Title: 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 Hölder 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.