# Lipschitz Solutions for the Gradient Flow of Polyconvex Functionals

## TL;DR

This paper constructs specific smooth polyconvex functions to demonstrate that the gradient flow of such functionals can be highly ill-posed, with solutions converging weakly to non-solutions.

## Contribution

It provides explicit examples of smooth polyconvex functions satisfying certain conditions that lead to ill-posed gradient flow problems.

## Key findings

- Gradient flow problems can be ill-posed for certain polyconvex functionals.
- Existence of Lipschitz weak solutions that do not converge to actual solutions.
- Ill-posedness persists even with smooth initial-boundary data.

## Abstract

In this sequel to a previous paper, we construct certain smooth strongly polyconvex functions $F$ on $\mathbb M^{2\times 2}$ such that $\sigma=DF$ satisfies the Condition (OC) in that paper. As a result, we show that the initial-boundary value problem for the gradient flow of such polyconvex energy functionals is highly ill-posed even for some smooth initial-boundary data in the sense that the problem possesses a weakly* convergent sequence of Lipschitz weak solutions whose limit is not a weak solution.