We analyze continuity equations with Stratonovich stochasticity on a smooth closed and compact Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms W_1(t), . . . ,W_N (t) driven by smooth spatially dependent vector fields a_1(x), . . . , a_N (x) on M . The velocity u belongs to L^1_t W^{1,2}_x with div_h u bounded in L^p_{t,x} for p > d + 2, where d is the dimension of M (we do not assume div_h u ∈ L^∞_{t,x}).
We show that by carefully choosing the noise vector fields a_i (and the number N of them), the initial-value problem is well-posed in the class of weak L^2 solutions, although the problem can be ill-posed in the deterministic case because of concentration effects.
The proof of this “regularization by noise” result reveals a link between the nonlinear structure of the underlying domain M and the noise, a link that is somewhat hidden in the Euclidean case (a_i constant). To our knowledge, this is the first instance of “regularization by noise” phenomena beyond R^d. The proof is based on an a priori estimate in L^2, which is obtained by a duality method, and a weak compactness argument.
This is a joint work with Kenneth Karlsen (UiO).