We recall the state of the art and the role of polyconvexity in the calculus of variations. Then we keep our focus on a particular polyconvex function, applicable to the study of Euler incompressible fluids.
We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional 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.
We introduce a minimizing movement scheme to construct Lr-solutions of the Navier-Stokes equation (NSE) for a short time interval. Our scheme is an improved version of the split scheme introduced in Ebin–Marsden in 1970, and allows us to solve the equation with less regular initial data as opposed to more regular initial data requirement in the 1970 Ebin–Marsden’s work.
(Most of the material of these lectures is based on a joint work with M. Jacobs and I. Kim).
Wilfrid Gangbo is a professor of mathematics at the University of California, Los Angeles. His areas of research interest are nonlinear analysis, PDEs and the calculus of variations. His special interests are elasticity theory, optimal transport theory and most recently Mean Field Games.