Well-posedness of nonlinear transport equation by stochastic perturbation

TL;DR

This paper proves the existence and uniqueness of solutions for multidimensional nonlinear stochastic transport equations with irregular fluxes, showing that stochastic perturbations can regularize and make the equations well-posed.

Contribution

It introduces stochastic BGK approximations and commutator estimates to establish well-posedness for irregular fluxes and demonstrates regularization effects of multiplicative Brownian noise.

Findings

Existence and uniqueness of stochastic entropy solutions for irregular fluxes.

Regularity results for solutions with BV initial data.

Stochastic perturbation renders the transport equation well-posed, unlike the deterministic case.

Abstract

We are concerned with multidimensional nonlinear stochastic transport equation driven by Brownian motions. For irregular fluxes, by using stochastic BGK approximations and commutator estimates, we gain the existence and uniqueness of stochastic entropy solutions. Besides, for $BV$ initial data, the $BV$ and H\"older regularities are also derived for the unique stochastic entropy solution. Particularly, for the transport equation, we gain a regularization result, i.e. while the existence fails for the transport equation, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be another explicit example (the first example is given in [22]) of a PDE of fluid dynamics that becomes well-posed under the influence of a multiplicative Brownian type noise.