Slow motion for the 1D Swift-Hohenberg equation

2016/04/08 by Gurgen Hayrapetyan, Hayrapetyan, Gurgen, Matteo Rinaldi +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1604.02407

arxiv created 2016/04/08 · arxiv updated 2016/04/11

Abstract

The goal of this paper is to study the behavior of certain solutions to the Swift-Hohenberg equation on a one-dimensional torus \mathbbT. Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L¹-close to a jump function v, remain close to v for large time. This can be achieved by regarding the equation as the L²-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.

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