共查询到20条相似文献,搜索用时 15 毫秒
1.
The existence of a pullback attractor is established for a stochastic reaction-diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction-diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions. 相似文献
2.
First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain. 相似文献
3.
4.
Z. Brzeźniak T. Caraballo J.A. Langa Y. Li G. Łukaszewicz J. Real 《Journal of Differential Equations》2013
We show that the stochastic flow generated by the 2-dimensional Stochastic Navier–Stokes equations with rough noise on a Poincaré-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron–Martin (or reproducing kernel Hilbert) space. Our results complement the result by Brze?niak and Li (2006) [10] who showed that the corresponding flow is asymptotically compact and also generalize Caraballo et al. (2006) [12] who proved existence of a unique attractor for the time-dependent deterministic Navier–Stokes equations. 相似文献
5.
We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor. 相似文献
6.
Ricardo M.S. Rosa 《Journal of Differential Equations》2006,229(1):257-269
The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H. 相似文献
7.
The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rn when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero. 相似文献
8.
We obtain the existence and the structure of the weak uniform (with respect to the initial time) global attractor and construct a trajectory attractor for the 3D Navier–Stokes equations (NSE) with a fixed time-dependent force satisfying a translation boundedness condition. Moreover, we show that if the force is normal and every complete bounded solution is strongly continuous, then the uniform global attractor is strong, strongly compact, and solutions converge strongly toward the trajectory attractor. Our method is based on taking a closure of the autonomous evolutionary system without uniqueness, whose trajectories are solutions to the nonautonomous 3D NSE. The established framework is general and can also be applied to other nonautonomous dissipative partial differential equations for which the uniqueness of solutions might not hold. It is not known whether previous frameworks can also be applied in such cases as we indicate in open problems related to the question of uniqueness of the Leray–Hopf weak solutions. 相似文献
9.
Shengfan Zhou 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(5):2793-2805
In this paper, we first present some conditions for the upper semicontinuity of perturbed random attractors to a limiting random attractor. Then we apply this result to establish the upper semicontinuity of random attractors for the first order stochastic lattice differential equations with random coupled coefficients and multiplicative/additive white noises in the weighted space of infinite sequences as the coefficient of the white noise term tends to zero. 相似文献
10.
Bixiang Wang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7252-7260
This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed. 相似文献
11.
Janusz Mierczyński Wenxian Shen Xiao-Qiang Zhao 《Journal of Differential Equations》2004,204(2):471-510
The purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic Kolmogorov systems via the skew-product semiflows approach. It is first shown that the uniform persistence of the skew-product semiflow associated with a nonautonomous (random) parabolic Kolmogorov system implies that of the system. Various sufficient conditions in terms of the so-called unsaturatedness and/or Lyapunov exponents for uniform persistence of the skew-product semiflows are then provided. Among others, it is shown that if the associated skew-product semiflow has a global attractor and its restriction to the boundary of the state space has a Morse decomposition which is unsaturated or whose external Lyapunov exponents are positive, then it is uniformly persistent. More specific conditions are discussed for uniform persistence in n-species, particularly 3-species, random competitive systems. 相似文献
12.
Yuncheng You 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(5):1969-1986
In this work the existence of a global attractor is proved for the solution semiflow of the coupled two-compartment Gray-Scott equations with the homogeneous Neumann boundary condition on a bounded domain of space dimension n≤3. The grouping estimation method combined with a new decomposition approach is introduced to overcome the difficulties in proving the absorbing property and the asymptotic compactness of this four-component reaction-diffusion systems with cubic autocatalytic nonlinearity and linear coupling. The finite dimensionality of the global attractor is also proved. 相似文献
13.
A.V. Kapustyan 《Journal of Differential Equations》2007,240(2):249-278
We study in this paper the asymptotic behaviour of the weak solutions of the three-dimensional Navier-Stokes equations. On the one hand, using the weak topology of the usual phase space H (of square integrable divergence free functions) we prove the existence of a weak attractor in both autonomous and nonautonomous cases. On the other, we obtain a conditional result about the existence of the strong attractor, which is valid under an unproved hypothesis. Also, with this hypothesis we obtain continuous weak solutions with respect to the strong topology of H. 相似文献
14.
Dynamics of systems on infinite lattices 总被引:1,自引:0,他引:1
Bixiang Wang 《Journal of Differential Equations》2006,221(1):224-245
The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l2 space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems. 相似文献
15.
Martin Meyries 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(5):2922-2935
For a class of quasilinear parabolic systems with nonlinear Robin boundary conditions we construct a compact local solution semiflow in a nonlinear phase space of high regularity. We further show that a priori estimates in lower norms are sufficient for the existence of a global attractor in this phase space. The approach relies on maximal Lp-regularity with temporal weights for the linearized problem. An inherent smoothing effect due to the weights is employed for obtaining gradient estimates. In several applications we can improve the convergence to an attractor by one regularity level. 相似文献
16.
Jürgen Sprekels 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3028-3048
We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate. 相似文献
17.
Yang Lu 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4012-4025
In this paper, we first introduce the concept of a closed process in a Banach space, and we obtain the structure of a uniform attractor of the closed process by constructing a skew product-flow on the extended phase space. Then, the properties of the kernel section of closed process are investigated. Moreover, we prove the existence and structure of the uniform attractor for the reaction-diffusion equation with a dynamical boundary condition in Lp without any restriction on the growth order of the nonlinear term. 相似文献
18.
We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem
without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined
weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in
the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the
Hausdorff dimension of the attractor is finite. 相似文献
19.
Jong Yeoul Park Sun Hye Park 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):4046-4057
The existence of a pullback attractor is proven for a non-autonomous generalized 2D parabolic system in an unbounded domain. The asymptotic compactness of the solution operator is obtained by the uniform estimates on the tails of solutions. 相似文献
20.
GUOBOLING WANGBIXIANG 《高校应用数学学报(英文版)》1996,11(2):125-136
Abstract. In the present paper, we deal with the long-time behavior of dissipative partial differenttial equations, and we construct the approximate inertial mardfolds for the nonlbaear Stringer equation with a zero order dlssipation. The order of approximation of these manlfolde to the global attractor is derived. 相似文献