---
res:
  bibo_abstract:
  - '<jats:p> We introduce a new phase field model for tumor growth where viscoelastic
    effects are taken into account. The model is derived from basic thermodynamical
    principles and consists of a convected Cahn–Hilliard equation with source terms
    for the tumor cells and a convected reaction–diffusion equation with boundary
    supply for the nutrient. Chemotactic terms, which are essential for the invasive
    behavior of tumors, are taken into account. The model is completed by a viscoelastic
    system consisting of the Navier–Stokes equation for the hydrodynamic quantities,
    and a general constitutive equation with stress relaxation for the left Cauchy–Green
    tensor associated with the elastic part of the total mechanical response of the
    viscoelastic material. For a specific choice of the elastic energy density and
    with an additional dissipative term accounting for stress diffusion, we prove
    existence of global-in-time weak solutions of the viscoelastic model for tumor
    growth in two space dimensions [Formula: see text] by the passage to the limit
    in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula:
    see text], is required. </jats:p><jats:p> Moreover, in arbitrary dimensions [Formula:
    see text], we show stability and existence of solutions for the fully-discrete
    finite element scheme, where positive definiteness of the discrete Cauchy–Green
    tensor is proved with a regularization technique that was first introduced by
    Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B
    model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve
    the regularity results in arbitrary dimensions [Formula: see text] and in two
    dimensions [Formula: see text], where a CFL condition is required. Then, in two
    dimensions [Formula: see text], we pass to the limit in the discretization parameters
    and show that subsequences of discrete solutions converge to a global-in-time
    weak solution. Finally, we present numerical results in two dimensions [Formula:
    see text]. </jats:p>@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Harald
      foaf_name: Garcke, Harald
      foaf_surname: Garcke
  - foaf_Person:
      foaf_givenName: Balázs
      foaf_name: Kovács, Balázs
      foaf_surname: Kovács
      foaf_workInfoHomepage: http://www.librecat.org/personId=100441
    orcid: 0000-0001-9872-3474
  - foaf_Person:
      foaf_givenName: Dennis
      foaf_name: Trautwein, Dennis
      foaf_surname: Trautwein
  bibo_doi: 10.1142/s0218202522500634
  bibo_issue: '13'
  bibo_volume: 32
  dct_date: 2022^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/0218-2025
  - http://id.crossref.org/issn/1793-6314
  dct_language: eng
  dct_publisher: World Scientific Pub Co Pte Ltd@
  dct_subject:
  - Applied Mathematics
  - Modeling and Simulation
  dct_title: Viscoelastic Cahn–Hilliard models for tumor growth@
...