Boundedness of weak solutions to cross-diffusion population systems with Laplacian structure

Abstract

The global-in-time existence of bounded weak solutions to general cross-diffusion systems describing the evolution of $n$ population species is proved. The equations are considered in a bounded domain with no-flux boundary conditions. The system possesses a Laplacian structure, which allows for the derivation of uniform $L^\infty$ bounds, and an entropy structure, which yields suitable gradient estimates. Because of the boundedness, no growth conditions for the transition and interaction rates need to be assumed. The existence proof is based on a fixed-point argument first used by Desvillettes et al.\ and the Stampacchia truncation method for the approximate system. As a by-product, the boundedness of weak solutions to population models of Shigesada-Kawasaki-Teramoto type are deduced for the first time under natural conditions on the coefficients.