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Publication 124

Slow motion for the 1D Swift-Hohenberg equation


Gurgen Hayrapetyan
Ohio University
Athens, OH, USA

CMUMatteo Rinaldi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA, USA

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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