Well-posedness of nonlinear transport equation by stochastic perturbation

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.