Well-posedness and Regularity for a Polyconvex Energy

2020/11/08 by Gangbo, Wilfrid, Jacobs, Matt, Kim, Inwon
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2011.03876

Abstract

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the H¹ 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.

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