17 research outputs found
Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains
We show that the stochastic flow generated by the Stochastic Navier-Stokes
equations in a 2-dimensional Poincar\'e domain has a unique random attractor.
This result complements a recent result by Brze\'zniak and Li [10] who showed
that the flow is asymptotically compact and generalizes a recent result by
Caraballo et al. [12] who proved existence of a unique pullback attractor for
the time-dependent deterministic Navier-Stokes equations in a 2-dimensional
Poincar\'e domain
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
Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains
This paper is concerned with the asymptotic behavior of solutions to a class of non-autonomous stochastic nonlinear wave equations with dispersive and viscosity dissipative terms driven by operator-type noise defined on the entire space Rn. The existence, uniqueness, time-semi-uniform compactness and asymptotically autonomous robustness of pullback random attractors are proved in H1(Rn) _ H1(Rn)
when the growth rate of the nonlinearity has a subcritical range, the density of the noise is suitably controllable, and the time-dependent force converges to a time-independent function in some sense. The main difficulty to establish the time-semi-uniform pullback asymptotic compactness of the solutions in H1(Rn) _ H1(Rn) is caused by the lack of compact Sobolev embeddings on Rn, as well as the weak
dissipativeness of the equations is surmounted at light of the idea of uniform tail-estimates and a spectral decomposition approach. The measurability of random attractors is proved by using an argument which considers two attracting universes developed by Wang and Li (Phys. D 382: 46-57, 2018)
Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations
In this paper, we study the existence, stability and bifurcation of random
complete and periodic solutions for stochastic parabolic equations with
multiplicative noise. We first prove the existence and uniqueness of tempered
random attractors for the stochastic equations and characterize the structures
of the attractors by random complete solutions. We then examine the existence
and stability of random complete quasi-solutions and establish the relations of
these solutions and the structures of tempered attractors. When the stochastic
equations are incorporated with periodic forcing, we obtain the existence and
stability of random periodic solutions. For the stochastic Chafee-Infante
equation, we further establish the multiplicity and stochastic bifurcation of
complete and periodic solutions.Comment: Work was reported at IMA workshop in October 201