Global Behavior of Solutions to Chevron Pattern Equations

TL;DR

This paper analyzes the chevron pattern equations, proving existence, uniqueness, and stability of solutions, and explores their long-term behavior through reduction to dynamical systems.

Contribution

It establishes the well-posedness and existence of a global attractor for the chevron pattern equations, and links PDE behavior to simpler ODE systems.

Findings

Unique weak solutions depend continuously on initial data

Existence of a global attractor in the phase space

Reduction to ODE systems provides insights into system behavior

Abstract

Considering a system of equations modeling the chevron pattern dynamics, we show that the corresponding initial boundary value problem has a unique weak solution that continuously depends on initial data, and the semigroup generated by this problem in the phase space $X^0:= L^2(\Omega)\times L^2(\Omega)$ has a global attractor. We also provide some insight to the behavior of the system, by reducing it under special assumptions to systems of ODEs, that can in turn be studied as dynamical systems.