Calculus of Variations and Geometric Measure Theory

G. Crippa

Lagrangian flows and the one dimensional Peano phenomenon for ODEs

created by crippa on 06 Aug 2010
modified on 24 Mar 2011

[BibTeX]

Published Paper

Inserted: 6 aug 2010
Last Updated: 24 mar 2011

Journal: Journal of Differential Equations
Volume: 250
Number: 7
Pages: 3135-3149
Year: 2011

Abstract:

We consider the one dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative, and satisfying a condition on the amount of zeros. Although it is classically known that this problem lacks uniqueness of classical trajectories, we show that there is uniqueness for the so-called regular Lagrangian flow (the by now usual notion of flow in nonsmooth situations), as well as uniqueness of distributional solutions for the associated continuity equation. The proof relies on a space reparametrization argument around the zeros of the vector field.

Keywords: continuity equation, One-dimensional ODEs, Peano phenomenon, Regular Lagrangian flows, Lipschitz functions


Download: