# 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.