Random attractors for singular stochastic evolution equations

Gess, Benjamin

doi:10.1016/j.jde.2013.04.023

Cited by 64 publications

(46 citation statements)

References 38 publications

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“…SPDE of the form (1.7) (actually also allowing more general, multiplicative noise B(X t )dW t ) have been considered in [GT13] where the existence and uniqueness of solutions (for all d ∈ N) as well as ergodicity (for d = 1 and additive noise) has been shown. Based on the results developed in [Ges13a] one may expect that this implies the existence of a random attractor consisting of a single random point, which we expect to prove in subsequent work.…”

Section: Introductionmentioning

confidence: 61%

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

Gess

2014

Commun. Math. Phys.

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Section: Introductionmentioning

confidence: 61%

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

Gess

2014

Commun. Math. Phys.

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“…These allow an integration by parts formula which is employed in order to get rid of the stochastic integrals and do a pathwise analysis of the equation. Apart from this, all results regarding random pullback attractors for SPDEs with additive or linear multiplicative noise employ transformations in a PDE with random coefficients as described above [33,16,45,69,88,87,89,46,156,165]. To our best knowledge, for nonlinear multiplicative noise, and SPDEs such as (2.26), there are no general existence results concerning random attractors.…”

Section: Random Attractorsmentioning

confidence: 99%

Dynamics of Stochastic Reaction-Diffusion Equations

Kuehn¹,

Neamţu²

2020

Infinite Dimensional and Finite Dimensional Stochastic Equations and Applications in Physics

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“…The existence of global attractors for the Navier-Stokes equations has been extensively studied in the literature, see, e.g., [5,7,12,13,19,34,46,48,51] for the deterministic case, and [11,22,24,56] for the stochastic case. More work on random attractors can be found in [8,9,14,15,16,17,20,21,22,24,25,26,27,29,37,42,47,53,54] for the autonomous stochastic equations; and in [18,23,30,31,55,57,58] for the nonautonomous stochastic systems. In this paper, we will investigate pullback random attractors of the non-autonomous random system (2) driven by colored noise.The random system (2) can be considered as an approximation of the following Stratonovich stochastic equations: ∂u ∂t − ν∆u + (u · ∇)u = f (t, x) − ∇p + G(t, x, u) • dW dt , x ∈ Q, t > τ, div u = 0, x ∈ Q, t > τ,…”

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Long term behavior of random Navier-Stokes equations driven by colored noise

Gu¹,

Guo²,

Wang³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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This paper is devoted to the study of long term behavior of the twodimensional random Navier-Stokes equations driven by colored noise defined in bounded and unbounded domains. We prove the existence and uniqueness of pullback random attractors for the equations with Lipschitz diffusion terms. In the case of additive noise, we show the upper semi-continuity of these attractors when the correlation time of the colored noise approaches zero. When the equations are defined on unbounded domains, we establish the pullback asymptotic compactness of the solutions by Ball's idea of energy equations in order to overcome the difficulty introduced by the noncompactness of Sobolev embeddings.Given τ ∈ R and ω ∈ Ω, consider the following non-autonomous two-dimensional random Navier-Stokes equations driven by colored noise:along with homogeneous Dirichlet boundary condition, where ν is a positive constant, f is a given function defined on R × O, G is a nonlinear function satisfying certain conditions, and ζ δ (θ t ω) is the colored noise with correlation time δ > 0.The Navier-Stokes equations are used to model the flow of an incompressible viscous fluid of constant density enclosed in a region O with rigid boundary ∂O, where u(t, x) ∈ R 2 and p(t, x) ∈ R are, respectively, the velocity and the pressure of the fluid at point x ∈ O and time t > τ , ν > 0 is the kinematic viscosity of the fluid and f = f (t, x) ∈ R 2 is the time dependent external force. The existence of global attractors for the Navier-Stokes equations has been extensively studied in the literature, see, e.g., [5,7,12,13,19,34,46,48,51] for the deterministic case, and [11,22,24,56] for the stochastic case. More work on random attractors can be found in [8,9,14,15,16,17,20,21,22,24,25,26,27,29,37,42,47,53,54] for the autonomous stochastic equations; and in [18,23,30,31,55,57,58] for the nonautonomous stochastic systems. In this paper, we will investigate pullback random attractors of the non-autonomous random system (2) driven by colored noise.The random system (2) can be considered as an approximation of the following Stratonovich stochastic equations: ∂u ∂t − ν∆u + (u · ∇)u = f (t, x) − ∇p + G(t, x, u) • dW dt , x ∈ Q, t > τ, div u = 0, x ∈ Q, t > τ,

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Random attractors for singular stochastic evolution equations

Cited by 64 publications

References 38 publications

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

Dynamics of Stochastic Reaction-Diffusion Equations

Long term behavior of random Navier-Stokes equations driven by colored noise

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