Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws
Analysis of PDEs
2025-08-07 v1
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.
Cite
@article{arxiv.2508.04023,
title = {Nonlinear stability of two-dimensional periodic waves in parabolic systems with conservation laws},
author = {L. Miguel Rodrigues and Aric Wheeler},
journal= {arXiv preprint arXiv:2508.04023},
year = {2025}
}