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arXiv · 1604.02407

Slow motion for the 1D Swift-Hohenberg equation

Gurgen Hayrapetyan·Matteo Rinaldi
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Abstract

The goal of this paper is to study the behavior of certain solutions to the Swift-Hohenberg equation on a one-dimensional torus $\mathbb{T}$. Combining results from $\Gamma$-convergence and ODE theory, it is shown that solutions corresponding to initial data that is $L^1$-close to a jump function $v$, remain close to $v$ for large time. This can be achieved by regarding the equation as the $L^2$-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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Gurgen Hayrapetyan, Matteo Rinaldi. 2016-04-08. Slow motion for the 1D Swift-Hohenberg equation. https://arxiv.org/abs/1604.02407

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