Inserted: 3 sep 2009
Last Updated: 4 oct 2010
Journal: Contemporary Mathematics series AMS
(editors H. Holden and K. H. Karlsen)
In this paper, we study the asymptotic behavior of the solutions of the system of non-linear partial differential equations studied in a paper of Evans-Gangbo-Savin 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 the paper of Evans-Gangbo-Savin, 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.
Keywords: Gradient Flow, polyconvex functional, asymptotic behavior