Renormalization of stochastic continuity equations on Riemannian manifolds

Abstract

We consider the initial-value problem for stochastic continuity equations of the form ∂tρ+divhρu(t,x)+∑i=1Nai(x)∘dWidt=0,defined on a smooth closed Riemannian manifold M with metric h, where the Sobolev regular velocity field u is perturbed by Gaussian noise terms Ẇi(t) driven by smooth spatially dependent vector fields ai(x) on M. Our main result is that weak (L2) solutions are renormalized solutions, that is, if ρ is a weak solution, then the nonlinear composition S(ρ) is a weak solution as well, for any “reasonable” function S:R→R. The proof consists of a systematic procedure for regularizing tensor fields on a manifold, a convenient choice of atlas to simplify technical computations linked to the Christoffel symbols, and several DiPerna–Lions type commutators Cɛ(ρ,D) between (first/second order) geometric differential operators D and the regularization device (ɛ is the scaling parameter). This work, which is related to the “Euclidean” result in Punshon-Smith (0000), reveals some structural effects that noise and nonlinear domains have on the dynamics of weak solutions.