Well-posedness of stochastic continuity equations on Riemannian manifolds

Abstract: We analyze continuity equations with Stratonovich stochasticity, $\partial \rho+ div_h \left[ \rho \circ\left(u(t,x)+\sum_{i=1}^N a_i(x) \dot W_i(t) \right) \right]=0$, defined on a smooth closed Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms $\dot W_1(t),\ldots,\dot W_N(t)$ driven by smooth spatially dependent vector fields $a_1(x),\ldots,a_N(x)$ 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 $div_h u \in L^\infty_{t,x}$). 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 Euclidian case ($a_i$ constant) \cite{Beck:2019,Flandoli-Gubinelli-Priola,Neves:2015aa}. The proof is based on an a priori estimate in $L^2$, which is obtained by a duality method, and a weak compactness argument.