Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion

TL;DR

This paper introduces a new normal form for degenerate cross-diffusion equations with incomplete diffusion matrices, enabling the analysis of their short-term solutions in multiple dimensions.

Contribution

It develops a hyperbolic-parabolic normal form for a class of degenerate cross-diffusion systems, facilitating the analysis of their well-posedness in higher dimensions.

Findings

Established a normal form revealing the structure of the equations.

Proved short-time existence of solutions in Sobolev spaces.

Extended analysis beyond the purely convective case.

Abstract

We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic--parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in $H^s(\mathbb{T}^d)$ for $s>d/2+1$.