Faedo-Galerkin Approximations for the nonlinear Deterministic Constrained Modified Swift-Hohenberg Equation

TL;DR

This paper establishes the well-posedness of a nonlinear deterministic constrained modified Swift-Hohenberg equation, which models pattern formation, using Faedo-Galerkin methods to prove existence and uniqueness of solutions.

Contribution

It provides the first rigorous proof of existence and uniqueness for this specific class of amplitude equations via Faedo-Galerkin techniques.

Findings

Proved existence of solutions using Faedo-Galerkin method.

Established uniqueness of solutions for the equation.

Demonstrated the equation's well-posedness in the deterministic setting.

Abstract

In this paper, we study the well-posedness of the nonlinear deterministic constrained modified Swift-Hohenberg equation; this equation belongs to class of amplitude equations which describe the appearance of pattern formation in nature. The proof for existence and uniqueness is based on the Faedo-Galerkin compactness method.