Lipschitz Solutions for the Gradient Flow of Polyconvex Functionals

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.