Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against $C_{\mathrm{ub}}^m$-perturbations

TL;DR

This paper proves the nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against bounded, non-localized perturbations using a novel method based on $L^ Infty$-estimates, extending stability results beyond localized disturbances.

Contribution

It introduces a new technique employing pure $L^ Infty$-estimates to establish stability of periodic solutions against bounded perturbations, applicable to systems with conservation laws.

Findings

Proved stability of periodic roll solutions against bounded perturbations.

Developed a new method using $L^ Infty$-estimates for nonlinear stability analysis.

Extended the method to pattern-forming systems with conservation laws.

Abstract

The real Ginzburg-Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure $L^\infty$-estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.