---
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 [@MR2593276] applicable to 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.'
author:
- Benjamin Gess and Scott Smith
bibliography:
- 'refs.bib'
title: Stochastic continuity equations with conservative noise
---

Introduction
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Classically, the continuity equation refers to the following Cauchy problem $$\label{eq:ContEq}
\partial_{t}\rho + \Div(\rho u)=0 \quad \text{on} \quad [0,T] \times \R^{d}, \quad \quad \rho \mid_{t=0}=\rho_{0} \quad \text{on} \quad \R^{d}.$$ Typically, $u(t,x) \in \R^{d}$ is interpreted as a velocity field, while $\rho(t,x)$ is a density for a mass distribution. Under classical assumptions for the inputs $u$ and $\rho_{0}$, the solution $\rho$ to admits a representation formula via characteristics. Indeed, consider the differential equation $$\label{eq:Char}
\dot{X}_{t}^{s,x}=u(t,X_{t}^{s,x}), \quad X_{s}^{s,x}=x.$$ If $u$ is sufficiently regular, admits a unique solution for each initial 