Relaxation to Equilibrium in the One-Dimensional Cahn--Hilliard Equation
Félix Otto, Maria G. Westdickenberg · SIAM Journal on Mathematical Analysis · 2014
We study the stability of a so-called kink profile for the one-dimensional Cahn--Hilliard problem on the real line. We derive optimal bounds on the decay to equilibrium under the assumption that the initial energy is less than three times the energy of a kink and that the initial $\dot{H}^{-1}$ distance to a kink is bounded. Working with the $\dot{H}^{-1}$ distance is natural, since the equation is a gradient flow with respect to this metric. Indeed, our method is to establish and exploit elementary algebraic and differential relationships among three natural quantities: the energy, the dissipation, and the $\dot{H}^{-1}$ distance to a kink. Along the way it is necessary and possible to control the time-dependent shift of the center of the $L^2$ closest kink. Our result is different from earlier results because we do not assume smallness of the initial distance to a kink; we assume only boundedness.