Calculus of Variations and Geometric Measure Theory
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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


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