 | =0 314 | 315 | . 316 | 317 | Remark 1. A simple linear algebra computation shows that for every v ∈ Rn and u ∈ S0n there exists q ∈ R such that (v, u, q) ∈ Λ, revealing that the wave cone is very large. Indeed, let V ⊥ ⊂ Rn be the linear space orthogonal to v and consider on V ⊥ the quadratic form ξ → ξ · uξ. Then, det U = 0 if and 318 | only if −q is an eigenvalue of this quadratic form. 319 | 320 | In order to exploit this fact for constructing irregular solutions to the nonlinear system, one needs plane wave–like solutions to (8) which are localized in space. Clearly an exact plane–wave as in (6) has compact support only if it is identically zero. Therefore this can only be done by introducing an error in the range of the wave, deviating from the line spanned by the wave state a ∈ Rd. However, this error can be made arbitrarily small. This is the content of the following proposition, which is the building block of our construction. 321 | 322 | Proposition 2.2 (Localized plane waves). Let a = (v0, u0, q0) ∈ Λ with v0 = 0, and denote by σ the line segment in Rn × S0n × R joining the points −a and a. For every ε > 0 there exists a smooth solution (v, u, q) of (8) with the properties: 323 | • the support of (v, u, q) is contained in B1(0) ⊂ Rnx × Rt, • the image of (v, u, q) is contained in the ε–neighborhood of σ, • |v(x, t)| dx dt ≥ α|v0|, 324 | where α > 0 is a dimensional constant. 325 | 326 | 3. Localized plane waves 327 | 328 | For the proof of Proposition 2.2 there are two main points. Firstly, we appeal to a particular large group of symmetries of the equations in order to reduce the problem to some special Λ-directions. Secondly, to achieve a cut-off which preserves the linear equations (8), we introduce a suitable potential. 329 | 330 | Definition 3.1. We denote by M the set of symmetric (n + 1) × (n + 1) matrices A such that A(n+1)(n+1) = 0. Clearly, the map 331 | 332 | Rn × S0n × R ∋ (v, u, q) 333 | 334 | → 335 | 336 | U= 337 | 338 | u + qIn v 339 | 340 | v 0 341 | 342 | ∈M 343 | 344 | (11) 345 | 346 | is a linear isomorphism. 347 | 348 | As already observed, in the variables y = (x, t) ∈ Rn+1, the equation (8) is equivalent to div U = 0. Therefore Proposition 2.2 follows immediately from 349 | 350 | 6 351 | 352 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 353 | 354 | Proposition 3.2. Let U ∈ M be such that det U = 0 and U en+1 = 0, and consider the line segment σ with endpoints −U and U . Then there exists a constant α > 0 such that for any ε > 0 there exists a smooth divergence–free matrix field U : Rn+1 → M with the properties 355 | (p1) supp U ⊂ B1(0), (p2) dist (U (y), σ) < ε for all y ∈ B1(0), (p3) |U (y)en+1|dy ≥ α|U en+1|, 356 | where α > 0 is a dimensional constant. 357 | 358 | The proof of Proposition 3.2 relies on two lemmas. The first deals with the symmetries of the equations. 359 | 360 | Lemma 3.3 (The Galilean group). Let G be the subgroup of GLn+1(R) defined by 361 | 362 | A ∈ R(n+1)×(n+1) : det A = 0, Aen+1 = en+1 . 363 | 364 | (12) 365 | 366 | For every divergence–free map U : Rn+1 → M and every A ∈ G the map 367 | 368 | V (y) := At · U (A−ty) · A 369 | 370 | is also a divergence–free map V : Rn+1 → M. 371 | 372 | The second deals with the potential. 373 | 374 | Lemma 3.4 (Potential in the general case). Let Eikjl ∈ C∞(Rn+1) be functions for i, j, k, l = 1, . . . , n + 1 so that the tensor E is skew–symmetric in 375 | 376 | ij and kl, that is 377 | 378 | Eikjl = −Eiljk = −Ejkil = Ejlki . 379 | 380 | (13) 381 | 382 | Then 383 | 384 | Uij 385 | 386 | = 387 | 388 | L(E) 389 | 390 | = 391 | 392 | 1 2 393 | 394 | ∂k2l(Ekilj + Ekjli) 395 | 396 | k,l 397 | 398 | (14) 399 | 400 | is symmetric and divergence–free. If in addition 401 | 402 | E((nn++11))ij = 0 403 | 404 | for every i and j, 405 | 406 | (15) 407 | 408 | then U takes values in M. 409 | 410 | Remark 2. A suitable potential in the case n = 2 can be obtained in a more direct way. Indeed, let w ∈ C∞(R3, R3) be a divergence–free vector field and consider the map U : R3 → M given by 411 | 412 |  ∂2w1 413 | 414 | U 415 | 416 | = 417 | 418 |  419 | 420 | 1 21 2 421 | 422 | ∂2w2 ∂2w3 423 | 424 | − 425 | 426 | 1 2 427 | 428 | ∂1w1 429 | 430 | 1 2 431 | 432 | ∂2 433 | 434 | w2 435 | 436 | − 437 | 438 | 1 2 439 | 440 | ∂1w1 441 | 442 | −∂1w2 443 | 444 | − 445 | 446 | 1 2 447 | 448 | ∂1w3 449 | 450 | −21 ∂212∂w13w3 451 | 452 |  . 453 | 454 | 0 455 | 456 | (16) 457 | 458 | Then it can be readily checked that U is divergence–free. Moreover, w is the 459 | curl of a vector field ω. However, this is just a particular case of Lemma 3.4. Indeed, given E as in the Lemma define the tensor Dikj = l ∂lEikjl. Note 460 | 461 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 462 | 463 | 7 464 | 465 | that D is skew–symmetric in ij and for each ij, the vector (Dikj)k=1,...,n+1 is divergence–free. Moreover, 466 | 467 | Uij 468 | 469 | = 470 | 471 | 1 2 472 | 473 | ∂k(Dki j + Dkji) . 474 | 475 | k 476 | 477 | Then the vector field w above is simply the special choice where D1k2 = −D2k1 = wk and all other D’s are zero, and a corresponding relation can be found for E and ω. 478 | 479 | The proofs of the two Lemmas will be postponed until the end of the section and we now come to the proof of the Proposition. 480 | 481 | Proof of Proposition 3.2. Step 1. First we treat the case when U ∈ M is 482 | 483 | such that 484 | 485 | U e1 = 0, U en+1 = 0. 486 | 487 | (17) 488 | 489 | Let 490 | 491 | Eij11 492 | 493 | = 494 | 495 | −E1ji1 496 | 497 | = 498 | 499 | −Ei11j 500 | 501 | = 502 | 503 | E11ij 504 | 505 | = 506 | 507 | U 508 | 509 | ij 510 | 511 | sin(N y1) N2 512 | 513 | (18) 514 | 515 | and all the other entries equal to 0. Note that by our assumption U ij = 0 whenever one index is 1 or both of them are n + 1. This ensures that the 516 | 517 | tensor E is well defined and satisfies the properties of Lemma 3.4. 518 | 519 | We remark that in the case n = 2 the matrix U takes necessarily the form 520 | 521 | 0 0 0 522 | 523 | U =0 a b 524 | 525 | (19) 526 | 527 | 0b 0 528 | 529 | with b = 0, and we can use the potential of Remark 2 by simply setting 530 | 531 | w 532 | 533 | = 534 | 535 | 1 N 536 | 537 | (0, 538 | 539 | a 540 | 541 | cos(N 542 | 543 | y1), 544 | 545 | 2b 546 | 547 | cos(N 548 | 549 | y1)) 550 | 551 | , 552 | 553 | ω 554 | 555 | = 556 | 557 | 1 N2 558 | 559 | (0, 560 | 561 | 2b 562 | 563 | sin(N 564 | 565 | y1 566 | 567 | ), 568 | 569 | −a 570 | 571 | sin(N 572 | 573 | y1)) 574 | 575 | . 576 | 577 | We come back to the general case. Let E be defined as in (18), fix a 578 | 579 | smooth cutoff function ϕ such that 580 | 581 | • |ϕ| ≤ 1, 582 | • ϕ = 1 on B1/2(0), • supp (ϕ) ⊂ B1(0), 583 | 584 | and consider the map 585 | 586 | U = L(ϕE). 587 | 588 | Clearly, U is smooth and supported in B1(0). By Lemma 3.4, U is M–valued and divergence–free. Moreover 589 | 590 | U (y) = U sin(N y1) for y ∈ B1/2(0), and in particular 591 | 592 | |U (y)en+1|dy ≥ |U en+1| 593 | 594 | | sin(N y1)| dy ≥ 2α|U en+1|, 595 | 596 | B1/2 (0) 597 | 598 | 8 599 | 600 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 601 | 602 | for some positive dimensional constant α = α(n) for sufficiently large N . Finally, observe that 603 | U − ϕU˜ = L(ϕE) − ϕL(E) 604 | 605 | is a sum of products of first–order derivatives of ϕ with first–order derivatives of components of E and of second–order derivatives of ϕ with components of E. Thus, 606 | 607 | U − ϕU˜ 608 | 609 | ∞ 610 | 611 | ≤ 612 | 613 | C 614 | 615 | ϕ 616 | 617 | C2 618 | 619 | E 620 | 621 | C1 622 | 623 | ≤ 624 | 625 | C′ N 626 | 627 | ϕ 628 | 629 | C2 , 630 | 631 | and by choosing N sufficiently large we obtain U − ϕU˜ ∞ < ε. On the other hand, since |ϕ| ≤ 1 and U˜ takes values in σ, the image of ϕU˜ is also contained in σ. This shows that the image of U is contained in the ε–neighborhood of σ. 632 | 633 | Step 2. We treat the general case by reducing to the situation above. Let U ∈ M be as in the Proposition, so that 634 | 635 | U f = 0, U en+1 = 0, 636 | where f ∈ Rn+1 \ {0} is such that {f, en+1} are linearly independent. Let f1, . . . , fn+1 be a basis for Rn+1 such that f1 = f and fn+1 = en+1 and consider the matrix A such that 637 | 638 | Aei = fi for i = 1, . . . , n + 1. Then A ∈ G (cf. with the definition of G given in Lemma 3.3), and the map 639 | 640 | T : X → (A−1)tXA−1 641 | 642 | (20) 643 | 644 | is a linear isomorphism of Rn+1. Set 645 | 646 | V = AtU A, 647 | 648 | (21) 649 | 650 | so that V ∈ M satisfies 651 | 652 | V e1 = 0, V en+1 = 0. 653 | Given ε > 0, using Step 1 we construct a smooth map V : Rn+1 → M supported in B1(0) with the image lying in the T −1ε–neighborhood of the line segment τ with endpoints −V and V , and such that 654 | 655 | V (y) = V sin(N y1). Let U be the M–valued map 656 | U (y) = (A−1)tV (Aty)A−1. 657 | 658 | By our discussion above the isomorphism T : X → (A−1)tXA−1 maps the line segment τ onto σ. Therefore: 659 | • U is supported in A−t(B1(0)) and it is smooth, • U is divergence–free thanks to Lemma 3.3, • U takes values in an ε–neighborhood of the segment σ, 660 | 661 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 662 | 663 | 9 664 | 665 | and furthermore 666 | 667 | |U (y)en+1|dy = 668 | 669 | |A−tV (Aty)en+1|dy 670 | 671 | A−t (B1 (0)) 672 | 673 | A−t (B1 (0)) 674 | 675 | = 676 | 677 | B1 (0) 678 | 679 | |A−tV 680 | 681 | (z)en+1| 682 | 683 | | 684 | 685 | dz det At| 686 | 687 | ≥ 688 | 689 | 2α|A−tV en+1| | det A| 690 | 691 | = 692 | 693 | | 694 | 695 | 2α det A| 696 | 697 | |U 698 | 699 | en+1 700 | 701 | |. 702 | 703 | (22) 704 | 705 | To complete the proof we appeal to a standard covering/rescaling argument. 706 | That is, we can find a finite number of points yk ∈ B1(0) and radii rk > 0 so that the rescaled and translated sets A−t(Brk (yk)) are pairwise disjoint, all contained in B1(0), and 707 | 708 | A−t(Brk (yk)) 709 | 710 | ≥ 711 | 712 | 1 2 713 | 714 | |B1(0)|. 715 | 716 | (23) 717 | 718 | k 719 | 720 | Let 721 | 722 | Uk (y ) 723 | 724 | = 725 | 726 | U 727 | 728 | ( 729 | 730 | y−yk rk 731 | 732 | ) 733 | 734 | and 735 | 736 | U˜ 737 | 738 | = 739 | 740 | k Uk. Then U˜ : Rn+1 → M is smooth, 741 | 742 | clearly satisfies (p1) and (p2), and 743 | 744 | |U˜ (y)en+1|dy = 745 | 746 | |Uk (y )en+1 |dy 747 | k A−tBrk (yk) 748 | 749 | (22) 750 | ≥ 751 | 752 | k 753 | 754 | 2α|U 755 | 756 | en+1 757 | 758 | || 759 | 760 | det 761 | 762 | A|−1 763 | 764 | |Brk (yk)| |B1(0)| 765 | 766 | = 767 | 768 | 2α|U en+1| 769 | 770 | k A−t(Brk (yk)) |B1(0)| 771 | 772 | (23) 773 | ≥ α|U en+1|. 774 | 775 | This completes the proof. 776 | 777 | Proof of Lemma 3.3. First of all we check that whenever B ∈ M, then AtBA ∈ M for all A ∈ G. Indeed, AtBA is symmetric, and since A satisfies 778 | Aen+1 = en+1, we have 779 | 780 | (AtBA)(n+1)(n+1) = en+1 · AtBAen+1 = Aen+1 · BAen+1 = en+1 · Ben+1 = B(n+1)(n+1) = 0. 781 | 782 | (24) 783 | 784 | Now, let A, U and V be as in the statement. The argument above shows 785 | that V is M–valued. It remains to check that if U is divergence–free, then V is also divergence–free. To this end let φ ∈ Cc∞(Rn+1; Rn+1) be a compactly supported test function and consider φ˜ ∈ Cc∞(Rn+1; Rn+1) defined by 786 | 787 | φ˜(x) = Aφ(Atx). 788 | 789 | 10 790 | 791 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 792 | 793 | Then ∇φ˜(x) = A∇φ(Atx)At, and by a change of variables we obtain 794 | 795 | tr V(y)∇φ(y) dy = tr AtU(A−ty)A∇φ(y) dy 796 | 797 | = tr U(A−ty)A∇φ(y)At dy 798 | 799 | = tr U(x)A∇φ(Atx)At (det A)−1dx 800 | 801 | =(det A)−1 tr U(x)∇φ˜(x) dx = 0, since U is divergence–free. But this implies that V is also divergence-free. 802 | 803 | Proof of Lemma 3.4. First of all, U is clearly symmetric and U(n+1)(n+1) = 0. Hence U takes values in M. To see that U is divergence–free, we calculate 804 | 805 | ∂j Uij 806 | 807 | = 808 | 809 | 1 2 810 | 811 | ∂j3kl(Ekilj + Ekjli) 812 | 813 | j 814 | 815 | k,l 816 | 817 | = 818 | 819 | 1 2 820 | 821 | ∂l 822 | 823 | ∂j2k Ekilj 824 | 825 | + 826 | 827 | 1 2 828 | 829 | ∂k 830 | 831 | l 832 | 833 | jk 834 | 835 | k 836 | 837 | ∂j2lEkjli 838 | jl 839 | 840 | (1=3) 0 . 841 | 842 | This completes the proof of the lemma. 843 | 844 | 4. Proof of the main results 845 | For clarity we now state the precise form of our main result. Theorems 1.1, 1.2 and 1.3 are direct corollaries. 846 | Theorem 4.1. Let Ω ⊂ Rnx × Rt be a bounded open domain. There exists (v, p) ∈ L∞(Rnx × Rt) solving the Euler equations 847 | ∂tv + div (v ⊗ v) + ∇p = 0 div v = 0 , 848 | such that • |v(x, t)| = 1 for a.e. (x, t) ∈ Ω, • v(x, t) = 0 and p(x, t) = 0 for a.e. (x, t) ∈ (Rnx × Rt) \ Ω. 849 | Moreover, there exists a sequence of functions (vk, pk, fk) ∈ Cc∞(Ω) such that 850 | ∂tvk + div (vk ⊗ vk) + ∇pk = fk div vk = 0 , 851 | and • fk converges to 0 in H−1, • vk ∞ + pk ∞ is uniformly bounded, • (vk, pk) → (v, p) in Lq for every q < ∞. 852 | 853 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 854 | 855 | 11 856 | 857 | We remark that the statements of Theorem 1.1 and Theorem 1.3 are just subsets of the statement of Theorem 4.1. As for Theorem 1.2, note that it suffices to choose, for instance, Ω = Br(0)×] − 1, 1[, where Br(0) is the ball of Rn with volume 1. 858 | We recall from Lemma 2.1 that for the first half of the theorem it suffices to prove that there exist 859 | (v, u, q) ∈ L∞(Rnx × Rt; Rn × S0n × R) 860 | with support in Ω, such that |v| = 1 a.e. in Ω and (8) and (9) are satisfied. In Proposition 2.2 we constructed compactly supported solutions (v, u, q) to (8). The point is thus to find solutions which satisfy in addition the pointwise constraint (9). The main idea is to consider the sets 861 | 862 | K= 863 | 864 | (v, u) 865 | 866 | ∈ 867 | 868 | Rn × S0n 869 | 870 | : 871 | 872 | u 873 | 874 | = 875 | 876 | v⊗v− 877 | 878 | 1 n 879 | 880 | |v|2 881 | 882 | In 883 | 884 | , 885 | 886 | |v| = 1 887 | 888 | , 889 | 890 | (25) 891 | 892 | and 893 | 894 | U = int (Kco × [−1, 1]), 895 | 896 | (26) 897 | 898 | where int denotes the topological interior of the set in Rn × S0n × R, and Kco denotes the convex hull of K. Thus, a triple (v, u, q) solving (8) and taking values in the convex extremal points of U is indeed a solution to (9). We will prove that 0 ∈ U, and therefore there exist plane waves taking values in U. The goal is to add them so to get an infinite sum 899 | 900 | ∞ 901 | (v, u, q) = (vi, ui, qi) 902 | i=1 903 | 904 | with the properties that 905 | 906 | • the partial sums ki=0(vi, ui, qi) take values in U , • (v, u, q) is supported in Ω, • (v, u, q) takes values in the convex extremal points of U a.e. in Ω, • (v, u, q) solves the linear partial differential equations (8). 907 | 908 | There are two important reasons why this construction is possible. First of all, since the wave cone Λ is very large, we can always get closer and closer to the extremal point of U with the sequence (vk, uk, pk). Secondly, because the waves are localized in space–time, by choosing the supports smaller and smaller we can achieve strong convergence of the sequence. In view of Lemma 2.1 this gives the solution of Euler that we we are looking for. The partial sums give the approximating sequence of the theorem. 909 | 910 | This sketch of the proof is philosophically closer to the method of convex integration, where the difficulty is to ensure strong convergence of the partial sums. The Baire category argument avoids this difficulty by introducing a metric for the space of solutions to (8) with values in U, and proving that in its closure a generic element takes values in the convex extreme points. An interesting corollary of the Baire category argument is that, within the class of solutions to the Euler equations with driving force in some particular 911 | 912 | 12 913 | 914 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 915 | 916 | bounded subset of H−1, the typical (in the sense of category) element has the properties of Theorem 4.1 . 917 | We split the proof of Theorem 4.1 into several lemmas and a short concluding argument, which is given at the beginning of Section 4.3. For the purpose of this section, we could have presented a shorter proof, avoiding Lemma 4.3 and without giving the explicit bound (30) of Lemma 4.6. However, these statements will be needed in the convex integration proof of Section 5. 918 | 919 | 4.1. The geometric setup. 920 | 921 | Lemma 4.2. Let K and U be defined as in (25) and (26), i.e. 922 | 923 | K= 924 | 925 | (v, u) 926 | 927 | ∈ 928 | 929 | Sn−1 930 | 931 | × 932 | 933 | S0n 934 | 935 | : 936 | 937 | u 938 | 939 | = 940 | 941 | v 942 | 943 | ⊗v 944 | 945 | − 946 | 947 | In n 948 | 949 | . 950 | 951 | Then 0 ∈ int Kco and hence 0 ∈ U . 952 | 953 | Proof. Let µ be the Haar measure on Sn−1 and consider the linear map 954 | 955 | T : C(Sn−1) → Rn × S0n, Clearly, if 956 | 957 | φ→ 958 | Sn−1 959 | 960 | v, 961 | 962 | v 963 | 964 | ⊗ 965 | 966 | v 967 | 968 | − 969 | 970 | In n 971 | 972 | φ(v) dµ . 973 | 974 | φ ≥ 0 and 975 | 976 | φ dµ = 1 , 977 | 978 | (27) 979 | 980 | Sn−1 981 | 982 | then T (φ) ∈ Kco. Notice that 983 | 984 | T (1) = 985 | 986 | v, v ⊗ v − In 987 | 988 | Sn−1 989 | 990 | n 991 | 992 | dµ = 0, 993 | 994 | and hence 0 ∈ Kco. Moreover, whenever ψ ∈ C(Sn−1) is such that 995 | 996 | α = 1− 997 | 998 | ψ dµ ≥ ψ C(Sn−1), 999 | 1000 | Sn−1 1001 | 1002 | (28) 1003 | 1004 | φ = α + ψ satisfies (27) and hence T (ψ) = T (φ) ∈ Kco. Since (28) holds 1005 | 1006 | whenever ψ C(Sn−1) < 1/2, it suffices to show that T is surjective to prove that Kco contains a neighborhood of 0. 1007 | The surjectivity of T follows from orthogonality in L2(Sn−1). Indeed, 1008 | 1009 | letting φ = vi for each i, we obtain 1010 | 1011 | T (φ) = β1(ei, 0), where β1 = 1012 | 1013 | v12dµ. 1014 | 1015 | Sn−1 1016 | 1017 | Furthermore, setting φ = vivj with i = j, we obtain 1018 | 1019 | T (φ) = β2 0, ei ⊗ ej + ej ⊗ ei , where β2 = 1020 | 1021 | v12v22dµ. 1022 | 1023 | Sn−1 1024 | 1025 | Finally, 1026 | 1027 | setting 1028 | 1029 | φ 1030 | 1031 | = 1032 | 1033 | vi2 1034 | 1035 | − 1036 | 1037 | 1 n 1038 | 1039 | we 1040 | 1041 | obtain 1042 | 1043 | T (φ) = β3 1044 | 1045 | 0, 1046 | 1047 | ei 1048 | 1049 | ⊗ 1050 | 1051 | ei 1052 | 1053 | − 1054 | 1055 | (n 1056 | 1057 | 1 − 1058 | 1059 | 1) 1060 | 1061 | ej ⊗ ej , 1062 | 1063 | j=i 1064 | 1065 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 1066 | 1067 | 13 1068 | 1069 | where 1070 | 1071 | β3 = 1072 | 1073 | Sn−1 1074 | 1075 | v12 1076 | 1077 | − 1078 | 1079 | 1 n 1080 | 1081 | 2 1082 | dµ. 1083 | 1084 | This shows that elements, hence 1085 | 1086 | the image of T a basis for Rn 1087 | 1088 | contains × S0n. 1089 | 1090 | n+ 1091 | 1092 | 1 2 1093 | 1094 | n(n+1)−1 1095 | 1096 | linearly 1097 | 1098 | independent 1099 | 1100 | Lemma 4.3. There exists a dimensional constant C > 0 such that for any (v, u, q) ∈ U there exists (v¯, u¯) ∈ Rn × S0n such that (v¯, u¯, 0) ∈ Λ, the line segment with endpoints (v, u, q) ± (v¯, u¯, 0) is contained in U, and 1101 | 1102 | |v¯| ≥ C(1 − |v|2). 1103 | 1104 | Proof. Let z = (v, u) ∈ int Kco. By Carath´eodory’s theorem (v, u) lies in the interior of a simplex in Rn × S0n spanned by elements of K. In other words 1105 | N +1 1106 | z = λizi, 1107 | i=1 1108 | 1109 | where λi ∈ ]0, 1[ , zi = (vi, ui) ∈ K, 1110 | 1111 | N +1 i=1 1112 | 1113 | λi 1114 | 1115 | = 1116 | 1117 | 1, 1118 | 1119 | and 1120 | 1121 | N 1122 | 1123 | = 1124 | 1125 | n(n + 3)/2 − 1 1126 | 1127 | is 1128 | 1129 | the dimension of Rn × S0n. Assume that the coefficients are ordered so that 1130 | 1131 | λ1 = maxi λi. Then for any j > 1 1132 | 1133 | z 1134 | 1135 | ± 1136 | 1137 | 1 2 1138 | 1139 | λj 1140 | 1141 | (zj 1142 | 1143 | − z1) 1144 | 1145 | ∈ 1146 | 1147 | int 1148 | 1149 | K co. 1150 | 1151 | Indeed, 1152 | 1153 | z 1154 | 1155 | ± 1156 | 1157 | 1 2 1158 | 1159 | λj 1160 | 1161 | (zj 1162 | 1163 | − z1) = 1164 | 1165 | µizi, 1166 | 1167 | i 1168 | 1169 | where 1170 | 1171 | µ1 1172 | 1173 | = λ1 ∓ 1174 | 1175 | 1 2 1176 | 1177 | λj 1178 | 1179 | , 1180 | 1181 | µj 1182 | 1183 | = 1184 | 1185 | λj 1186 | 1187 | ± 1188 | 1189 | 1 2 1190 | 1191 | λj 1192 | 1193 | and 1194 | 1195 | µi 1196 | 1197 | = λi 1198 | 1199 | for 1200 | 1201 | i 1202 | 1203 | ∈/ 1204 | 1205 | {1, j}. 1206 | 1207 | It 1208 | 1209 | is 1210 | 1211 | easy 1212 | 1213 | to 1214 | 1215 | see that µi ∈ ]0, 1[ for all i = 1 . . . N + 1. 1216 | 1217 | On the other hand z − z1 = 1218 | 1219 | N +1 i=2 1220 | 1221 | λi(zi 1222 | 1223 | − 1224 | 1225 | z1), 1226 | 1227 | so 1228 | 1229 | that 1230 | 1231 | in 1232 | 1233 | particular 1234 | 1235 | |v 1236 | 1237 | − 1238 | 1239 | v1| 1240 | 1241 | ≤ 1242 | 1243 | N 1244 | 1245 | max 1246 | i=2...N +1 1247 | 1248 | λi|vi 1249 | 1250 | − 1251 | 1252 | v1| 1253 | 1254 | Let j > 1 be such that λj|vj − v1| = maxi=2...N+1 λi|vi − v1|, and let 1255 | 1256 | (v¯, u¯) 1257 | 1258 | = 1259 | 1260 | 1 2 1261 | 1262 | λj 1263 | 1264 | (zj 1265 | 1266 | − 1267 | 1268 | z1). 1269 | 1270 | The line segment with endpoints (v, u) ± (v¯, u¯) is contained in the interior of Kco and hence also the line segment (v, u, q) ± (v¯, u¯, 0) is contained in U . 1271 | Furthermore 1272 | 1273 | 1 4N 1274 | 1275 | (1 1276 | 1277 | − 1278 | 1279 | |v|2) 1280 | 1281 | ≤ 1282 | 1283 | 1 2N 1284 | 1285 | (1 1286 | 1287 | − 1288 | 1289 | |v|) 1290 | 1291 | ≤ 1292 | 1293 | 1 2N 1294 | 1295 | (|v 1296 | 1297 | − 1298 | 1299 | v1|) 1300 | 1301 | ≤ 1302 | 1303 | |v¯|. 1304 | 1305 | 14 1306 | 1307 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 1308 | 1309 | Finally, we show that (v¯, u¯, 0) ∈ Λ. This amounts to showing that whenever a, b ∈ Sn−1, the matrix 1310 | 1311 | a 1312 | 1313 | ⊗ 1314 | 1315 | a 1316 | 1317 | − 1318 | 1319 | In n 1320 | 1321 | a 1322 | 1323 | a 0 1324 | 1325 | − 1326 | 1327 | b 1328 | 1329 | ⊗ 1330 | 1331 | b 1332 | 1333 | − 1334 | 1335 | In n 1336 | 1337 | b 1338 | 1339 | b 0 1340 | 1341 | has zero determinant and hence lies in the wave cone Λ defined in (7). Let 1342 | 1343 | P ∈ GLn(R) with P a = e1 and P b = e2. Note that 1344 | 1345 | P0 01 1346 | 1347 | a⊗a a a0 1348 | 1349 | Pt 0 1350 | 1351 | 0 1 1352 | 1353 | = 1354 | 1355 | Pa ⊗ Pa Pa 1356 | 1357 | Pa 1 1358 | 1359 | , 1360 | 1361 | so that it suffices to check the determinant of 1362 | 1363 | e1 ⊗ e1 e1 1364 | 1365 | e1 0 1366 | 1367 | − 1368 | 1369 | e2 ⊗ e2 e2 1370 | 1371 | e2 0 1372 | 1373 | . 1374 | 1375 | Since e1 + e2 − en+1 is in the kernel of this matrix, it has indeed determinant zero. This completes the proof. 1376 | 1377 | 4.2. The functional setup. We define the complete metric space X as follows. Let 1378 | X0 := (v, u, q) ∈ C∞(Rnx × Rt) : (i), (ii) and (iii) below hold 1379 | (i) supp (v, u, q) ⊂ Ω, (ii) (v, u, q) solves (8) in Rnx × Rt, (iii) (v(x, t), u(x, t), q(x, t)) ∈ U for all (x, t) ∈ Rnx × Rt. We equip X0 with the topology of L∞-weak* convergence of (v, u, q) and we let X be the closure of X0 in this topology. 1380 | Lemma 4.4. The set X with the topology of L∞ weak* convergence is a nonempty compact metrizable space. Moreover, if (v, u, q) ∈ X is such that 1381 | 1382 | |v(x, t)| = 1 for almost every (x, t) ∈ Ω, 1383 | 1384 | then 1385 | 1386 | v 1387 | 1388 | and 1389 | 1390 | p 1391 | 1392 | := 1393 | 1394 | q 1395 | 1396 | − 1397 | 1398 | 1 n 1399 | 1400 | |v|2 1401 | 1402 | is a weak solution 1403 | 1404 | of 1405 | 1406 | (1) in Rnx × Rt 1407 | 1408 | such that 1409 | 1410 | v(x, t) = 0 and p(x, t) = 0 for all (x, t) ∈ Rnx × Rt \ Ω. 1411 | 1412 | Proof. In Lemma 4.2 we showed that 0 ∈ U, hence X is nonempty. Moreover, X is a bounded and closed subset of L∞(Ω), hence with the weak* topology 1413 | it becomes a compact metrizable space. Since U is a compact convex set, any (v, u, q) ∈ X satisfies 1414 | 1415 | supp (v, u, q) ⊂ Ω, (v, u, q) solves (8) and takes values in U. 1416 | 1417 | In particular (v, u)(x, t) ∈ Kco almost everywhere. Finally, observe also that if (v, u)(x, t) ∈ Kco, then 1418 | 1419 | (v, u)(x, t) ∈ K if and only if |v(x, t)| = 1. 1420 | 1421 | In light of Lemma 2.1 this concludes the proof. 1422 | 1423 | Fix a metric d∗∞ inducing the weak* topology of L∞ in X, so that (X, d∗∞) is a complete metric space. 1424 | 1425 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 1426 | 1427 | 15 1428 | 1429 | Lemma 4.5. The identity map 1430 | 1431 | I : (X, d∗∞) → L2(Rnx × Rt) defined by (v, u, q) → (v, u, q) 1432 | is a Baire-1 map and therefore the set of points of continuity is residual in (X, d∗∞). 1433 | 1434 | Proof. Let φr(x, t) = r−(n+1)φ(rx, rt) be any regular space-time convolution kernel. For each fixed (v, u, q) ∈ X we have 1435 | 1436 | (φr ∗ v, φr ∗ u, φr ∗ q) → (v, u, q) strongly in L2 as r → 0. On the other hand, for each r > 0 and (vk, uk, qk) ∈ X 1437 | 1438 | (vk, uk, qk) ⇀∗ (v, u, q) in L∞ =⇒ φr ∗ (vk, uk, qk) → φr ∗ (v, u, q) in L2. Therefore each map Ir : (X, d∗∞) → L2 defined by 1439 | Ir : (u, v, q) → (φr ∗ v, φr ∗ u, φr ∗ q) 1440 | 1441 | is continuous, and 1442 | 1443 | I (v, 1444 | 1445 | u, 1446 | 1447 | q) 1448 | 1449 | = 1450 | 1451 | lim 1452 | r→0 1453 | 1454 | Ir (v, 1455 | 1456 | u, 1457 | 1458 | q) 1459 | 1460 | for all (v, u, q) ∈ X . 1461 | 1462 | This shows that I : (X, d∗∞) → L2 is a pointwise limit of continuous maps, hence it is a Baire-1 map. Therefore the set of points of continuity of I is 1463 | residual in (X, d∗∞), see [20]. 1464 | 1465 | 4.3. Points of continuity of the identity map. The proof of Theorem 4.1 will follow from Lemmas 4.4 and 4.5 once we prove the following 1466 | 1467 | Claim: If (v, u, q) ∈ X is a point of continuity of I, then 1468 | 1469 | |v(x, t)| = 1 for almost every (x, t) ∈ Ω . 1470 | 1471 | (29) 1472 | 1473 | Indeed, if the claim is true, then the set of (v, u, q) ∈ X such that |v| = 1 1474 | 1475 | a.e. is nonempty, yielding solutions of (1). Furthermore, any such (v, u, q) 1476 | 1477 | must be the strong L2 limit of some sequence {(vk, uk, qk)} ⊂ X0. Therefore, 1478 | 1479 | with 1480 | 1481 | pk 1482 | 1483 | = 1484 | 1485 | qk 1486 | 1487 | − 1488 | 1489 | 1 n 1490 | 1491 | |vk 1492 | 1493 | |2, 1494 | 1495 | and 1496 | 1497 | fk = div 1498 | 1499 | vk 1500 | 1501 | ⊗ 1502 | 1503 | vk 1504 | 1505 | − 1506 | 1507 | 1 n 1508 | 1509 | |vk 1510 | 1511 | |2I 1512 | 1513 | d 1514 | 1515 | − 1516 | 1517 | uk 1518 | 1519 | , 1520 | 1521 | we obtain div vk = 0 and 1522 | 1523 | ∂tvk + div vk ⊗ vk + ∇pk = fk. 1524 | 1525 | Moreover, fk → 0 in H−1. 1526 | 1527 | Therefore it remains to prove our claim. Observe that since |v(x, t)| ≤ 1 a.e. (x, t) ∈ Ω, (29) is equivalent to 1528 | v L2(Ω) = |Ω|, 1529 | 1530 | 16 1531 | 1532 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 1533 | 1534 | where |Ω| denotes the (n + 1)-dimensional Lebesgue measure of Ω. To prove the claim we prove the following lemma, from which the claim immediately follows: 1535 | 1536 | Lemma 4.6. There exists a dimensional constant β > 0 with the following 1537 | 1538 | property. Given (v0, u0, q0) ∈ X0 there exists a sequence (vk, uk, qk) ∈ X0 1539 | 1540 | such that 1541 | 1542 | vk 1543 | 1544 | 2 L2 (Ω) 1545 | 1546 | ≥ 1547 | 1548 | v0 1549 | 1550 | 2 L2 (Ω) 1551 | 1552 | + 1553 | 1554 | β 1555 | 1556 | |Ω| − 1557 | 1558 | v0 1559 | 1560 | 2 L2 (Ω) 1561 | 1562 | 2 1563 | , 1564 | 1565 | (30) 1566 | 1567 | and (vk, uk, qk) ⇀∗ (v0, u0, q0) in L∞(Ω). 1568 | 1569 | Indeed, assume for a moment that (v, u, q) is a point of continuity of I. Fix a sequence {(vk, uk, qk} ⊂ X0 converges weakly∗ to (v, u, q). Using 1570 | Lemma 4.6 and a standard diagonal argument, we can produce a second sequences (v˜k, u˜k, q˜k) which converges weakly∗ to (v, u, q) and such that 1571 | 1572 | lim inf 1573 | k→∞ 1574 | 1575 | v˜k 1576 | 1577 | 2 2 1578 | 1579 | ≥ lim inf 1580 | k→∞ 1581 | 1582 | vk 1583 | 1584 | 2 2 1585 | 1586 | + 1587 | 1588 | β 1589 | 1590 | |Ω| − 1591 | 1592 | vk 1593 | 1594 | 2 2 1595 | 1596 | 2 1597 | 1598 | . 1599 | 1600 | (31) 1601 | 1602 | Since I is continuous at (v, u, q), both vk and v˜k converge strongly to v. 1603 | 1604 | Therefore 1605 | 1606 | v 1607 | 1608 | 2 2 1609 | 1610 | ≥ 1611 | 1612 | v 1613 | 1614 | 2 2 1615 | 1616 | + 1617 | 1618 | β 1619 | 1620 | |Ω| − 1621 | 1622 | v 1623 | 1624 | 2 2 1625 | 1626 | 2. 1627 | 1628 | (32) 1629 | 1630 | Therefore, 1631 | 1632 | v 1633 | 1634 | 2 2 1635 | 1636 | = |Ω|. 1637 | 1638 | On the other 1639 | 1640 | hand, since v = 0 a.e. 1641 | 1642 | outside Ω and 1643 | 1644 | |v| ≤ 1 a.e. on Ω, this implies (29). 1645 | 1646 | Proof of Lemma 4.6. Step 1. Let (v0, u0, q0) ∈ X0. By Lemma 4.3 for any (x, t) ∈ Ω there exists a direction 1647 | v¯(x, t), u¯(x, t) ∈ Rn × S0n 1648 | such that the line segment with endpoints 1649 | 1650 | v0(x, t), u0(x, t), q0(x, t) ± v¯(x, t), u¯(x, t), 0 1651 | is contained in U, and |v¯(x, t)| ≥ C(1 − |v0(x, t)|2). 1652 | Moreover, since (v0, u0, q0) is uniformly continuous, there exists ε > 0 such that for any (x, t), (x0, t0) ∈ Ω with |x−x0|+|t−t0| < ε, the ε-neighbourhood of the line segment with endpoints 1653 | 1654 | v0(x, t), u0(x, t), q0(x, t) ± v¯(x0, t0), u¯(x0, t0), 0 is also contained in U. 1655 | 1656 | Step 2. Fix (x0, t0) ∈ Ω for the moment. Use Proposition 2.2 with 1657 | 1658 | a = (v¯(x0, t0), u¯(x0, t0), 0) ∈ Λ 1659 | 1660 | and ε > 0 to obtain a smooth solution (v, u, q) of (8) with the properties stated in the Proposition, and for any r < ε let 1661 | 1662 | (vr, ur, qr)(x, t) = (v, u, q) 1663 | 1664 | x 1665 | 1666 | − r 1667 | 1668 | x0 1669 | 1670 | , 1671 | 1672 | t 1673 | 1674 | − r 1675 | 1676 | t0 1677 | 1678 | . 1679 | 1680 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 1681 | 1682 | 17 1683 | 1684 | Then (vr, ur, qr) is also a smooth solution of (8), with the properties • the support of (vr, ur, qr) is contained in Br(x0, t0) ⊂ Rnx × Rt, • the image of (vr, ur, qr) is contained in the ε–neighborhood of the line-segment with endpoints ±(v¯(x, t), u¯(x, t), 0), • and 1685 | |vr(x, t)| dx dt ≥ α|v¯(x0, t0)||Br(x0, r0)|. 1686 | 1687 | In particular, for any r < ε we have (v0, u0, q0) + (vr, ur, qr) ∈ X0. 1688 | Step 3. Next, observe that since v0 is uniformly continuous, there exists r0 > 0 such that for any r < r0 there exists a finite family of pairwise disjoint balls Brj (xj, tj) ⊂ Ω with rj < r such that 1689 | 1690 | (1 − |v0(x, t)|2)dxdt ≤ 2 (1 − |v0(xj, tj)|2)|Br(xj, tj)| (33) 1691 | 1692 | Ω 1693 | 1694 | j 1695 | 1696 | Fix 1697 | 1698 | k 1699 | 1700 | ∈ 1701 | 1702 | N with 1703 | 1704 | 1 k 1705 | 1706 | < 1707 | 1708 | min{r0, ε} 1709 | 1710 | and 1711 | 1712 | choose a 1713 | 1714 | finite family 1715 | 1716 | of 1717 | 1718 | pair- 1719 | 1720 | wise 1721 | 1722 | disjoint 1723 | 1724 | balls 1725 | 1726 | Brk,j (xk,j , tk,j ) 1727 | 1728 | ⊂ 1729 | 1730 | Ω 1731 | 1732 | with 1733 | 1734 | radii 1735 | 1736 | rk,j 1737 | 1738 | < 1739 | 1740 | 1 k 1741 | 1742 | such 1743 | 1744 | that 1745 | 1746 | (33) 1747 | 1748 | holds. In each ball Brk,j (xk,j, tk,j) we apply the construction above to ob- 1749 | 1750 | tain (vk,j, uk,j, qk,j), and in particular we then have 1751 | 1752 | (vk, uk, qk) := (v0, u0, q0) + (vk,j, uk,j, qk,j) ∈ X0, 1753 | j 1754 | and 1755 | 1756 | |vk(x, t) − v0(x, t)|dxdt = 1757 | j 1758 | 1759 | |vk,j(x, t)|dxdt 1760 | 1761 | ≥ α |v¯(xk,j, tk,j)||Brk,j (xk,j, tk,j)| 1762 | j 1763 | 1764 | ≥ Cα (1 − |v0(xk,j, tk,j)|2)|Brk,j (xk,j, tk,j)| 1765 | j 1766 | 1767 | ≥ 1768 | 1769 | 1Cα 2 1770 | 1771 | (1 − |v0(x, t)|2)dxdt. 1772 | Ω 1773 | 1774 | (34) 1775 | 1776 | Finally observe that by letting k → ∞, the above construction yields a sequence (vk, uk, qk) ∈ X0 such that 1777 | 1778 | (vk, uk, qk) ⇀∗ (v0, u0, q0). 1779 | 1780 | (35) 1781 | 1782 | Hence, 1783 | 1784 | lim inf 1785 | k→∞ 1786 | 1787 | vk L2(Ω) 1788 | 1789 | = 1790 | 1791 | v0 1792 | 1793 | 2 2 1794 | 1795 | + 1796 | 1797 | lim inf 1798 | k→∞ 1799 | 1800 | v0, (vk − v0) 2 + 1801 | 1802 | vk − v0 1803 | 1804 | 2 2 1805 | 1806 | (3=5) 1807 | 1808 | |v0 1809 | 1810 | 2 2 1811 | 1812 | + 1813 | 1814 | lim inf 1815 | k→∞ 1816 | 1817 | vk − v0 1818 | 1819 | 2 2 1820 | 1821 | ≥ 1822 | 1823 | v0 1824 | 1825 | 2 2 1826 | 1827 | + 1828 | 1829 | |Ω| 1830 | 1831 | lim inf 1832 | k→∞ 1833 | 1834 | vk − v0 L1(Ω) 2 1835 | 1836 | (36) 1837 | 1838 | 18 1839 | 1840 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 1841 | 1842 | Combining (34) and (36) we get 1843 | 1844 | lim inf 1845 | k→∞ 1846 | 1847 | vk 1848 | 1849 | L2 (Ω) 1850 | 1851 | ≥ 1852 | 1853 | v0 1854 | 1855 | 2 L2 (Ω) 1856 | 1857 | + 1858 | 1859 | |Ω|C 2 α2 4 1860 | 1861 | which 1862 | 1863 | gives 1864 | 1865 | (30) 1866 | 1867 | with 1868 | 1869 | β 1870 | 1871 | = 1872 | 1873 | 1 4 1874 | 1875 | |Ω|C 1876 | 1877 | 2α2 1878 | 1879 | . 1880 | 1881 | |Ω| − 1882 | 1883 | v0 1884 | 1885 | 2 L2 (Ω) 1886 | 1887 | 2 1888 | , 1889 | 1890 | 5. A proof of Theorem 4.1 using convex integration 1891 | In this section we provide an alternative, more direct proof for Theorem 4.1, following the method of convex integration as presented for example in [17]. 1892 | In fact the two approaches (i.e. Baire category methods and convex integration) can be unified to a large extent. For a discussion comparing the two approaches we refer to the end of Section 3.3 in [14], see also the paper [24] for a different point of view. Nevertheless, in order to get a feeling for the type of solution that Theorem 4.1 produces, it helps to see the direct construction of the convex integration method. 1893 | We will freely refer to the notation of the previous sections. In particular the proof relies on Lemmas 4.2, 4.3, 4.4 and 4.6. These results enable us to construct an approximating sequence, as explained briefly at the beginning of Section 4, by adding (almost-)plane-waves on top of each other. It is only the limiting step that is more explicit in this approach. The following argument is essentially from Section 3.3 in [17]. 1894 | 1895 | Alternative proof of Theorem 4.1. Using Lemma 4.6, we construct inductively a sequence (vk, uk, qk) ∈ X0 and a sequence of numbers ηk > 0 as follows. Let ρε be a standard mollifying kernel in Rn+1 = Rnx × Rt and set (v1, u1, q1) ≡ 0 in Rnx × Rt. 1896 | 1897 | Having obtained zj := (vj, uj, qj) for j ≤ k and η1, . . . , ηk−1 we choose 1898 | 1899 | ηk < 2−k 1900 | 1901 | (37) 1902 | 1903 | in such a way that 1904 | 1905 | zk − zk ∗ ρηk L2(Ω) < 2−k. 1906 | 1907 | (38) 1908 | 1909 | Then we apply Lemma 4.6 to obtain zk+1 = (vk+1, uk+1, qk+1) ∈ X0 such 1910 | 1911 | that 1912 | 1913 | vk+1 1914 | 1915 | 2 L2 (Ω) 1916 | 1917 | ≥ 1918 | 1919 | vk 1920 | 1921 | 2 L2 (Ω) 1922 | 1923 | + 1924 | 1925 | β 1926 | 1927 | |Ω| − 1928 | 1929 | vk 1930 | 1931 | 2 L2 (Ω) 1932 | 1933 | 2 1934 | , 1935 | 1936 | (39) 1937 | 1938 | and 1939 | 1940 | (zk+1 − zk) ∗ ρηj L2(Ω) < 2−k for all j ≤ k. 1941 | 1942 | (40) 1943 | 1944 | The sequence {zk} is bounded in L∞(Rnx × Rt), therefore by passing to a suitable subsequence we may assume without loss of generality that 1945 | zk ⇀∗ z in L∞(Rnx × Rt) 1946 | 1947 | THE EULER EQUATIONS AS A DIFFERENTIAL INCLUSION 1948 | 1949 | 19 1950 | 1951 | for some z = (v, u, q) ∈ X, and that the sequence {zk} and the corresponding sequence {ηk} satisfies the properties (37),(38),(39) and (40). Then for every k∈N 1952 | 1953 | ∞ 1954 | 1955 | zk ∗ ρηk − z ∗ ρηk L2(Ω) ≤ 1956 | 1957 | zk+j ∗ ρηk − zk+j+1 ∗ ρηk L2(Ω) 1958 | 1959 | j=0 1960 | 1961 | ∞ 1962 | ≤ 2−(k+j) ≤ 2−k+1, 1963 | 1964 | j=0 1965 | 1966 | and since zk − z L2(Ω) ≤ zk − zk ∗ ρηk L2(Ω) + zk ∗ ρηk − z ∗ ρηk L2(Ω) + z ∗ ρηk − z L2(Ω), 1967 | we deduce that vk → v strongly in L2(Ω). Therefore, passing into the limit in (39) we conclude 1968 | 1969 | v 1970 | 1971 | 2 L2 (Ω) 1972 | 1973 | ≥ 1974 | 1975 | v 1976 | 1977 | 2 L2 (Ω) 1978 | 1979 | + 1980 | 1981 | β 1982 | 1983 | |Ω| − 1984 | 1985 | v 1986 | 1987 | 2 L2 (Ω) 1988 | 1989 | 2 1990 | 1991 | (41) 1992 | 1993 | and hence 1994 | 1995 | v 1996 | 1997 | 2 2 1998 | 1999 | = 2000 | 2001 | |Ω|. 2002 | 2003 | Since v vanishes outside Ω and |v| ≤ 1 in Ω, we 2004 | 2005 | conclude that |v| = 1Ω. Since (v, u, q) ∈ X, we also have that (v, u)(x, t) ∈ 2006 | 2007 | Kco for a.e. (x, t) ∈ Ω. From this we deduce that (v, u)(x, t) ∈ K for a.e. 2008 | 2009 | (x, t) ∈ Ω, thus concluding the proof. 2010 | 2011 | References 2012 | [1] Bressan, A., and Flores, F. On total differential inclusions. Rend. Sem. Mat. Univ. Padova 92 (1994), 9–16. 2013 | [2] Cellina, A. On the differential inclusion x′ ∈ [−1, +1]. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 69, 1-2 (1980), 1–6 (1981). 2014 | [3] Chorin, A. J. Vorticity and turbulence, vol. 103 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994. 2015 | [4] Constantin, P., E, W., and Titi, E. S. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Comm. Math. Phys. 165, 1 (1994), 207–209. 2016 | [5] Dacorogna, B., and Marcellini, P. General existence theorems for HamiltonJacobi equations in the scalar and vectorial cases. Acta Math. 178 (1997), 1–37. 2017 | [6] Dafermos, C. M. Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000. 2018 | [7] DiPerna, R. J. Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292, 2 (1985), 383–420. 2019 | [8] DiPerna, R. J., and Majda, A. J. Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40, 3 (1987), 301–345. 2020 | [9] Duchon, J., and Robert, R. Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations Nonlinearity, 13 (2000), 249–255. 2021 | [10] Eyink, G. L. Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Phys. D 78, 3-4 (1994), 222–240. 2022 | [11] Frisch, U. Turbulence. Cambridge University Press, Cambridge, 1995. The legacy of A. N. Kolmogorov. 2023 | [12] Gromov, M. Partial differential relations, vol. 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1986. 2024 | 2025 | 20 2026 | 2027 | CAMILLO DE LELLIS AND LA´ SZLO´ SZE´ KELYHIDI JR. 2028 | 2029 | [13] Kirchheim, B. Deformations with finitely many gradients and stability of quasiconvex hulls. C. R. Acad. Sci. Paris S´er. I Math. 332, 3 (2001), 289–294. 2030 | [14] Kirchheim, B. Rigidity and Geometry of microstructures. Habilitation thesis, University of Leipzig, 2003. 2031 | [15] Kirchheim, B., Mu¨ller, S., and Sˇvera´k, V. Studying nonlinear PDE by geometry in matrix space. In Geometric analysis and Nonlinear partial differential equations, S. Hildebrandt and H. Karcher, Eds. Springer-Verlag, 2003, pp. 347–395. 2032 | [16] Majda, A. J., and Bertozzi, A. L. Vorticity and incompressible flow, vol. 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. 2033 | [17] Mu¨ller, S., and Sˇvera´k, V. Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157, 3 (2003), 715–742. 2034 | [18] Mu¨ller, S., and Sychev, M. Optimal existence theorems for nonhomogeneous differential inclusions. J. Funct. Anal. 181, 2 (2001), 447–475. 2035 | [19] Onsager, L. Statistical hydrodynamics. Nuovo Cimento (9) 6, Supplemento, 2(Convegno Internazionale di Meccanica Statistica) (1949), 279–287. 2036 | [20] Oxtoby, J. C. Measure and category, second ed., vol. 2 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1980. A survey of the analogies between topological and measure spaces. 2037 | [21] Scheffer, V. An inviscid flow with compact support in space-time. J. Geom. Anal. 3, 4 (1993), 343–401. 2038 | [22] Shnirelman, A. On the nonuniqueness of weak solution of the Euler equation. Comm. Pure Appl. Math. 50, 12 (1997), 1261–1286. 2039 | [23] Shnirelman, A. Weak solutions with decreasing energy of incompressible Euler equations. Comm. Math. Phys. 210, 3 (2000), 541–603. 2040 | [24] Sychev, M. A. A few remarks on differential inclusions. Proc. Roy. Soc. Edinburgh Sect. A, 3 (2006), 649-668. 2041 | [25] Tao, T. Nonlinear dispersive equations, vol. 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. 2042 | [26] Tartar, L. Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Res. Notes in Math. Pitman, Boston, Mass., 1979, pp. 136–212. 2043 | [27] Tartar, L. The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear partial differential equations (Oxford, 1982), vol. 111 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Reidel, Dordrecht, 1983, pp. 263–285. 2044 | [28] Temam, R. Navier-Stokes equations, third ed., vol. 2 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis, With an appendix by F. Thomasset. 2045 | Institut fu¨r Mathematik, Universita¨t Zu¨rich, CH-8057 Zu¨rich E-mail address: camillo.delellis@math.unizh.ch 2046 | Departement Mathematik, ETH Zu¨rich, CH-8092 Zu¨rich E-mail address: szekelyh@math.ethz.ch 2047 | 2048 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/PIMHYJGK/.zotero-ft-info: -------------------------------------------------------------------------------- 1 | Title: 2 | Subject: 3 | Keywords: 4 | Author: 5 | Creator: LaTeX with hyperref package 6 | Producer: dvips + GPL Ghostscript SVN PRE-RELEASE 8.62 7 | CreationDate: Sat Feb 2 07:26:26 2008 8 | ModDate: Sat Feb 2 07:26:26 2008 9 | Tagged: no 10 | Form: none 11 | Pages: 20 12 | Encrypted: no 13 | Page size: 612 x 792 pts (letter) (rotated 0 degrees) 14 | File size: 277364 bytes 15 | Optimized: no 16 | PDF version: 1.2 17 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/PIMHYJGK/.zotero-pdf-state: -------------------------------------------------------------------------------- 1 | {"pageIndex":0,"scale":"page-width","top":792,"left":-6,"scrollMode":0,"spreadMode":0} -------------------------------------------------------------------------------- /tests/resources/sql/storage/PIMHYJGK/De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/storage/PIMHYJGK/De Lellis and Székelyhidi - 2009 - The Euler equations as a differential inclusion.pdf -------------------------------------------------------------------------------- /tests/resources/sql/storage/WN7WJBGS/.zotero-ft-info: -------------------------------------------------------------------------------- 1 | Title: () 2 | Subject: () 3 | Keywords: () 4 | Author: () 5 | Creator: LaTeX with hyperref package 6 | Producer: dvips + GPL Ghostscript GIT PRERELEASE 9.08 7 | CreationDate: Wed Nov 20 21:03:39 2013 8 | ModDate: Wed Nov 20 21:03:39 2013 9 | Tagged: no 10 | Form: none 11 | Pages: 28 12 | Encrypted: no 13 | Page size: 595 x 842 pts (A4) (rotated 0 degrees) 14 | File size: 351391 bytes 15 | Optimized: no 16 | PDF version: 1.4 17 | -------------------------------------------------------------------------------- /tests/resources/sql/storage/WN7WJBGS/.zotero-pdf-state: -------------------------------------------------------------------------------- 1 | {"pageIndex":0,"scale":"page-width","top":845,"left":-6,"scrollMode":0,"spreadMode":0} -------------------------------------------------------------------------------- /tests/resources/sql/storage/WN7WJBGS/Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/storage/WN7WJBGS/Svärd and Nordström - 2014 - Review of summation-by-parts schemes for initial–b.pdf -------------------------------------------------------------------------------- /tests/resources/sql/zotero.sqlite: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/papis/papis-zotero/b4f8296202630ba11929ec793da2c3e5dbcfc879/tests/resources/sql/zotero.sqlite -------------------------------------------------------------------------------- /tests/resources/sql_out.yaml: -------------------------------------------------------------------------------- 1 | author: Svärd, Magnus and Nordström, Jan 2 | author_list: 3 | - family: Svärd 4 | given: Magnus 5 | - family: Nordström 6 | given: Jan 7 | doi: 10.1016/j.jcp.2014.02.031 8 | files: 9 | - WN7WJBGS.pdf 10 | issn: 00219991 11 | journal: Journal of Computational Physics 12 | journalAbbreviation: Journal of Computational Physics 13 | language: en 14 | libraryCatalog: DOI.org (Crossref) 15 | month: 7 16 | pages: 17-38 17 | ref: ReviewOfSummaSvard2014 18 | time-added: 2023-02-26-09:05:00 19 | title: Review of summation-by-parts schemes for initial–boundary-value problems 20 | type: article 21 | url: https://linkinghub.elsevier.com/retrieve/pii/S002199911400151X 22 | volume: '268' 23 | year: 2014 24 | -------------------------------------------------------------------------------- /tests/test_bibtex.py: -------------------------------------------------------------------------------- 1 | import os 2 | import yaml 3 | import pytest 4 | 5 | import papis.id 6 | import papis.yaml 7 | import papis.database 8 | import papis.document 9 | import papis_zotero.bibtex 10 | from papis.testing import TemporaryLibrary 11 | 12 | 13 | @pytest.mark.skipif(os.name == "nt", reason="encoding is incorrect on windows") 14 | @pytest.mark.library_setup(populate=False) 15 | def test_simple(tmp_library: TemporaryLibrary) -> None: 16 | bibpath = os.path.join(os.path.dirname(__file__), 17 | "resources", "bibtex", "zotero-library.bib") 18 | papis_zotero.bibtex.add_from_bibtex(bibpath, tmp_library.libname, link=False) 19 | 20 | db = papis.database.get() 21 | db.clear() 22 | db.initialize() 23 | 24 | doc, = db.query_dict({"author": "Magnus"}) 25 | with open(doc.get_info_file(), encoding="utf-8") as fd: 26 | data = yaml.load( 27 | fd, Loader=papis.yaml.Loader) # type: ignore[attr-defined] 28 | del data[papis.id.key_name()] 29 | 30 | info_name = os.path.join(os.path.dirname(__file__), "resources", "bibtex_out.yaml") 31 | with open(info_name, encoding="utf-8") as fd: 32 | expected_data = yaml.load( 33 | fd, Loader=papis.yaml.Loader) # type: ignore[attr-defined] 34 | 35 | assert data == expected_data 36 | -------------------------------------------------------------------------------- /tests/test_sql.py: -------------------------------------------------------------------------------- 1 | import os 2 | import glob 3 | import yaml 4 | import pytest 5 | 6 | import papis.yaml 7 | import papis.document 8 | import papis_zotero.sql 9 | from papis.testing import TemporaryLibrary 10 | 11 | 12 | @pytest.mark.skipif(os.name == "nt", reason="encoding is incorrect on windows") 13 | @pytest.mark.library_setup(populate=False) 14 | def test_simple(tmp_library: TemporaryLibrary) -> None: 15 | sqlpath = os.path.join(os.path.dirname(__file__), "resources", "sql") 16 | papis.config.set("add-folder-name", "{doc[author]}") 17 | papis_zotero.sql.add_from_sql(sqlpath) 18 | 19 | folders = os.listdir(tmp_library.libdir) 20 | assert len(folders) == 5 21 | assert len(glob.glob(tmp_library.libdir + "/**/*.pdf")) == 4 22 | 23 | doc = papis.document.from_folder( 24 | os.path.join( 25 | tmp_library.libdir, 26 | "svard-magnus-and-nordstrom-jan" 27 | ) 28 | ) 29 | 30 | info_name = os.path.join(os.path.dirname(__file__), "resources", "sql_out.yaml") 31 | with open(info_name, encoding="utf-8") as fd: 32 | data = yaml.load(fd, Loader=papis.yaml.Loader) # type: ignore[attr-defined] 33 | expected_doc = papis.document.from_data(data) 34 | 35 | assert expected_doc["author"] == doc["author"] 36 | 37 | # FIXME: currently fails on windows 38 | # assert doc.get_files() 39 | -------------------------------------------------------------------------------- /tools/update-pypi.sh: -------------------------------------------------------------------------------- 1 | #! /usr/bin/env bash 2 | set -ex 3 | 4 | DIST_DIR=dist 5 | 6 | 7 | rm -rf distenv 8 | virtualenv -p python3 distenv 9 | source ./distenv/bin/activate 10 | pip install . 11 | pip install .[develop] 12 | 13 | rm -rf ${DIST_DIR} 14 | python3 setup.py sdist 15 | 16 | pip install twine 17 | read -p "Do you want to push? (y/N)" -n 1 -r 18 | echo 19 | if [[ $REPLY =~ ^[Yy]$ ]]; then 20 | twine upload ${DIST_DIR}/*.tar.gz 21 | fi 22 | REPLY= # unset REPLY after using it 23 | --------------------------------------------------------------------------------