论文标题
粘弹性Cahn-肿瘤生长模型
Viscoelastic Cahn--Hilliard models for tumour growth
论文作者
论文摘要
我们引入了一种新的相位场模型，以考虑粘弹性效应。该模型源自基本的热力学原理，由带有肿瘤细胞源项的对流的cahn--hilliard方程组成，以及具有养分边界供应的对流反应扩散方程。考虑到对于肿瘤的侵入性行为必不可少的趋化项。该模型是由构成Navier-Stokes方程的粘弹性系统完成的，以及与左Cauchy的一般组成型方程，其应力松弛 - 与粘弹性材料总机械响应的弹性部分相关的绿色张量。 为了具体选择弹性能量密度，并有了额外的耗散期限来考虑应力扩散，我们证明存在粘弹性模型的全球弱化解决方案在两个空间尺寸$ d = 2 $中的粘弹性模型$ d = 2 $，在完全消耗的有限元方案中限制了cfl条件，即cfl条件，即$Δt\ eude qude。 此外，在\ {2,3 \} $中的任意维度中，我们显示了完全差异有限元方案的稳定性和解决方案的稳定性和解决方案，在该方案中，离散的cauchy-green Tensor的积极确定性是用J. W. Barrett，S。W. Barrett，S。Barrett（S. barret of Boyaval（of）的旧模型（“存在）和近似模型”（“存在”（“存在）”和“现有”（“存在）”（“存在）”（“存在”（“存在）， M3AS。21.9（2011），第1783--1837页。之后，我们改善规律性会导致任意维度$ d \ in \ {2,3 \} $，在二维中$ d = 2 $，其中需要CFL条件。然后，对于$ d = 2 $，我们在离散参数中传递到极限，并显示离散解决方案的子序列会收敛到全球弱解决方案。 最后，我们提出数值结果。
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.
