Nonlinear Stability at the Zigzag Boundary

2020/12/08 by Haberle, Mason, Chowdhary, Abhijit, Wu, Qiliang
#35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.04154

Abstract

We investigate the dynamics of roll solutions at the zigzag boundary of the planar Swift-Hohenberg equation. Linear analysis shows an algebraic decay of small perturbation with a t^{- 1/4} rate, instead of the classical t^{- 1/2} diffusive decay rate, due to the degeneracy of the quadratic term of the continuation of the translational mode of the linearized operator in the Bloch-Fourier spaces. The proof is based on a decomposition of the neutral mode and the faster decaying modes in the Bloch-Fourier space, and a fixed-point argument, demonstrating the irrelevancy of the nonlinear terms.

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