Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - S. Lisini

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

created by lisini on 03 Sep 2009
modified on 04 Oct 2010


Published Paper

Inserted: 3 sep 2009
Last Updated: 4 oct 2010

Journal: Contemporary Mathematics series AMS
Volume: 526
Pages: 37-51
Year: 2010

(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