A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2019-03-20

AUTHORS

Clément Cancès, Daniel Matthes, Flore Nabet

ABSTRACT

We study a non-local version of the Cahn–Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. Differently to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes—but not necessarily the fluxes themselves—annihilate each other. Our main result is a rigorous proof of the existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn–Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields. More... »

PAGES

1-30

References to SciGraph publications

  • 2017. Finite Volume Approximation of a Degenerate Immiscible Two-Phase Flow Model of Cahn–Hilliard Type in FINITE VOLUMES FOR COMPLEX APPLICATIONS VIII - METHODS AND THEORETICAL ASPECTS
  • 2009-02. A new class of transport distances between measures in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2017-02. A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation in FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
  • 2000-01. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem in NUMERISCHE MATHEMATIK
  • 2013. A User’s Guide to Optimal Transport in MODELLING AND OPTIMISATION OF FLOWS ON NETWORKS
  • 2015. Optimal Transport for Applied Mathematicians, Calculus of Variations, PDEs, and Modeling in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00205-019-01369-6

    DOI

    http://dx.doi.org/10.1007/s00205-019-01369-6

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1112900989


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "French National Centre for Scientific Research", 
              "id": "https://www.grid.ac/institutes/grid.4444.0", 
              "name": [
                "Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlev\u00e9, 59000, Lille, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Canc\u00e8s", 
            "givenName": "Cl\u00e9ment", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Technical University Munich", 
              "id": "https://www.grid.ac/institutes/grid.6936.a", 
              "name": [
                "Zentrum f\u00fcr Mathematik, Technische Universit\u00e4t M\u00fcnchen, 85747, Garching, Germany"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Matthes", 
            "givenName": "Daniel", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Centre de Math\u00e9matiques Appliqu\u00e9es", 
              "id": "https://www.grid.ac/institutes/grid.462265.1", 
              "name": [
                "CMAP, Centre de Math\u00e9matiques Appliqu\u00e9es, \u00c9cole Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Nabet", 
            "givenName": "Flore", 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s10208-015-9284-6", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003649973", 
              "https://doi.org/10.1007/s10208-015-9284-6"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1080/03605300903296256", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012885737"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s002110050002", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016132384", 
              "https://doi.org/10.1007/s002110050002"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00526-008-0182-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019877631", 
              "https://doi.org/10.1007/s00526-008-0182-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00526-008-0182-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019877631", 
              "https://doi.org/10.1007/s00526-008-0182-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1051/proc/201654001", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1020924927"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/num.1690110205", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021198317"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/num.1690110205", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021198317"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s1570-8659(00)07005-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024807654"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-20828-2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028296622", 
              "https://doi.org/10.1007/978-3-319-20828-2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-20828-2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028296622", 
              "https://doi.org/10.1007/978-3-319-20828-2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.jde.2012.04.004", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1032105082"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/j.bulsci.2011.12.004", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1033734076"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0167-2789(95)00173-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1034265114"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-32160-3_1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1036211003", 
              "https://doi.org/10.1007/978-3-642-32160-3_1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1017/s0956792500002369", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1053917375"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1051/m2an:2003062", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057033060"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.439809", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1058017838"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.474153", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1058056876"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1088/0951-7715/24/4/016", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1059110088"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1090/mcom/2997", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1059342798"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/070683337", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062850947"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/s0036141094267662", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062876273"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/s0036141096303359", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062876406"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/s0036142997331669", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062877529"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1142/s0218202510004799", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1062963227"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1137/16m1056560", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1084226067"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-319-57397-7_36", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1085578268", 
              "https://doi.org/10.1007/978-3-319-57397-7_36"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2140/apde.2017.10.1845", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1091642446"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1090/gsm/014", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1098740269"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2019-03-20", 
        "datePublishedReg": "2019-03-20", 
        "description": "We study a non-local version of the Cahn\u2013Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. Differently to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes\u2014but not necessarily the fluxes themselves\u2014annihilate each other. Our main result is a rigorous proof of the existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn\u2013Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/s00205-019-01369-6", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1047617", 
            "issn": [
              "0003-9527", 
              "1432-0673"
            ], 
            "name": "Archive for Rational Mechanics and Analysis", 
            "type": "Periodical"
          }
        ], 
        "name": "A Two-Phase Two-Fluxes Degenerate Cahn\u2013Hilliard Model as Constrained Wasserstein Gradient Flow", 
        "pagination": "1-30", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "c585b60d35f80bba5ab612c7be16982033bf0022a87f69e8e67cf7cebc4f64c3"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00205-019-01369-6"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1112900989"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00205-019-01369-6", 
          "https://app.dimensions.ai/details/publication/pub.1112900989"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-11T12:42", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000363_0000000363/records_70056_00000003.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://link.springer.com/10.1007%2Fs00205-019-01369-6"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00205-019-01369-6'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00205-019-01369-6'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00205-019-01369-6'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00205-019-01369-6'


     

    This table displays all metadata directly associated to this object as RDF triples.

    159 TRIPLES      21 PREDICATES      51 URIs      16 LITERALS      5 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00205-019-01369-6 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N3ca45bfb22b1444fbc5d71bffc2c6e30
    4 schema:citation sg:pub.10.1007/978-3-319-20828-2
    5 sg:pub.10.1007/978-3-319-57397-7_36
    6 sg:pub.10.1007/978-3-642-32160-3_1
    7 sg:pub.10.1007/s002110050002
    8 sg:pub.10.1007/s00526-008-0182-5
    9 sg:pub.10.1007/s10208-015-9284-6
    10 https://doi.org/10.1002/num.1690110205
    11 https://doi.org/10.1016/0167-2789(95)00173-5
    12 https://doi.org/10.1016/j.bulsci.2011.12.004
    13 https://doi.org/10.1016/j.jde.2012.04.004
    14 https://doi.org/10.1016/s1570-8659(00)07005-8
    15 https://doi.org/10.1017/s0956792500002369
    16 https://doi.org/10.1051/m2an:2003062
    17 https://doi.org/10.1051/proc/201654001
    18 https://doi.org/10.1063/1.439809
    19 https://doi.org/10.1063/1.474153
    20 https://doi.org/10.1080/03605300903296256
    21 https://doi.org/10.1088/0951-7715/24/4/016
    22 https://doi.org/10.1090/gsm/014
    23 https://doi.org/10.1090/mcom/2997
    24 https://doi.org/10.1137/070683337
    25 https://doi.org/10.1137/16m1056560
    26 https://doi.org/10.1137/s0036141094267662
    27 https://doi.org/10.1137/s0036141096303359
    28 https://doi.org/10.1137/s0036142997331669
    29 https://doi.org/10.1142/s0218202510004799
    30 https://doi.org/10.2140/apde.2017.10.1845
    31 schema:datePublished 2019-03-20
    32 schema:datePublishedReg 2019-03-20
    33 schema:description We study a non-local version of the Cahn–Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. Differently to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes—but not necessarily the fluxes themselves—annihilate each other. Our main result is a rigorous proof of the existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn–Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.
    34 schema:genre research_article
    35 schema:inLanguage en
    36 schema:isAccessibleForFree false
    37 schema:isPartOf sg:journal.1047617
    38 schema:name A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow
    39 schema:pagination 1-30
    40 schema:productId N3c64df102d83444ebeb39f337fc9b6e0
    41 N753e05205e2348b2aea8f4a477bafa76
    42 N82345a0793ed42719d1e663267e7ff52
    43 schema:sameAs https://app.dimensions.ai/details/publication/pub.1112900989
    44 https://doi.org/10.1007/s00205-019-01369-6
    45 schema:sdDatePublished 2019-04-11T12:42
    46 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    47 schema:sdPublisher N68f4074240734ee9b6514fcce59d84f3
    48 schema:url https://link.springer.com/10.1007%2Fs00205-019-01369-6
    49 sgo:license sg:explorer/license/
    50 sgo:sdDataset articles
    51 rdf:type schema:ScholarlyArticle
    52 N3c64df102d83444ebeb39f337fc9b6e0 schema:name doi
    53 schema:value 10.1007/s00205-019-01369-6
    54 rdf:type schema:PropertyValue
    55 N3ca45bfb22b1444fbc5d71bffc2c6e30 rdf:first Na5f9c5d53cf54ca0b9dbdc9f91a1a640
    56 rdf:rest N81593ef8229a4a0f94b75bc4b7f5c573
    57 N68f4074240734ee9b6514fcce59d84f3 schema:name Springer Nature - SN SciGraph project
    58 rdf:type schema:Organization
    59 N738e863f6f6a4969a807744e5698ecac schema:affiliation https://www.grid.ac/institutes/grid.6936.a
    60 schema:familyName Matthes
    61 schema:givenName Daniel
    62 rdf:type schema:Person
    63 N753e05205e2348b2aea8f4a477bafa76 schema:name dimensions_id
    64 schema:value pub.1112900989
    65 rdf:type schema:PropertyValue
    66 N81593ef8229a4a0f94b75bc4b7f5c573 rdf:first N738e863f6f6a4969a807744e5698ecac
    67 rdf:rest N87ca669c9a094409a6f7b88808a3f50e
    68 N82345a0793ed42719d1e663267e7ff52 schema:name readcube_id
    69 schema:value c585b60d35f80bba5ab612c7be16982033bf0022a87f69e8e67cf7cebc4f64c3
    70 rdf:type schema:PropertyValue
    71 N87ca669c9a094409a6f7b88808a3f50e rdf:first Ne544c442e4f140768c65700f9ccb762f
    72 rdf:rest rdf:nil
    73 Na5f9c5d53cf54ca0b9dbdc9f91a1a640 schema:affiliation https://www.grid.ac/institutes/grid.4444.0
    74 schema:familyName Cancès
    75 schema:givenName Clément
    76 rdf:type schema:Person
    77 Ne544c442e4f140768c65700f9ccb762f schema:affiliation https://www.grid.ac/institutes/grid.462265.1
    78 schema:familyName Nabet
    79 schema:givenName Flore
    80 rdf:type schema:Person
    81 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    82 schema:name Mathematical Sciences
    83 rdf:type schema:DefinedTerm
    84 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    85 schema:name Pure Mathematics
    86 rdf:type schema:DefinedTerm
    87 sg:journal.1047617 schema:issn 0003-9527
    88 1432-0673
    89 schema:name Archive for Rational Mechanics and Analysis
    90 rdf:type schema:Periodical
    91 sg:pub.10.1007/978-3-319-20828-2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028296622
    92 https://doi.org/10.1007/978-3-319-20828-2
    93 rdf:type schema:CreativeWork
    94 sg:pub.10.1007/978-3-319-57397-7_36 schema:sameAs https://app.dimensions.ai/details/publication/pub.1085578268
    95 https://doi.org/10.1007/978-3-319-57397-7_36
    96 rdf:type schema:CreativeWork
    97 sg:pub.10.1007/978-3-642-32160-3_1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1036211003
    98 https://doi.org/10.1007/978-3-642-32160-3_1
    99 rdf:type schema:CreativeWork
    100 sg:pub.10.1007/s002110050002 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016132384
    101 https://doi.org/10.1007/s002110050002
    102 rdf:type schema:CreativeWork
    103 sg:pub.10.1007/s00526-008-0182-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019877631
    104 https://doi.org/10.1007/s00526-008-0182-5
    105 rdf:type schema:CreativeWork
    106 sg:pub.10.1007/s10208-015-9284-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003649973
    107 https://doi.org/10.1007/s10208-015-9284-6
    108 rdf:type schema:CreativeWork
    109 https://doi.org/10.1002/num.1690110205 schema:sameAs https://app.dimensions.ai/details/publication/pub.1021198317
    110 rdf:type schema:CreativeWork
    111 https://doi.org/10.1016/0167-2789(95)00173-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1034265114
    112 rdf:type schema:CreativeWork
    113 https://doi.org/10.1016/j.bulsci.2011.12.004 schema:sameAs https://app.dimensions.ai/details/publication/pub.1033734076
    114 rdf:type schema:CreativeWork
    115 https://doi.org/10.1016/j.jde.2012.04.004 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032105082
    116 rdf:type schema:CreativeWork
    117 https://doi.org/10.1016/s1570-8659(00)07005-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024807654
    118 rdf:type schema:CreativeWork
    119 https://doi.org/10.1017/s0956792500002369 schema:sameAs https://app.dimensions.ai/details/publication/pub.1053917375
    120 rdf:type schema:CreativeWork
    121 https://doi.org/10.1051/m2an:2003062 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057033060
    122 rdf:type schema:CreativeWork
    123 https://doi.org/10.1051/proc/201654001 schema:sameAs https://app.dimensions.ai/details/publication/pub.1020924927
    124 rdf:type schema:CreativeWork
    125 https://doi.org/10.1063/1.439809 schema:sameAs https://app.dimensions.ai/details/publication/pub.1058017838
    126 rdf:type schema:CreativeWork
    127 https://doi.org/10.1063/1.474153 schema:sameAs https://app.dimensions.ai/details/publication/pub.1058056876
    128 rdf:type schema:CreativeWork
    129 https://doi.org/10.1080/03605300903296256 schema:sameAs https://app.dimensions.ai/details/publication/pub.1012885737
    130 rdf:type schema:CreativeWork
    131 https://doi.org/10.1088/0951-7715/24/4/016 schema:sameAs https://app.dimensions.ai/details/publication/pub.1059110088
    132 rdf:type schema:CreativeWork
    133 https://doi.org/10.1090/gsm/014 schema:sameAs https://app.dimensions.ai/details/publication/pub.1098740269
    134 rdf:type schema:CreativeWork
    135 https://doi.org/10.1090/mcom/2997 schema:sameAs https://app.dimensions.ai/details/publication/pub.1059342798
    136 rdf:type schema:CreativeWork
    137 https://doi.org/10.1137/070683337 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062850947
    138 rdf:type schema:CreativeWork
    139 https://doi.org/10.1137/16m1056560 schema:sameAs https://app.dimensions.ai/details/publication/pub.1084226067
    140 rdf:type schema:CreativeWork
    141 https://doi.org/10.1137/s0036141094267662 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062876273
    142 rdf:type schema:CreativeWork
    143 https://doi.org/10.1137/s0036141096303359 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062876406
    144 rdf:type schema:CreativeWork
    145 https://doi.org/10.1137/s0036142997331669 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062877529
    146 rdf:type schema:CreativeWork
    147 https://doi.org/10.1142/s0218202510004799 schema:sameAs https://app.dimensions.ai/details/publication/pub.1062963227
    148 rdf:type schema:CreativeWork
    149 https://doi.org/10.2140/apde.2017.10.1845 schema:sameAs https://app.dimensions.ai/details/publication/pub.1091642446
    150 rdf:type schema:CreativeWork
    151 https://www.grid.ac/institutes/grid.4444.0 schema:alternateName French National Centre for Scientific Research
    152 schema:name Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000, Lille, France
    153 rdf:type schema:Organization
    154 https://www.grid.ac/institutes/grid.462265.1 schema:alternateName Centre de Mathématiques Appliquées
    155 schema:name CMAP, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France
    156 rdf:type schema:Organization
    157 https://www.grid.ac/institutes/grid.6936.a schema:alternateName Technical University Munich
    158 schema:name Zentrum für Mathematik, Technische Universität München, 85747, Garching, Germany
    159 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...