Stochastic continuity equations with conservative noise

TL;DR

This paper proves the existence and uniqueness of solutions to a stochastic continuity equation with non-linear, degenerate noise, extending previous linear noise results using duality and non-linear PDE techniques.

Contribution

It introduces a novel approach to well-posedness for non-linear stochastic continuity equations with weaker assumptions on the velocity field.

Findings

Established existence and uniqueness of entropy solutions.

Extended linear noise results to non-linear stochastic perturbations.

Utilized duality and non-linear PDE regularity theory.

Abstract

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence and uniqueness of entropy solutions under hypotheses on the velocity field which are weaker than those required in the deterministic setting. This extends related results of [Flandoli, Gubinelli, Priola; Invent. Math., 2010] applicable for linear multiplicative noise to a non-linear setting. The existence proof relies on a duality argument which makes use of the regularity theory for fully non-linear parabolic equations.