  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 $v$.
