Viscoelastic Cahn–Hilliard models for tumour growth
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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. Δ t≤ Ch^2, is required. Moreover, in arbitrary dimensions d∈{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∈{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.
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