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.

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 Retarded Lattice Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Xiaoquan Ding ◽  
Jifa Jiang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
White Noise ◽  
Random Attractor ◽  
Additive White Noise ◽  
Lattice Dynamical System ◽  
Random Attractors ◽  
Lattice Dynamical Systems ◽  
Bounded Sets

This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise. We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove the existence of a compact global random attractor within the set of tempered random bounded sets.

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.

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 Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kush Kinra ◽  
Manil T. Mohan
Keyword(s):  
White Noise ◽  
Random Perturbation ◽  
Three Dimensional ◽  
Random Attractor ◽  
Sufficient Evidence ◽  
Dimensional Torus ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
2D And 3D ◽  
Upper And Lower Semicontinuity

<p style='text-indent:20px;'>This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r&lt;\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r&lt;\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.</p>

scholarly journals A Stochastic Weakly Damped, Forced KdV-BO Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Guolian Wang
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Global Solution ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Energy Type ◽  
Bo Equation

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the spaceH1(ℝ).

scholarly journals Random Attractors for the Stochastic Discrete Long Wave-Short Wave Resonance Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Jie Xin ◽  
Hong Lu
Keyword(s):  
Dynamical System ◽  
Short Wave ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Asymptotic Compactness ◽  
Infinite Lattice ◽  
Long Wave ◽  
Wave Resonance ◽  
Random Attractors

We prove the existence of the random attractor for the stochastic discrete long wave-short wave resonance equations in an infinite lattice. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor.

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.

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