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Well-posedness of stochastic continuity equations on Riemannian manifolds

Luca Galimberti, Kenneth H. Karlsen

Abstract

We analyze continuity equations with Stratonovich stochasticity, $\partial ρ+ div_h \left[ ρ\circ\left(u(t,x)+\sum_{i=1}^N a_i(x) \dot W_i(t) \right) \right]=0$, 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.

Well-posedness of stochastic continuity equations on Riemannian manifolds

Abstract

We analyze continuity equations with Stratonovich stochasticity, , defined on a smooth closed Riemannian manifold with metric . The velocity field is perturbed by Gaussian noise terms driven by smooth spatially dependent vector fields on . The velocity belongs to with bounded in for , where is the dimension of (we do not assume ). We show that by carefully choosing the noise vector fields (and the number of them), the initial-value problem is well-posed in the class of weak 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 and the noise, a link that is somewhat hidden in the Euclidian case ( constant) \cite{Beck:2019,Flandoli-Gubinelli-Priola,Neves:2015aa}. The proof is based on an a priori estimate in , which is obtained by a duality method, and a weak compactness argument.

Paper Structure

This paper contains 22 sections, 16 theorems, 179 equations.

Table of Contents

  1. Introduction and main results
  2. Background material
  3. Geometric framework
  4. Stochastic framework
  5. Smooth data and strong solutions
  6. Strong solution
  7. Elementary $L^p$ bound
  8. Time-dependent test functions
  9. Irregular test functions
  10. On the ellipticity of $\sum_i a_i(a_i)$, proof of Lemma \ref{['lem:ellipticity']}
  11. Test function for duality method
  12. $L^2$ estimate and uniqueness for weak solutions
  13. Proof of main result, Theorem \ref{['thm:main-result']}
  14. Smoothing of velocity vector field $u$
  15. Weak compactness of approximate solutions
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume eq:u-ass-1 and consider a weak $L^2$ solution $\rho$ of eq:target with initial datum $\rho_0\in L^2(M)$, according to Definition def:L2-weak-sol-Ito. Then $\rho$ is renormalizable in the sense of Definition def:renorm.

Theorems & Definitions (32)

  • Definition 1.1: weak $L^2$ solution, Stratonovich formulation
  • Definition 1.2: weak $L^2$ solution, Itô formulation
  • Definition 1.3: renormalization, Itô formulation
  • Theorem 1.1: renormalization property GK
  • Lemma 1.2: non-degenerate second order operator
  • Theorem 1.3: well-posedness
  • Proposition 2.1
  • Remark 3.1
  • Lemma 3.1: strong solution, smooth data
  • Lemma 3.2: $L^p$ estimates, smooth data
  • ...and 22 more