Well-posedness of the non-local conservation law by stochastic perturbation

Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an $L^{1}\cap L^{2}$ setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), It\^o-Wentzell-Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper [20] where the linear case was addressed.Comment: several corrections were made with respect to the first version. arXiv admin note: text overlap with arXiv:1702.0597

Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition

We consider the stochastic divergence-free continuity equations with Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of uniqueness realies on stochastic characteristic method and the generalized Ito-Ventzel-Kunita formula. Moreover, we prove a stability property for the solution with respect to the initial datum.Comment: To appears in Nonlinear Differential Equations and Applications NoDE