Transport equations and flows with one-sided Lipschitz velocity fields

TL;DR

This paper investigates transport equations and flows with velocity fields satisfying a one-sided Lipschitz condition, providing a comprehensive theory for solutions, flows, and stochastic equations in both compressive and expansive regimes.

Contribution

It offers a complete characterization of solutions and flows for first- and second-order transport equations with one-sided Lipschitz velocity fields, including existence, uniqueness, and stability results.

Findings

Characterization of stable solutions in the compressive regime.

Complete theory for conservative and nonconservative equations in Lebesgue spaces.

Existence and uniqueness of regular Lagrangian ODE flows and analogous stochastic results.

Abstract

We study first- and second-order linear transport equations, as well as ODE and SDE flows, with velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In the compressive regime, we characterize the stable continuous distributional solutions of both the first and second-order nonconservative transport equations as the unique viscosity solution. Our results in the expansive regime complement the theory of Bouchut, James, and Mancini, and we provide a complete theory for both the conservative and nonconservative equations in Lebesgue spaces, as well as proving the existence, uniqueness, and stability of the regular Lagrangian ODE flow. We also provide analogous results in this context for second order equations and SDEs with degenerate noise coefficients that are constant in the spatial variable.