Local Well-Posedness of Free-Boundary Incompressible Elastodynamics with Surface Tension via Vanishing Viscosity Limit View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2022-07-11

AUTHORS

Xumin Gu, Zhen Lei

ABSTRACT

In this paper, we consider the free boundary problem of incompressible elastodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with surface tension, we establish the well-posedness theory for a short time interval. Our method uses the vanishing viscosity limit by establishing a uniform a priori estimate with respect to the viscosity. As a byproduct, the inviscid limit of the incompressible viscoelasticity (the system coupled with the Navier-Stokes equations) is also justified. We point out that the framework of establishing the well-posedness for water wave equations simply does not apply here, even for obtaining a priori estimates. Based on a crucial new observation about the inherent structure of the elastic term on the free boundary, we successfully develop a new estimate for the pressure and then manage to apply an induction method to control normal derivatives. This strategy also allows us to establish the vanishing viscosity limit in standard Sobolev spaces, rather than only in the conormal ones (cf. Masmoudi and Rousset in Arch Ration Mech Anal 223(1):301–417, 2017; Wang and Xin in Vanishing viscosity and surface tension limits of incompressible viscous surface waves, 2015). More... »

PAGES

1285-1338

References to SciGraph publications

  • 2016-09-07. Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2009-02-06. Almost global wellposedness of the 2-D full water wave problem in INVENTIONES MATHEMATICAE
  • 2013-03-27. Global Solvability of a Free Boundary Three-Dimensional Incompressible Viscoelastic Fluid System with Surface Tension in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2020-05-22. Stability of Multidimensional Thermoelastic Contact Discontinuities in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2005-06-22. The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2007-11-03. Global Solutions for Incompressible Viscoelastic Fluids in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2014-05-27. Global solutions for the gravity water waves system in 2d in INVENTIONES MATHEMATICAE
  • 2010-10-20. Global wellposedness of the 3-D full water wave problem in INVENTIONES MATHEMATICAE
  • 1997-09. Well-posedness in Sobolev spaces of the full water wave problem in 2-D in INVENTIONES MATHEMATICAE
  • 2008-12-05. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows in BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, NEW SERIES
  • 2014-03-18. Zero Surface Tension Limit of Viscous Surface Waves in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2019-01-18. Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients in MATHEMATISCHE ANNALEN
  • 2006-12-19. Remarks about the Inviscid Limit of the Navier–Stokes System in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2010-09-16. Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid in JOURNAL OF MATHEMATICAL FLUID MECHANICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00205-022-01806-z

    DOI

    http://dx.doi.org/10.1007/s00205-022-01806-z

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1149412206


    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/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "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/2008/0102", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Applied Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/grid.443531.4", 
              "name": [
                "School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Gu", 
            "givenName": "Xumin", 
            "id": "sg:person.016555034375.87", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016555034375.87"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "School of Mathematical Sciences, Fudan University, 200433, Shanghai, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/grid.8547.e", 
              "name": [
                "School of Mathematical Sciences, Fudan University, 200433, Shanghai, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Lei", 
            "givenName": "Zhen", 
            "id": "sg:person.014464147007.76", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014464147007.76"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s00205-007-0089-x", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1038988213", 
              "https://doi.org/10.1007/s00205-007-0089-x"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00205-020-01531-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1127837917", 
              "https://doi.org/10.1007/s00205-020-01531-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00205-013-0615-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1023889787", 
              "https://doi.org/10.1007/s00205-013-0615-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s002220050177", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1043801768", 
              "https://doi.org/10.1007/s002220050177"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-006-0171-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1012520035", 
              "https://doi.org/10.1007/s00220-006-0171-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00220-014-1986-0", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1042396076", 
              "https://doi.org/10.1007/s00220-014-1986-0"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00222-014-0521-4", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1002639983", 
              "https://doi.org/10.1007/s00222-014-0521-4"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00222-009-0176-8", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1034432727", 
              "https://doi.org/10.1007/s00222-009-0176-8"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00222-010-0288-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1009499580", 
              "https://doi.org/10.1007/s00222-010-0288-1"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00205-005-0385-2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019954649", 
              "https://doi.org/10.1007/s00205-005-0385-2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00205-016-1036-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1025662291", 
              "https://doi.org/10.1007/s00205-016-1036-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00021-010-0029-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039037680", 
              "https://doi.org/10.1007/s00021-010-0029-7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00208-018-01798-w", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1111506605", 
              "https://doi.org/10.1007/s00208-018-01798-w"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00574-008-0001-9", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019647844", 
              "https://doi.org/10.1007/s00574-008-0001-9"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2022-07-11", 
        "datePublishedReg": "2022-07-11", 
        "description": "In this paper, we consider the free boundary problem of incompressible elastodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with surface tension, we establish the well-posedness theory for a short time interval. Our method uses the vanishing viscosity limit by establishing a uniform a priori estimate with respect to the viscosity. As a byproduct, the inviscid limit of the incompressible viscoelasticity (the system coupled with the Navier-Stokes equations) is also justified. We point out that the framework of establishing the well-posedness for water wave equations simply does not apply here, even for obtaining a priori estimates. Based on a crucial new observation about the inherent structure of the elastic term on the free boundary, we successfully develop a new estimate for the pressure and then manage to apply an induction method to control normal derivatives. This strategy also allows us to establish the vanishing viscosity limit in standard Sobolev spaces, rather than only in the conormal ones (cf. Masmoudi and Rousset in Arch Ration Mech Anal 223(1):301\u2013417, 2017; Wang and Xin in Vanishing viscosity and surface tension limits of incompressible viscous surface waves, 2015).", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00205-022-01806-z", 
        "isAccessibleForFree": false, 
        "isFundedItemOf": [
          {
            "id": "sg:grant.8131420", 
            "type": "MonetaryGrant"
          }
        ], 
        "isPartOf": [
          {
            "id": "sg:journal.1047617", 
            "issn": [
              "0003-9527", 
              "1432-0673"
            ], 
            "name": "Archive for Rational Mechanics and Analysis", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "245"
          }
        ], 
        "keywords": [
          "surface tension", 
          "incompressible elastodynamics", 
          "free boundary", 
          "fluid motion", 
          "water wave equation", 
          "deformation tensor", 
          "transport equation", 
          "coupling system", 
          "elastic term", 
          "viscosity limit", 
          "elastodynamics", 
          "boundary problem", 
          "balance laws", 
          "Euler equations", 
          "free boundary problem", 
          "inviscid limit", 
          "wave equation", 
          "equations", 
          "viscoelasticity", 
          "boundaries", 
          "normal derivative", 
          "viscosity", 
          "tension", 
          "short time interval", 
          "limit", 
          "motion", 
          "inherent structure", 
          "method", 
          "uniform", 
          "pressure", 
          "byproducts", 
          "time interval", 
          "structure", 
          "system", 
          "induction method", 
          "tensor", 
          "estimates", 
          "law", 
          "problem", 
          "respect", 
          "terms", 
          "well posedness", 
          "one", 
          "observations", 
          "theory", 
          "new observations", 
          "space", 
          "posedness", 
          "strategies", 
          "framework", 
          "posedness theory", 
          "new estimates", 
          "interval", 
          "derivatives", 
          "Local Well-Posedness", 
          "Sobolev spaces", 
          "standard Sobolev space", 
          "paper"
        ], 
        "name": "Local Well-Posedness of Free-Boundary Incompressible Elastodynamics with Surface Tension via Vanishing Viscosity Limit", 
        "pagination": "1285-1338", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1149412206"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00205-022-01806-z"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00205-022-01806-z", 
          "https://app.dimensions.ai/details/publication/pub.1149412206"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-12-01T06:45", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20221201/entities/gbq_results/article/article_953.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00205-022-01806-z"
      }
    ]
     

    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/s00205-022-01806-z'

    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/s00205-022-01806-z'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00205-022-01806-z'

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

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00205-022-01806-z'


     

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

    187 TRIPLES      21 PREDICATES      97 URIs      74 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00205-022-01806-z schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 anzsrc-for:0102
    4 schema:author Na22a33f5db424017bafac9f5b61d207d
    5 schema:citation sg:pub.10.1007/s00021-010-0029-7
    6 sg:pub.10.1007/s00205-005-0385-2
    7 sg:pub.10.1007/s00205-007-0089-x
    8 sg:pub.10.1007/s00205-013-0615-y
    9 sg:pub.10.1007/s00205-016-1036-5
    10 sg:pub.10.1007/s00205-020-01531-5
    11 sg:pub.10.1007/s00208-018-01798-w
    12 sg:pub.10.1007/s00220-006-0171-5
    13 sg:pub.10.1007/s00220-014-1986-0
    14 sg:pub.10.1007/s00222-009-0176-8
    15 sg:pub.10.1007/s00222-010-0288-1
    16 sg:pub.10.1007/s00222-014-0521-4
    17 sg:pub.10.1007/s002220050177
    18 sg:pub.10.1007/s00574-008-0001-9
    19 schema:datePublished 2022-07-11
    20 schema:datePublishedReg 2022-07-11
    21 schema:description In this paper, we consider the free boundary problem of incompressible elastodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with surface tension, we establish the well-posedness theory for a short time interval. Our method uses the vanishing viscosity limit by establishing a uniform a priori estimate with respect to the viscosity. As a byproduct, the inviscid limit of the incompressible viscoelasticity (the system coupled with the Navier-Stokes equations) is also justified. We point out that the framework of establishing the well-posedness for water wave equations simply does not apply here, even for obtaining a priori estimates. Based on a crucial new observation about the inherent structure of the elastic term on the free boundary, we successfully develop a new estimate for the pressure and then manage to apply an induction method to control normal derivatives. This strategy also allows us to establish the vanishing viscosity limit in standard Sobolev spaces, rather than only in the conormal ones (cf. Masmoudi and Rousset in Arch Ration Mech Anal 223(1):301–417, 2017; Wang and Xin in Vanishing viscosity and surface tension limits of incompressible viscous surface waves, 2015).
    22 schema:genre article
    23 schema:isAccessibleForFree false
    24 schema:isPartOf Ne90072dfadea4ca486f8faae473a9d68
    25 Nfc00825078114b20b8d53c532da576d0
    26 sg:journal.1047617
    27 schema:keywords Euler equations
    28 Local Well-Posedness
    29 Sobolev spaces
    30 balance laws
    31 boundaries
    32 boundary problem
    33 byproducts
    34 coupling system
    35 deformation tensor
    36 derivatives
    37 elastic term
    38 elastodynamics
    39 equations
    40 estimates
    41 fluid motion
    42 framework
    43 free boundary
    44 free boundary problem
    45 incompressible elastodynamics
    46 induction method
    47 inherent structure
    48 interval
    49 inviscid limit
    50 law
    51 limit
    52 method
    53 motion
    54 new estimates
    55 new observations
    56 normal derivative
    57 observations
    58 one
    59 paper
    60 posedness
    61 posedness theory
    62 pressure
    63 problem
    64 respect
    65 short time interval
    66 space
    67 standard Sobolev space
    68 strategies
    69 structure
    70 surface tension
    71 system
    72 tension
    73 tensor
    74 terms
    75 theory
    76 time interval
    77 transport equation
    78 uniform
    79 viscoelasticity
    80 viscosity
    81 viscosity limit
    82 water wave equation
    83 wave equation
    84 well posedness
    85 schema:name Local Well-Posedness of Free-Boundary Incompressible Elastodynamics with Surface Tension via Vanishing Viscosity Limit
    86 schema:pagination 1285-1338
    87 schema:productId N1057c285cca94688bf064442521082a6
    88 Nb1e0f5e1659a41e2a22dbb63f4ce83cb
    89 schema:sameAs https://app.dimensions.ai/details/publication/pub.1149412206
    90 https://doi.org/10.1007/s00205-022-01806-z
    91 schema:sdDatePublished 2022-12-01T06:45
    92 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    93 schema:sdPublisher N9954624d941e439fba47118db0693647
    94 schema:url https://doi.org/10.1007/s00205-022-01806-z
    95 sgo:license sg:explorer/license/
    96 sgo:sdDataset articles
    97 rdf:type schema:ScholarlyArticle
    98 N00aee212b7794aa0ba8485d947d2ad96 rdf:first sg:person.014464147007.76
    99 rdf:rest rdf:nil
    100 N1057c285cca94688bf064442521082a6 schema:name dimensions_id
    101 schema:value pub.1149412206
    102 rdf:type schema:PropertyValue
    103 N9954624d941e439fba47118db0693647 schema:name Springer Nature - SN SciGraph project
    104 rdf:type schema:Organization
    105 Na22a33f5db424017bafac9f5b61d207d rdf:first sg:person.016555034375.87
    106 rdf:rest N00aee212b7794aa0ba8485d947d2ad96
    107 Nb1e0f5e1659a41e2a22dbb63f4ce83cb schema:name doi
    108 schema:value 10.1007/s00205-022-01806-z
    109 rdf:type schema:PropertyValue
    110 Ne90072dfadea4ca486f8faae473a9d68 schema:volumeNumber 245
    111 rdf:type schema:PublicationVolume
    112 Nfc00825078114b20b8d53c532da576d0 schema:issueNumber 3
    113 rdf:type schema:PublicationIssue
    114 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    115 schema:name Mathematical Sciences
    116 rdf:type schema:DefinedTerm
    117 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    118 schema:name Pure Mathematics
    119 rdf:type schema:DefinedTerm
    120 anzsrc-for:0102 schema:inDefinedTermSet anzsrc-for:
    121 schema:name Applied Mathematics
    122 rdf:type schema:DefinedTerm
    123 sg:grant.8131420 http://pending.schema.org/fundedItem sg:pub.10.1007/s00205-022-01806-z
    124 rdf:type schema:MonetaryGrant
    125 sg:journal.1047617 schema:issn 0003-9527
    126 1432-0673
    127 schema:name Archive for Rational Mechanics and Analysis
    128 schema:publisher Springer Nature
    129 rdf:type schema:Periodical
    130 sg:person.014464147007.76 schema:affiliation grid-institutes:grid.8547.e
    131 schema:familyName Lei
    132 schema:givenName Zhen
    133 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014464147007.76
    134 rdf:type schema:Person
    135 sg:person.016555034375.87 schema:affiliation grid-institutes:grid.443531.4
    136 schema:familyName Gu
    137 schema:givenName Xumin
    138 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016555034375.87
    139 rdf:type schema:Person
    140 sg:pub.10.1007/s00021-010-0029-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039037680
    141 https://doi.org/10.1007/s00021-010-0029-7
    142 rdf:type schema:CreativeWork
    143 sg:pub.10.1007/s00205-005-0385-2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019954649
    144 https://doi.org/10.1007/s00205-005-0385-2
    145 rdf:type schema:CreativeWork
    146 sg:pub.10.1007/s00205-007-0089-x schema:sameAs https://app.dimensions.ai/details/publication/pub.1038988213
    147 https://doi.org/10.1007/s00205-007-0089-x
    148 rdf:type schema:CreativeWork
    149 sg:pub.10.1007/s00205-013-0615-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1023889787
    150 https://doi.org/10.1007/s00205-013-0615-y
    151 rdf:type schema:CreativeWork
    152 sg:pub.10.1007/s00205-016-1036-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1025662291
    153 https://doi.org/10.1007/s00205-016-1036-5
    154 rdf:type schema:CreativeWork
    155 sg:pub.10.1007/s00205-020-01531-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1127837917
    156 https://doi.org/10.1007/s00205-020-01531-5
    157 rdf:type schema:CreativeWork
    158 sg:pub.10.1007/s00208-018-01798-w schema:sameAs https://app.dimensions.ai/details/publication/pub.1111506605
    159 https://doi.org/10.1007/s00208-018-01798-w
    160 rdf:type schema:CreativeWork
    161 sg:pub.10.1007/s00220-006-0171-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1012520035
    162 https://doi.org/10.1007/s00220-006-0171-5
    163 rdf:type schema:CreativeWork
    164 sg:pub.10.1007/s00220-014-1986-0 schema:sameAs https://app.dimensions.ai/details/publication/pub.1042396076
    165 https://doi.org/10.1007/s00220-014-1986-0
    166 rdf:type schema:CreativeWork
    167 sg:pub.10.1007/s00222-009-0176-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1034432727
    168 https://doi.org/10.1007/s00222-009-0176-8
    169 rdf:type schema:CreativeWork
    170 sg:pub.10.1007/s00222-010-0288-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1009499580
    171 https://doi.org/10.1007/s00222-010-0288-1
    172 rdf:type schema:CreativeWork
    173 sg:pub.10.1007/s00222-014-0521-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1002639983
    174 https://doi.org/10.1007/s00222-014-0521-4
    175 rdf:type schema:CreativeWork
    176 sg:pub.10.1007/s002220050177 schema:sameAs https://app.dimensions.ai/details/publication/pub.1043801768
    177 https://doi.org/10.1007/s002220050177
    178 rdf:type schema:CreativeWork
    179 sg:pub.10.1007/s00574-008-0001-9 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019647844
    180 https://doi.org/10.1007/s00574-008-0001-9
    181 rdf:type schema:CreativeWork
    182 grid-institutes:grid.443531.4 schema:alternateName School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, People’s Republic of China
    183 schema:name School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, People’s Republic of China
    184 rdf:type schema:Organization
    185 grid-institutes:grid.8547.e schema:alternateName School of Mathematical Sciences, Fudan University, 200433, Shanghai, People’s Republic of China
    186 schema:name School of Mathematical Sciences, Fudan University, 200433, Shanghai, People’s Republic of China
    187 rdf:type schema:Organization
     




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


    ...