Link to this article: http://journals.cambridge.org/abstract_S0956792514000436How to cite this article: SERGIO FRIGERI, MAURIZIO GRASSELLI and ELISABETTA ROCCA (2015). On a diffuse interface model of tumour growth. We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol. 67 1457-1485). This model consists of the Cahn-Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction-diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L 1 . The whole system has a highly nonlinear character.We address the existence of a weak notion of solution, referred to as "entropic", where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics, as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data.We prove our results by passing to the limit in a time-discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its "entropic" formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.for all t ∈ (0, T ] and almost all s ∈ (0, t), with ξ a selection in the (convex analysis) subdifferential ∂ β( χ ) = ∂I [0,+∞) ( χ ) of I [0,+∞) . In [47, Prop. 2.14] (see also [24]), we prove that, under additional regularity properties any weak solution in fact fulfills (1.3) pointwise.Let us also mention that other approaches to the weak solvability of coupled PDE systems with an L 1 -righthand side are available in the literature: in particular, we refer here to [54] and [49]. In [54], the notion of renormalized solution has been used in order to prove a global-in-time existence result for a nonlinear system in thermoviscoelasticity. In [49] the focus is on rate-independent processes coupled with viscosity and inertia in the displacement equation, and with the temperature equation. There the internal variable equation is not of gradient-flow type as (1.3), but instead features a 1-positively homogeneous dissipation potential. For the resulting PDE system, a weak solution concept partially mutuated from the theory of rate-independent processes by A. Mielke (cf., e.g., [39]) is analyzed. An existence result is proved combining techniques for rate-independent evolution, with Boccardo-Gallouët type estimates of the temperature gradient in the heat equation with L 1 -right-hand side. Our existence results. The main results of this paper, Theorems 1 and 2, state the existence of entropic solutions for system (1.1-1.3), supplemented with the boundary conditions (1.4) (cf. Remark 2.12), in the irreversible (µ = 1) and reversible (µ = 0) cases.More precisely, in the case of unidirectional evolution for χ we can prove the existence of a global-intime entropic solution (i.e. satisfying the entropy (1.6) and the total energy (1.7) inequalities, the (pointwise) momentum balance (1.2), the one-sided variational inequality (1.9) and the energy (1.10) inequalities for...

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.