Viscoelastic Cahn--Hilliard models for tumour growth
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913397444968448 |
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| author | Garcke, Harald Kovács, Balázs Trautwein, Dennis |
| author_facet | Garcke, Harald Kovács, Balázs Trautwein, Dennis |
| contents | We introduce a new phase field model for tumour 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 tumour cells and a convected reaction-diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behaviour of tumours, are taken into account. The model is completed by a viscoelastic system constisting 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 tumour growth in two space dimensions $d=2$ by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. $Δt\leq Ch^2$, is required.
Moreover, in arbitrary dimensions $d\in\{2,3\}$, 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 J. W. Barrett, S. Boyaval ("Existence and approximation of a (regularized) Oldroyd-B model". In: M3AS. 21.9 (2011), pp. 1783--1837). After that, we improve the regularity results in arbitrary dimensions $d\in\{2,3\}$ and in two dimensions $d=2$, where a CFL condition is required. Then, for $d=2$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_04147 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Viscoelastic Cahn--Hilliard models for tumour growth Garcke, Harald Kovács, Balázs Trautwein, Dennis Numerical Analysis Analysis of PDEs 35K35, 76A10, 76M10, 35Q92 We introduce a new phase field model for tumour 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 tumour cells and a convected reaction-diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behaviour of tumours, are taken into account. The model is completed by a viscoelastic system constisting 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 tumour growth in two space dimensions $d=2$ by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. $Δt\leq Ch^2$, is required. Moreover, in arbitrary dimensions $d\in\{2,3\}$, 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 J. W. Barrett, S. Boyaval ("Existence and approximation of a (regularized) Oldroyd-B model". In: M3AS. 21.9 (2011), pp. 1783--1837). After that, we improve the regularity results in arbitrary dimensions $d\in\{2,3\}$ and in two dimensions $d=2$, where a CFL condition is required. Then, for $d=2$, 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. |
| title | Viscoelastic Cahn--Hilliard models for tumour growth |
| topic | Numerical Analysis Analysis of PDEs 35K35, 76A10, 76M10, 35Q92 |
| url | https://arxiv.org/abs/2204.04147 |