Publication 16-CNA-005

Slow motion for the 1D Swift-Hohenberg equation

Gurgen Hayrapetyan
Ohio University
Athens, OH, USA
hayrapet@ohio.edu

Matteo Rinaldi
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
matteor@andrew.cmu.edu

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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