Inserted: 6 aug 2010
Last Updated: 24 mar 2011
Journal: Journal of Differential Equations
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