---
_id: '45970'
abstract:
- lang: eng
  text: '<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>'
author:
- first_name: Harald
  full_name: Garcke, Harald
  last_name: Garcke
- first_name: Balázs
  full_name: Kovács, Balázs
  id: '100441'
  last_name: Kovács
  orcid: 0000-0001-9872-3474
- first_name: Dennis
  full_name: Trautwein, Dennis
  last_name: Trautwein
citation:
  ama: Garcke H, Kovács B, Trautwein D. Viscoelastic Cahn–Hilliard models for tumor
    growth. <i>Mathematical Models and Methods in Applied Sciences</i>. 2022;32(13):2673-2758.
    doi:<a href="https://doi.org/10.1142/s0218202522500634">10.1142/s0218202522500634</a>
  apa: Garcke, H., Kovács, B., &#38; Trautwein, D. (2022). Viscoelastic Cahn–Hilliard
    models for tumor growth. <i>Mathematical Models and Methods in Applied Sciences</i>,
    <i>32</i>(13), 2673–2758. <a href="https://doi.org/10.1142/s0218202522500634">https://doi.org/10.1142/s0218202522500634</a>
  bibtex: '@article{Garcke_Kovács_Trautwein_2022, title={Viscoelastic Cahn–Hilliard
    models for tumor growth}, volume={32}, DOI={<a href="https://doi.org/10.1142/s0218202522500634">10.1142/s0218202522500634</a>},
    number={13}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World
    Scientific Pub Co Pte Ltd}, author={Garcke, Harald and Kovács, Balázs and Trautwein,
    Dennis}, year={2022}, pages={2673–2758} }'
  chicago: 'Garcke, Harald, Balázs Kovács, and Dennis Trautwein. “Viscoelastic Cahn–Hilliard
    Models for Tumor Growth.” <i>Mathematical Models and Methods in Applied Sciences</i>
    32, no. 13 (2022): 2673–2758. <a href="https://doi.org/10.1142/s0218202522500634">https://doi.org/10.1142/s0218202522500634</a>.'
  ieee: 'H. Garcke, B. Kovács, and D. Trautwein, “Viscoelastic Cahn–Hilliard models
    for tumor growth,” <i>Mathematical Models and Methods in Applied Sciences</i>,
    vol. 32, no. 13, pp. 2673–2758, 2022, doi: <a href="https://doi.org/10.1142/s0218202522500634">10.1142/s0218202522500634</a>.'
  mla: Garcke, Harald, et al. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.”
    <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 13, World
    Scientific Pub Co Pte Ltd, 2022, pp. 2673–758, doi:<a href="https://doi.org/10.1142/s0218202522500634">10.1142/s0218202522500634</a>.
  short: H. Garcke, B. Kovács, D. Trautwein, Mathematical Models and Methods in Applied
    Sciences 32 (2022) 2673–2758.
date_created: 2023-07-10T11:47:27Z
date_updated: 2024-04-03T09:15:35Z
department:
- _id: '841'
doi: 10.1142/s0218202522500634
intvolume: '        32'
issue: '13'
keyword:
- Applied Mathematics
- Modeling and Simulation
language:
- iso: eng
page: 2673-2758
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  issn:
  - 0218-2025
  - 1793-6314
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: Viscoelastic Cahn–Hilliard models for tumor growth
type: journal_article
user_id: '100441'
volume: 32
year: '2022'
...