Convex Integration for Diffusion Equations

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.