Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species

Abstract

The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager's principle in thermodynamics. Under detailed balance (and without reaction) the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold. Mathematics Subject Classification

A cross-diffusion system derived from a Fokker–Planck equation with partial averaging

“…In a similar way, we find that (and taking into account that M (3) is positive definite), we deduce that for z ∈ R 2 ,…”

Entropic structure and duality for multiple species cross-diffusion systems

“…These studies generally do not focus on the Cauchy problem but more on the existence of specific solutions (stationary or periodic solutions for instance). Examples of global existence of weak solutions for multiple species can be found in [DLMT15] (Section 5.2) and more importantly in [CDJ16]. One of the purposes of the paper is to show that weak solutions can be obtained for systems like (1) under mainly structural hypotheses using the methods derived in [DLMT15].…”

“…We try to get rid as much as possible of hypotheses that arise from technical issues. Concerning the entropy structure of more than 2 species system, a very recent work [CDJ16] identifies a structure for system with pressure of the form…”

“…The focus in [CDJ16] is made on the entropy and all the approximation is based on a suitable construction and extension of entropic variables. The authors exhibited the following detailed balance condition which for a matrix M = (m ij ) writes…”

“…Thus, the detailed balance condition can be seen as a way to avoid an assumption of the type self > cross, but applies only for sublinear exponents. The (dimension dependent) lower-bound on the exponent is related to the fact that the regularity and the integrability of the solution is derived from the entropy inequality (see end of Section 1 in [CDJ16]) and does not make advantage of the L 2 structure that one exhibits when exploiting Lemma 1.2. In this manuscript we identify in Section 4 natural extensions (including different exponents s i ) of the structure identified in [CDJ16].…”