Convex Integration for Diffusion Equations

TL;DR

This paper applies convex integration techniques to analyze nonmonotone diffusion equations, demonstrating nonuniqueness and instability of solutions under a new structural condition.

Contribution

It introduces Condition (OC) for diffusion flux functions and establishes nonuniqueness and instability results for solutions of certain diffusion equations.

Findings

Nonuniqueness of Lipschitz solutions under Condition (OC)

Instability of solutions in the initial-boundary value problem

Connection between rank-one convex hulls and solution behavior

Abstract

We study the initial-boundary value problem for a class of diffusion equations with nonmonotone diffusion flux functions, including forward-backward parabolic equations and the gradient flows of nonconvex energy functionals, under the framework of partial differential inclusions using the method of convex integration and Baire's category. In connection with rank-one convex hulls of the corresponding matrix sets, we introduce a structural condition on the diffusion flux function, called Condition (OC), and establish the nonuniqueness and instability of Lipschitz solutions to the initial-boundary value problem under this condition.