Finite-dimensional attractor for the viscous Cahn-Hilliard equation in an unbounded domain

Abstract

Abstract.We consider the viscous Cahn-Hilliard equation in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by this equation has the global attractor which has finite Hausdorff dimension.1. Introduction. Many equations arising from mechanics and physics possess a global attractor, which is a compact invariant set which uniformly attracts the trajectories as time goes to infinity, and thus appears as a suitable object for the study of the asymptotic behaviour of the system. An important issue is then to study the dimension, in the sense of the Hausdorff or fractal dimension, of the global attractor. A finite bound of the dimension of the attractor means that the system has an asymptotic behaviour determined by a finite number of degrees of freedom, indeed a remarkable improvement compared to the a priori infinite-dimensional dynamics (see [5] and [15]).For the equations on bounded domains, the known constructions of global attractors make use of some compactness properties in an essential manner, and more specifically of the compact embedding of H m 1 into H m 2 , when m 1 > m 2 . Such properties are no longer valid for equations on unbounded domains, and it is thus more difficult to develop a general theory of existence of the global attractor in this case. A possibility then consists of working in weighted Sobolev spaces (see In this article, we study, on an unbounded domain, the existence of global attractors and their Hausdorff dimensions for the viscous Cahn-Hilliard equation of the form

On the Existence of Global Attractor for 3D Viscous Cahn-Hilliard Equation

“…(1) in 1D case. We also noticed that some investigations of the viscous Cahn-Hilliard equation were studied, such as in [3,4,14].…”

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“…Concerning unbounded domains, in [9] the author considered the viscous Cahn-Hilliard model in a channel like unbounded domain in dimensions N = 2, 3. Using weighted spaces it was proved that the model has a finite dimensional attractor.…”

Global attractors for 2D Navier–Stokes–Voight equations in an unbounded domain

“…It is possible to overcome this difficulty by studying the problem in weighted Sobolev spaces [6][7][8][9][10][11]. In this article of Babin [7], the advantage of the smoothing property of parabolic systems has been used for both the proof of dissipation and the proof of compactness.…”