On the 2D Cahn–Hilliard Equation with Inertial Term

Abstract

P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.

On the 3D Cahn–Hilliard equation with inertial term

“…(1.1) and solutions which are more regular than the energy ones. We also report the statement of a result on the existence of exponential attractors which could be shown just by following the lines of [20,Sec. 5] and whose proof is thus omitted.…”

A Splitting Method for the Cahn–hilliard Equation With Inertial Term

“…The existence of more regular (global) solutions can be proven relatively easy in the case d = 1. This can also be done in dimension two for f with cubic controlled growth, though proofs are much more technical (see [17]). If d = 3, existence of smoother (global) solutions has been recently established in [16] for small enough and taking (smooth) initial data bounded by a constant which blows up as goes to 0.…”

“…As far as uniqueness is concerned, the proof is standard in one dimension (see again [6]). If d = 2, uniqueness is proven in [17] by means of a more refined argument which takes advantage of the so-called Brézis-Gallouet inequality (cf. [3]) to show the whole Galerkin approximating sequence converge to (the) solution.…”

“…The onedimensional case is rather well understood [2,6,13,26,27], but the two-dimensional [17] or the three-dimensional case [16,30] is far more complicated, because the so-called energy bounded solutions (of H 1 -type) are no longer bounded in L ∞ (see Theorem 2.1 below). We refer in particular the reader to the introduction of [16] for an analysis of these issues.…”

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…The results of this section complete the ones proven in [45] for Dirichlet-type boundary conditions; in particular, our construction allows us to establish some continuity properties of inertial manifolds with respect to . Equation (1.1) has recently been studied in more that one spatial dimension (see [6,[26][27][28]42]). In that case, the mathematical analysis is much more involved.…”