Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term

Abstract

This paper is devoted to a modification of the classical Cahn–Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient ε > 0, which is usually small in comparison with the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is ‘supercritical’ in the case of two and three space dimensions. Thus, the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small ε and arbitrary growth rate of the nonlinearity (see (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70)). The present contribution studies the long-time behaviour of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers (Grasselli et al 2009 J. Evol. Eqns 9 371–404, Grasselli et al 2009 Commun. Partial Diff. Eqns 34 137–70). In particular, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well. It is worth noting that the uniqueness of energy solutions in the 3D case is not known yet, so we have to use the so-called trajectory approach which does not require uniqueness. Finally, we apply the obtained exponential regularization of the energy solutions for verifying the dissipativity of solutions of the 2D modified Cahn–Hilliard equation in the intermediate phase space of weak solutions (in between energy and strong solutions) without any restriction on ε.