Well-posedness and Regularity for a Polyconvex Energy

TL;DR

This paper establishes the existence, uniqueness, and regularity of minimizers for a polyconvex energy functional, introducing new criteria for uniqueness and applying the results to Navier-Stokes solutions.

Contribution

It proves well-posedness and regularity for a class of polyconvex functionals and develops a scheme to construct short-time solutions to Navier-Stokes equations.

Findings

Existence and uniqueness of minimizers under new criteria.

Regularity results for minimizers in 2D and 3D.

Construction of short-time Navier-Stokes solutions.

Abstract

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the $H^1$ projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct $L^r$ solutions of the Navier-Stokes equation for a short time interval.