Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws

TL;DR

This paper establishes a nonlinear stability theory for two-dimensional periodic waves in parabolic systems with conservation laws, extending previous results to more complex multidimensional and conservation law scenarios.

Contribution

It introduces a diffusive spectral stability assumption that guarantees nonlinear stability for a broad class of perturbations in 2D parabolic systems with conservation laws.

Findings

Proves nonlinear stability under spectral assumptions.

Extends stability results to 2D patterns and systems with conservation laws.

Addresses low spectral regularity issues such as conic-like and Jordan-block singularities.

Abstract

We develop a stability theory for two-dimensional periodic traveling waves of general parabolic systems, possibly including conservation laws. In particular, we identify a diffusive spectral stability assumption and prove that it implies nonlinear stability for variously-(non)localized perturbations, including critically nonlocalized perturbations. Thus we extend the stability parts of Johnson et al., Invent. Math. 2014, to two-dimensional patterns and of Melinand-Rodrigues, preprint 2024, to systems with conservation laws. In doing so we need to bypass two kinds of low spectral regularity, explicitly conic-like singularities due to multidimensionality and Jordan-block like singularities due to conservation laws.