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On the 3D Cahn-Hilliard equation with inertial term

Grasselli, M, Schimperna, G, Segatti, A and Zelik, S (2009) On the 3D Cahn-Hilliard equation with inertial term JOURNAL OF EVOLUTION EQUATIONS, 9 (2). pp. 371-404.

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We study the modified Cahn–Hilliard equation proposed by Galenko et al. in order to account for rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration multiplied by a (small) positive coefficient ε. Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of the 2D case has been recently carried out by Grasselli, Schimperna and Zelik. The 3D case is still rather poorly understood but for the existence of energy bounded solutions. Taking advantage of this fact, Segatti has investigated the asymptotic behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing the existence and uniqueness of a global weak solution, provided that ε is small enough. More precisely, we show that there exists ε0 > 0 such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of ε ∈ (0, ε0) which goes to +∞ as ε tends to 0. This result allows us to construct a semigroup Sε(t) on an appropriate (bounded) phase space and, besides, to prove the existence of a global attractor. Finally, we show a regularity result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors : Grasselli, M, Schimperna, G, Segatti, A and Zelik, S
Date : 1 June 2009
DOI : 10.1007/s00028-009-0017-7
Copyright Disclaimer : The final publication is available at Springer via
Uncontrolled Keywords : Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, MATHEMATICS, MATHEMATICS, APPLIED, Cahn-Hilliard equation, Weak solutions, Global existence, Global attractors, Exponential attractors, SEMILINEAR WAVE-EQUATIONS, LONG-TIME BEHAVIOR, PHASE-FIELD SYSTEM, SPINODAL DECOMPOSITION, EXPONENTIAL ATTRACTORS, HYPERBOLIC RELAXATION, GLOBAL ATTRACTORS, SINGULAR PERTURBATIONS, BINARY-SYSTEM, POTENTIALS
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 03 Nov 2016 13:32
Last Modified : 06 Jul 2019 05:15

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