Renormalization of stochastic continuity equations on Riemannian manifolds

We consider the initial-value problem for stochastic continuity equations of the form partial derivative t rho + div(h) [rho(u(t, x) + Sigma(N)(i=1) a(i)(x) o dw(i)/dt)] = 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 (W) over dot(i)(t) driven by smooth spatially dependent vector fields a(i)(x) on M. Our main result is that weak (L-2) solutions are renormalized solutions, that is, if rho is a weak solution, then the nonlinear composition S(rho) 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 C-epsilon(rho, D) between (first/second order) geometric differential operators D and the regularization device (epsilon 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. (C) 2021 The Author(s). Published by Elsevier B.V.