Abstract
The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.
Highlights
The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved
We present a general technique which allows us, under certain assumptions, to prove simultaneously the global existence of a weak solution as well as its boundedness from below and/or above
Le and Nguyen [44] proved that bounded weak solutions are Holder continuous if certain structural assumptions on the diffusion matrix are imposed
Summary
Chemistry, and biology can be modeled by reactiondiffusion systems with cross diffusion, which describe the temporal evolution of the densities or mass fractions of a multicomponent system. We refer to H as an entropy and to the integral of ∇w : B∇w as the corresponding entropy dissipation Under certain conditions, it gives gradient estimates for u needed to prove the global-in-time existence of solutions to (2) and (3). Le and Nguyen [44] proved that bounded weak solutions are Holder continuous if certain structural assumptions on the diffusion matrix are imposed. We identify the key elements of this idea and provide a general global existence result for bounded weak solutions to certain systems. We show that our technique can be adapted to situations in which variants of Hypotheses H2’ and H2” hold To illustrate this idea, we choose a class of cross-diffusion systems modeling the time evolution of two population species. We may consider reaction terms f (u) satisfying a particular structure; see Remark
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