Martingale Solutions of Stochastic Constrained Modified Swift-Hohenberg Equation

TL;DR

This paper proves the existence of global martingale solutions for a stochastic constrained modified Swift-Hohenberg equation, which models pattern formation in nature under multiplicative noise, using advanced stochastic analysis techniques.

Contribution

It establishes the existence of solutions in a Hilbert space framework for a complex stochastic PDE with multiplicative noise, applying stochastic Galerkin methods and energy estimates.

Findings

Existence of global martingale solutions proven.

Application of stochastic Galerkin method to this class of equations.

Use of tightness criterion and Skorokhod theorem for solution construction.

Abstract

In this paper, we aim to prove the existence of global Martingale solution to Stochastic Constrained Modified Swift-Hohenberg Equation driven by stratonovich multiplicative noise. This equation belongs to class of amplitude equations which describe the appearance of pattern formation in nature. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzezniak and collaborators, and Jakubowski's generalization of the Skorokhod theorem.