Stable patterns in a viscous diffusion equation

Abstract:

We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation ${u_t} = \Delta (f(u) + \nu {u_t})$, motivated by the problem of phase separation in a viscous binary mixture. The function $f$ is non-monotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution $u(x)$ satisfying $f(u) = {\text {constant}}$, $f’(u(x)) > 0$ a.e. is dynamically stable to perturbations in the sense of convergence in measure. In particular, smooth solutions may achieve discontinuous asymptotic states. Furthermore, stable states need not correspond to absolute minimizers of free energy, thus violating Gibbs’ principle of stability for phase mixtures.