Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

TL;DR

This paper proves that bounded weak solutions to certain degenerate parabolic cross-diffusion systems are unique and coincide with strong solutions when they exist, using a relative entropy estimate.

Contribution

It establishes a weak-strong uniqueness principle for a class of degenerate parabolic cross-diffusion systems based on a novel relative entropy estimate.

Findings

Weak solutions coincide with strong solutions when both exist.

The proof uses a relative entropy estimate specific to the system.

Uniqueness holds within the maximal existence interval.

Abstract

Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.