Well-posedness of stochastic continuity equations on Riemannian manifolds

Abstract

We analyze continuity equations with Stratonovich stochasticity, $\partial \rho +di{v}_h\left[\rho \circ \left(u(t,x)+{\sum }_{i=1}^N{a}_i(x){\dot{W}}_i(t)\right)\right]=0$, defined on a smooth closed Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms ${\dot{W}}_1(t),\dots ,{\dot{W}}_N(t)$ driven by smooth spatially dependent vector fields $a_1(x),\dots ,a_N(x)$ on $M$. The velocity $u$ belongs to $L_t^1{W}_x^{1,2}$ with $di{v}_{h}u$ bounded in $L_{t,x}^p$ for $p>d+2$, where $d$ is the dimension of $M$ (we do not assume $di{v}_{h}u\in L_{t,x}^{\infty }$). 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.