Well-posedness of the obstacle problem for generalized Dean-Kawasaki equation

TL;DR

This paper proves well-posedness for the obstacle problem in generalized Dean-Kawasaki equations with correlated noise, using a kinetic approach to handle reflection and stability in degenerate diffusion regimes.

Contribution

It introduces a kinetic characterization of the obstacle problem for degenerate stochastic PDEs, establishing existence, uniqueness, and stability under minimal regularity assumptions.

Findings

Established well-posedness for the obstacle problem in degenerate regimes

Extended obstacle problem theory to equations with singular diffusion coefficients

Provided a kinetic framework for explicit reflection mechanism

Abstract

We investigate the obstacle problem for generalized Dean--Kawasaki equations driven by correlated conservative noise, establishing the existence, uniqueness, and $L^1$-stability of stochastic kinetic solutions. Our core strategy combines a kinetic characterization of the Skorokhod condition with a precise description of the reflection measure term associated with the obstacle, in which the barrier substitutes the solution. This formulation makes the reflection mechanism explicit at the kinetic level and yields a stable framework adapted to $L^1$ doubling of variables method. Consequently, under a merely continuous obstacle and the same structural assumptions as in the obstacle-free setting, we obtain well-posedness over the full porous-medium regime, covering degenerate diffusion and the critical square-root noise coefficient. This extends the existing theory of obstacle problems for stochastic partial differential equations to a class of degenerate equations with singular diffusion coefficients.