75 research outputs found
Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains
We prove the existence of a compact random attractor for the stochastic
Benjamin-Bona-Mahony Equation defined on an unbounded domain. This random
attractor is invariant and attracts every pulled-back tempered random set under
the forward flow. The asymptotic compactness of the random dynamical system is
established by a tail-estimates method, which shows that the solutions are
uniformly asymptotically small when space and time variables approach infinity.Comment: 37 page
Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems
We study pullback attractors of non-autonomous non-compact dynamical systems
generated by differential equations with non-autonomous deterministic as well
as stochastic forcing terms. We first introduce the concepts of pullback
attractors and asymptotic compactness for such systems. We then prove a
sufficient and necessary condition for existence of pullback attractors. We
also introduce the concept of complete orbits for this sort of systems and use
these special solutions to characterize the structures of pullback attractors.
For random systems containing periodic deterministic forcing terms, we show the
pullback attractors are also periodic. As an application of the abstract
theory, we prove the existence of a unique pullback attractor for
Reaction-Diffusion equations on with both deterministic and random
external terms. Since Sobolev embeddings are not compact on unbounded domains,
the uniform estimates on the tails of solutions are employed to establish the
asymptotic compactness of solutions.Comment: References adde
Periodic Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Domains
This paper is concerned with the asymptotic behavior of solutions of the
two-dimensional Navier-Stokes equations with both non-autonomous deterministic
and stochastic terms defined on unbounded domains. We first introduce a
continuous cocycle for the equations and then prove the existence and
uniqueness of tempered random attractors. We also characterize the structures
of the random attractors by complete solutions. When deterministic forcing
terms are periodic, we show that the tempered random attractors are also
periodic. Since the Sobolev embeddings on unbounded domains are not compact, we
establish the pullback asymptotic compactness of solutions by Ball's idea of
energy equations.Comment: Title change
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