Abstract
In this paper, we first study the deterministic Swift-Hohenberg equation on a bounded domain. After obtaining some a priori estimates by the uniform Gronwall inequality, we prove the existence of an attractor by the Sobolev compact embeddings. Then, we consider the stochastic Swift-Hohenberg equation driven by additive noise on an unbounded domain and prove that the random dynamical system is asymptotically compact by uniform a priori estimates for the far-field values of the solution, which implies the existence of a random attractor for the random dynamical system associated with the stochastic Swift-Hohenberg equation.
Highlights
The Swift-Hohenberg (SH) equation describes the pattern formation in fluid layers confined between horizontal well-conducting boundaries, which was proposed by Swift and Hohenberg [ ] as a model for the convective instability in the Rayleigh-Bénard convection
The SwiftHohenberg equation has featured in different branches of physics, ranging from hydrodynamics to nonlinear optics, such as the Taylor-Couette flow [, ], study of lasers [ ], and so on
In Section, we prove the existence of a global attractor for the corresponding deterministic dynamical system on an bounded domain
Summary
The Swift-Hohenberg (SH) equation describes the pattern formation in fluid layers confined between horizontal well-conducting boundaries, which was proposed by Swift and Hohenberg [ ] as a model for the convective instability in the Rayleigh-Bénard convection. We have found that there are few results about the existence of a global attractor for the local Swift-Hohenberg equation. The existence of a global attractor for the local Swift-Hohenberg equation on a bounded domain will be given in Section .
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