Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows

TL;DR

This paper demonstrates that certain irregular diffusion equations, reformulated via convex integration and geometric structures, admit infinitely many weak solutions with no regularity, challenging the uniqueness and smoothness expectations.

Contribution

It introduces $ abla$-configurations and Condition $O_N$ to construct irregular solutions for polyconvex gradient flows, revealing non-uniqueness and loss of regularity.

Findings

Existence of infinitely many Lipschitz weak solutions with no $C^1$ regularity.

Construction of irregular solutions for polyconvex energy gradient flows.

Verification of structural conditions ensuring solution irregularity.

Abstract

We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction scheme based on new geometric structures, referred to as $\mathcal{T}_N$-configurations, together with a simplified structural hypothesis on the diffusion functions, termed Condition $O_N$. Under this condition, we show that the associated initial and boundary value problems with certain smooth initial-boundary data admit infinitely many Lipschitz weak solutions that are nowhere $C^1$. We further analyze specific $\mathcal{T}_N$-configurations and establish nondegeneracy conditions that are essential for verifying Condition $O_N$. As an application, we construct examples of strongly polyconvex energy functionals whose gradient flows generate irregular diffusion equations, thereby revealing a failure of regularity and uniqueness even within the class of polyconvex gradient flows.