Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

TL;DR

This paper establishes global existence and long-term convergence of solutions for certain singular quasilinear stochastic PDEs, using energy and Poincaré inequalities, and analyzes their stability and regularity properties.

Contribution

It proves global solvability and convergence to stationary solutions for a class of singular quasilinear SPDEs, extending previous results with new uniform estimates and stability analysis.

Findings

Proved global-in-time solvability of specific singular SPDEs.

Established convergence of solutions to stationary states as time approaches infinity.

Demonstrated the uniformity of Poincaré constants in approximations.

Abstract

We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincar\'e inequality. It is essential that the Poincar\'e constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in \cite{FHSX} except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.