The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

TL;DR

This paper revisits the well-posedness of certain quasi-linear parabolic systems with non-uniformly elliptic diffusion matrices, extending results to Sobolev and Besov spaces and applying them to the SKT system.

Contribution

It extends classical well-posedness results to endpoint Besov spaces and demonstrates the existence of solutions for the SKT system.

Findings

Revised well-posedness results in Sobolev spaces.

Extension to endpoint Besov space $B_{p,1}^{d/p}$.

Existence of local, non-negative solutions for the SKT system.

Abstract

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on $L^2$ and then explore the endpoint Besov case $B_{p,1}^{d/p}$. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.