Lagrangian flows and the one-dimensional Peano phenomenon for ODEs

Abstract

We consider the one-dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative, and satisfies 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 (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.