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On the asymptotic behavior of the gradient flow of a polyconvex functional

In this paper, we study the asymptotic behavior of the solutions of the system of nonlinear partial differential equations studied in [ESG05] for the evolution of a family of diffeomorphisms. We prove existence and regularity of the asymptotic state of solutions and we find an explicit rate of convergence of the time dependent solution to the corresponding final state. We study also a system not considered in [ESG05], linked to a linear Fokker-Planck equation. For this system we show existence of solutions, of the asymptotic state, the regularity and the rate of convergence of the solution to a final state. In both cases, the final states are obtained from the composition of the limit in time of the flow map with the initial data. This structure of the limiting stationary states allows a way of constructing maps with given jacobians as in [ASMV03].