Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

The authors analyze continuity equations with Stratonovich stochasticity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \rho + {\rm{di}}{{\rm{v}}_h}\left[ {\rho \circ \left( {u(t,x) + \sum\limits_{i = 1}^N {{a_i}(x){{\dot W}_i}(t)} } \right)} \right] = 0$$\end{document} defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ1(t), …, ẆN(t) driven by smooth spatially dependent vector fields a1(x), …, aN(x) on M. The velocity u belongs to Lt1Wx1,2 with divhu bounded in Lt,xp for p > d + 2, where d is the dimension of M (they do not assume divhu ∈ Lt,x∞). For carefully chosen noise vector fields ai (and the number N of them), they show that the initial-value problem is well-posed in the class of weak L2 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 L2 estimate, which is obtained by a duality method, and a weak compactness argument.