On nonuniqueness and nonregularity for gradient flows of polyconvex functionals

TL;DR

This paper demonstrates that gradient flows of certain polyconvex functionals can have multiple weak solutions with varying regularity, challenging assumptions of uniqueness and smoothness in such problems.

Contribution

It introduces counterexamples showing nonuniqueness and nonregularity for gradient flows of polyconvex functionals using convex integration techniques.

Findings

Existence of multiple weak solutions with different regularity levels.

Counterexamples to uniqueness of solutions.

Application of convex integration to construct solutions.

Abstract

We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a trivial classical solution as well as infinitely many weak solutions that are nowhere smooth. Such polyconvex functions have been constructed in the previous work, and the nonuniqueness and nonregularity will be achieved by reformulating the gradient flow as a space-time partial differential relation and then using the convex integration method to construct certain strongly convergent sequences of subsolutions that have a uniform control on the local essential oscillations of their spatial gradients.