Random attractors for 3D Benjamin–Bona–Mahony equations derived by a Laplace-multiplier noise

2017 ◽  
Vol 18 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Yangrong Li ◽  
Renhai Wang
Keyword(s):  
Dynamical System ◽  
Multiplicative Noise ◽  
Careful Analysis ◽  
Random Dynamical System ◽  
Stochastic Pde ◽  
Limit Set ◽  
Cutoff Function ◽  
Random System ◽  
Random Attractors ◽  
Bbm Equation

This paper contributes the dynamics for stochastic Benjamin–Bona–Mahony (BBM) equations on an unbounded 3D-channel with a multiplicative noise. An interesting feature is that the noise has a Laplace-operator multiplier, which seems not to appear in any literature for the study of stochastic PDE. After translating the stochastic BBM equation into a random equation and deducing a random dynamical system, we obtain both existence and semi-continuity of random attractors for this random system in the Sobolev space. The convergence of the system can be verified without the lower bound assumption of the nonlinear derivative. The tail-estimate is achieved by using a square of the usual cutoff function and by a careful analysis of the solution’s biquadrate. A spectrum method is also applied to prove the collective limit-set compactness.

Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

2015 ◽  
Vol 16 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Yanfeng Guo ◽  
Chunxiao Guo ◽  
Yongqian Han
Keyword(s):  
Dynamical System ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Coefficients ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Hydrodynamical Equation ◽  
Priori Estimates ◽  
Heisenberg Paramagnet ◽  
Truncation Function

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire [Formula: see text] is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Anhui Gu ◽  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Infinite Lattices

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

RANDOM ATTRACTORS FOR STOCHASTIC EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION

2010 ◽  
Vol 20 (09) ◽  
pp. 2761-2782 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
B. MASLOWSKI ◽  
B. SCHMALFUß
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Stochastic Equations ◽  
Hurst Parameter ◽  
Random Dynamical System ◽  
Random Attractors

In this paper, the asymptotic behavior of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 is studied. In particular, it is shown that the corresponding solutions generate a random dynamical system for which the existence and uniqueness of a random attractor is proved.

scholarly journals Random Attractors of Stochastic Three-Component Reversible Gray-Scott System on Unbounded Domains

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Anhui Gu
Keyword(s):  
Dynamical System ◽  
A Priori Estimates ◽  
A Priori ◽  
Unbounded Domains ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Far Field ◽  
Asymptotic Compactness ◽  
Priori Estimates ◽  
Random Attractors

The existence of a pullback random attractor is established for a stochastic three-component reversible Gray-Scott system on unbounded domains. The Gray-Scott system is recast as a random dynamical system and asymptotic compactness which is illustrated by using uniform, a priori estimates for far-field values of solutions and a cutoff technique.

scholarly journals Random uniform exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise in ℝ3

2019 ◽  
Vol 17 (1) ◽  
pp. 1411-1434
Author(s):  
Zongfei Han ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Multiplicative Noise ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Random Dynamical System ◽  
Exponential Attractor

Abstract We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ3.

RANDOM ATTRACTORS OF STOCHASTIC NON-NEWTONIAN FLUIDS ON UNBOUNDED DOMAIN

2013 ◽  
Vol 14 (01) ◽  
pp. 1350008 ◽  
Author(s):  
CHUNXIAO GUO ◽  
BOLING GUO ◽  
YANFENG GUO
Keyword(s):  
Dynamical System ◽  
Unbounded Domain ◽  
Random Attractor ◽  
Newtonian Fluids ◽  
Random Dynamical System ◽  
Two Dimensional ◽  
Random Set ◽  
Random Attractors

We consider the stochastic non-Newtonian fluids defined on a two-dimensional Poincaré unbounded domain, and prove that it generates an asymptotically compact random dynamical system. Then, we establish the existence of random attractor for the corresponding random dynamical system. Random attractor is invariant and attracts every pullback tempered random set.

scholarly journals RANDOM ATTRACTORS OF STOCHASTIC LATTICE DYNAMICAL SYSTEMS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350041 ◽  
Author(s):  
ANHUI GU
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Random Attractor ◽  
Hurst Parameter ◽  
Random Dynamical System ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Random Attractors ◽  
Lattice Dynamical Systems

This paper is devoted to consider stochastic lattice dynamical systems (SLDS) driven by fractional Brownian motions with Hurst parameter bigger than 1/2. Under usual dissipativity conditions these SLDS are shown to generate a random dynamical system for which the existence and uniqueness of a random attractor are established. Furthermore, the random attractor is, in fact, a singleton sets random attractor.

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Anhui Gu
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise

The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.

scholarly journals Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

2016 ◽  
Vol 14 (1) ◽  
pp. 586-602 ◽  
Author(s):  
Xiaoyao Jia ◽  
Juanjuan Gao ◽  
Xiaoquan Ding
Keyword(s):  
Dynamical System ◽  
Neumann Boundary Condition ◽  
Multiplicative Noise ◽  
Space Dimension ◽  
Stochastic Equation ◽  
Uniform Estimate ◽  
Neumann Boundary ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Homogeneous Neumann Boundary Condition

Abstract In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

scholarly journals Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Junwei Feng ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Dynamical System ◽  
Bounded Domain ◽  
Multiplicative Noise ◽  
Random Dynamical System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Long Time ◽  
Uniform Attractors

<p style='text-indent:20px;'>In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.</p>

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