Linear and nonlinear transport equations with coordinate-wise increasing velocity fields

TL;DR

This paper establishes well-posedness and uniqueness results for linear and nonlinear transport equations with coordinate-wise increasing velocity fields, with applications to mean field games and selection of solutions via noise.

Contribution

It introduces new well-posedness results for transport equations with irregular, coordinate-increasing velocity fields and applies these to mean field game models, including solution selection via noise.

Findings

Well-posedness of linear transport equations in Lebesgue spaces.

Existence and uniqueness of regular Lagrangian flows.

Noise can select among multiple solutions in the vanishing noise limit.

Abstract

We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which a unique minimal and maximal solution exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit.