A priori bounds for stochastic porous media equations via regularity structures

TL;DR

This paper establishes a priori bounds for solutions to singular stochastic porous media equations with multiplicative noise, using regularity structures, kinetic formulation, and a new energy inequality, advancing understanding of these complex stochastic PDEs.

Contribution

It introduces a novel approach combining regularity structures and kinetic formulation to derive bounds for singular stochastic porous media equations with multiplicative noise.

Findings

Proved modelledness of solutions in the regularity structures framework.

Established a new renormalized energy inequality for these equations.

Balanced degeneracy of diffusion with noise damping for small solutions.

Abstract

We prove a priori bounds for solutions of singular stochastic porous media equations with multiplicative noise in their natural $L^1$-based regularity class. We consider the first singular regime, i.e.~noise of space-time regularity $\alpha-2$ for $\alpha\in(2/3,1)$, and prove modelledness of the solution in the sense of regularity structures with respect to the solution of the corresponding linear stochastic heat equation. The proof relies on the kinetic formulation of the equation and a novel renormalized energy inequality. A careful analysis allows to balance the degeneracy of the diffusion coefficient against sufficiently strong damping of the multiplicative noise for small values of the solution.