Slow motion for the 1D Swift-Hohenberg equation

TL;DR

This paper investigates the long-term behavior of solutions to the 1D Swift-Hohenberg equation, demonstrating that solutions starting near jump functions stay close over time by analyzing energy functionals and gradient flows.

Contribution

It combines $ ext{Gamma}$-convergence and ODE theory to show stability of solutions near jump functions in the 1D Swift-Hohenberg equation.

Findings

Solutions near jump functions remain close over time.

Energy bounds relate to the number of jumps.

Solutions behave as gradient flows of an energy functional.

Abstract

The goal of this paper is to study the behavior of certain solutions to the Swift-Hohenberg equation on a one-dimensional torus $\mathbb{T}$. Combining results from $\Gamma$-convergence and ODE theory, it is shown that solutions corresponding to initial data that is $L^1$-close to a jump function $v$, remain close to $v$ for large time. This can be achieved by regarding the equation as the $L^2$-gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of $v$.