Slow motion for the 1D Swift–Hohenberg equation

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus TT. Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L1L1-close to a jump function v, remain close to v   for large time. This can be achieved by regarding the equation as the L2L2-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.