Analysis and numerical approximation of degenerate parabolic systems arising in thermodynamics and biology

Abstract

Since degenerate parabolic systems are quite peculiar, there is no general theory available to obtain the existence of solutions. Thus, we take a closer look at three different models which are motivated applications.The first model comes from thermodynamics and describes the evolution of multicomponent fluids. We extend the literature by proposing a model which includes non isothermaltemperature as well as Soret/Du four effects, and prove the global existence by using the entropy structure of the system.The second model is derived from biology and describes the development of biofilms. We develop a finite–volume scheme for which we prove the existence of discrete solutions and, under additional assumptions, the uniqueness. The main difficulty comes from the degenerate–singular diffusion term and the proof of lower/upper bounds for the biomass fraction since we cannot apply a comparison principle as in the continuous case. We overcome this challenge by introducing an entropy variable which guarantees these bounds. Furthermore, we prove that discrete solutions converge towards a weak solution under mesh refinement.The last model we discuss is obtained from biology as well and models the growth of biofilms by considering the biomass/solvent as fluid mixture. This system consists of a degenerate reaction–diffusion equation and a local fourth order Cahn–Hilliard equation with degenerate mobility, singular potential and nonlinear source terms. We prove global existence by applying a suitable truncation and a galerkin approximation. Since we do not find optimal estimates due to the degeneracy of the mobility, we perform a Browder–Minty trick for the identification of the source term in the deregularization limit.