Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

Abstract

The authors analyze continuity equations with Stratonovich stochasticity, $\partial \rho + {\rm{di}}{{\rm{v}}_h}\left[ {\rho \circ \left( {u(t,x) + \sum\limits_{i = 1}^N {{a_i}(x){{\dot W}_i}(t)} } \right)} \right] = 0$ defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ₁(t), …, Ẇ _N(t) driven by smooth spatially dependent vector fields a ₁(x), …, a _N(x) on M. The velocity u belongs to L _t ¹ W _x ^1,2 with div _h u bounded in L _t,x ^p for p > d + 2, where d is the dimension of M (they do not assume div _h u ∈ L _t,x ^∞). For carefully chosen noise vector fields a _i (and the number N of them), they show that 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 is based on a L ² estimate, which is obtained by a duality method, and a weak compactness argument.

Keywords

Stochastic continuity equation / Riemannian manifold / Hyperbolic equation / Non-smooth velocity field / Weak solution / Existence / Uniqueness