Calculus of Variations and Geometric Measure Theory

L. Ambrosio - S. Lisini - G. Savaré

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

created by ambrosio on 09 Nov 2005
modified by lisini on 21 Feb 2008


Published Paper

Inserted: 9 nov 2005
Last Updated: 21 feb 2008

Journal: manuscripta math.
Volume: 121
Pages: 1-50
Year: 2006


In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient.

We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type.

Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum.

Keywords: Gradient flows, Optimal transport maps, Stability of flows, Nonlinear heat flows