On the Global solution and Invariance of stochastic constrained Modified Swift-Hohenberg Equation on a Hilbert manifold

TL;DR

This paper studies a stochastic version of the constrained modified Swift-Hohenberg equation on a Hilbert manifold, proving global solutions and invariance of the manifold, which advances understanding of pattern formation under randomness.

Contribution

It introduces a stochastic generalization of the constrained Swift-Hohenberg equation and proves global well-posedness and invariance of the Hilbert submanifold.

Findings

Proved global existence and uniqueness of solutions.

Established invariance of the Hilbert submanifold.

Applied Khasminskii test to ensure non-explosiveness.

Abstract

This paper aims to investigate the stochastic generalization of the projected deterministic constrained modified Swift-Hohenberg equation. In particular, we prove the global well-posedness and its invariance of Hilbert submanifold i.e. if the initial condition are chosen from submanifold then trajectories of solutions are going to stay on manifold. The proof of global well-posedness is based on Khashminskii test for non-explosions test for no-explosions. Swift-Hohenberg equations belong to class of Amplitude equations that usually describe the pattern formation in nature.