scholarly journals H1-Random Attractors and Asymptotic Smoothing Effect of Solutions for Stochastic Boussinesq Equations with Fluctuating Dynamical Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yijin Zhang
Keyword(s):  
Dynamical System ◽  
Boundary Condition ◽  
Boussinesq Equations ◽  
Random Attractor ◽  
Boussinesq System ◽  
Random Dynamical System ◽  
Smoothing Effect ◽  
Dynamical Boundary Condition ◽  
Dynamical Boundary Conditions ◽  
White Noises

This work is concerned with the random dynamics of two-dimensional stochastic Boussinesq system with dynamical boundary condition. The white noises affect the system through a dynamical boundary condition. Using a method based on the theory of omega-limit compactness of a random dynamical system, we prove that theL2-random attractor for the generated random dynamical system is exactly theH1-random attractor. This improves a recent conclusion derived by Brune et al. on the existence of theL2-random attractor for the same system.

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.

scholarly journals Hausdorff Dimension of a Random Attractor for Stochastic Boussinesq Equations with Double Multiplicative White Noises

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Yin Li ◽  
Ruiying Wei ◽  
Donghong Cai
Keyword(s):  
Hausdorff Dimension ◽  
Boussinesq Equations ◽  
Random Attractor ◽  
White Noises

This paper investigates the existence of random attractor for stochastic Boussinesq equations driven by multiplicative white noises in both the velocity and temperature equations and estimates the Hausdorff dimension of the random attractor.

ON THE LAPLACE EQUATION WITH NON-LINEAR DYNAMICAL BOUNDARY CONDITIONS

2006 ◽  
Vol 93 (2) ◽  
pp. 418-446 ◽  
Author(s):  
ENZO VITILLARO
Keyword(s):  
Boundary Condition ◽  
Laplace Equation ◽  
Main Part ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Initial Condition ◽  
Dynamical Boundary Condition ◽  
Non Linear ◽  
Dynamical Boundary Conditions

The main part of the paper deals with local existence and global existence versus blow-up for solutions of the Laplace equation in bounded domains with a non-linear dynamical boundary condition. More precisely, we study the problem consisting in: (1) the Laplace equation in $(0, \infty) \times \Omega$; (2) a homogeneous Dirichlet condition $(0, \infty) \times \Gamma_0$; (3) the dynamical boundary condition $ \frac {\partial u}{\partial \nu} = - |u_t|^{m-2} u_t + |u|^{p - 2} u$ on $(0, \infty) \times \Gamma_1$; (4) the initial condition $u(0, x) = u_0 (x)$ on $\partial \Omega$. Here $\Omega$ is a regular and bounded domain in $\mathbb{R}^n$, with $n \ge 1$, and $\Gamma_0$ and $\Gamma_1$ endow a measurable partition of $\partial \Omega$. Moreover, $m>1$, $2 \le p < r$, where $r = 2 (n - 1) / (n - 2)$ when $n \ge 3$, $r = \infty$ when $n = 1,2$, and $u_0 \in H^{1/2} (\partial \Omega)$, $u_0 = 0$ on $\Gamma_0$.The final part of the paper deals with a refinement of a global non-existence result by Levine, Park and Serrin, which is applied to the previous problem.

scholarly journals On the Boussinesq system: local well-posedness of the strong solution and inviscid limits

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Lianhong Guo ◽  
Yuanfei Li ◽  
Chunjuan Hou
Keyword(s):  
Boundary Condition ◽  
Boussinesq Equations ◽  
Slip Boundary Condition ◽  
Boussinesq System ◽  
Well Posedness ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Navier Slip ◽  
Slip Boundary ◽  
Viscosity Limit

AbstractIn this paper, we consider the solvability, regularity and vanishing viscosity limit of the 3D viscous Boussinesq equations with a Navier-slip boundary condition. We also obtain the rate of convergence of the solution of viscous Boussinesq equations to the corresponding ideal Boussinesq equations.

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 PARABOLIC PROBLEMS WITH DYNAMICAL BOUNDARY CONDITIONS IN PERFORATED MEDIA

2003 ◽  
Vol 8 (4) ◽  
pp. 337-350 ◽  
Author(s):  
C. Timofte
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Dirichlet Condition ◽  
Parabolic Problems ◽  
Perforated Domain ◽  
Limit Equation ◽  
Dynamical Boundary Condition ◽  
Dynamical Boundary Conditions

The asymptotic behavior of the solution of a parabolic dynamical boundary‐value problem in a periodically perforated domain is analyzed. The perforations, which are identical and periodically distributed, are of size ϵ. In the perforated domain we consider a heat equation, with a Dirichlet condition on the exterior boundary and a dynamical boundary condition on the surface of the holes. The limit equation, as ϵ ? 0, is a heat equation with extra-terms coming from the influence of the non-homogeneous dynamical boundary condition.

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.

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