An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory

We consider a two-dimensional Navier–Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup.Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type bound on the time-averaged energy dissipation rate, independent of viscosity at large Reynolds numbers. Then we establish a version of the Lieb–Thirring inequality for a class of functions defined on the considered non-rectangular flow domain.This research is motivated by a problem from lubrication theory.     

Global attractor for a low order ODE model problem for transition to turbulence

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy‐conserving, non‐quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

一类长短波方程组的整体吸引子的存在性(英)

本文应用对时间的一致先验估计,证明了一类具有周期边值条件的长短波方程组的整体吸引子的存在性．     

Attractors for a plate equation with nonlocal nonlinearities

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of R ⁿ with smooth boundary, γ ,ρ ?0 and are nonlocal functions. The main result states that the dynamical system {S (t )}_{t ?0} associated with this problem has a compact global attractor. In addition, in the limit case γ  = 0, it is also shown that {S (t )}_{t ?0} has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.     

Long time behaviour of a flow in infinite pipe conforming to slip boundary conditions

The paper analyses long time behaviour of solutions of the Navier–Stokes equations in a two‐dimensional pipe‐like domain. The system is studied with perfect slip boundary conditions with arbitrary inflow conditions at infinity. The main results show the existence of global in time solutions and of an attractor for the dynamical system generated by the model. The paper also establishes an upper bound for the Hausdorff dimension of the attractor. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

Attractors for autonomous double‐diffusive convection systems based on Brinkman–Forchheimer equations

In this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension N≤4 which describe double‐diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds. Copyright © 2015 John Wiley & Sons, Ltd.     

Attractors for second order periodic lattices with nonlinear damping

We consider second order nonlinear lattices under the effect of nonlinear damping. The family we study is subject to cyclic boundary conditions and includes as distinguished examples the Fermi–Pasta–Ulam and sine-Gordon lattices. We prove global well posedness and existence of a global attractor.     

On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

In this paper, we prove the existence and uniqueness of a global solution for 2‐D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor. Copyright © 2006 John Wiley & Sons, Ltd.     

弱阻尼广义KdV方程的整体吸引子

证明了具有弱阻尼项的广义KdV方程约周期初边值问题解的存在唯—性及整体吸引子存在性,最后获得了吸引子的Hausdorff维数和分形维数的上界估计．     

On the 2D Cahn–Hilliard Equation with Inertial Term

Galenko et al. proposed a modified Cahn–Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.     

Pullback attractors for nonautonomous 2D Bénard problem in some unbounded domains

In this paper, we study the 2D Bénard problem, a system with the Navier–Stokes equations for the velocity field coupled with a convection–diffusion equation for the temperature, in an arbitrary domain (bounded or unbounded) satisfying the Poincaré inequality with nonhomogeneous boundary conditions and nonautonomous external force and heat source. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite‐dimensional pullback D_σ‐attractor for the process associated to the problem. Copyright © 2013 John Wiley & Sons, Ltd.     

On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction–diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction–diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.     

Global attractor for the semilinear thermoelastic problem

In this paper we consider a class of semilinear thermoelastic problems. The global attractor for this semilinear thermoelastic problem with Dirichlet boundary condition is obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Global attractors for 2D Navier–Stokes–Voight equations in an unbounded domain

We consider the 2D Navier–Stokes–Voight equation in an unbounded strip-like domain. It is shown that the semigroup generated by this equation has a global attractor in weighted Sobolev spaces.     

Global well‐posedness and global attractor of fourth order semilinear parabolic equation

We study the initial boundary value problem of a class of fourth order semilinear parabolic equations. Global existence and nonexistence of solutions with initial data in the potential well are derived. Moreover, by using the iteration technique for regularity estimates, we obtain that for any k ≥ 0, the semilinear parabolic possesses a global attractor in H^k(Ω), which attracts any bounded subsets of H^k(Ω) in the H^k‐norm. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence,asymptotic behavior,and uniform attractor for a nonautonomous equation

In this paper, we establish the global existence, asymptotic behavior, and uniform attractor for a nonautonomous viscoelastic equation with a delay term. Under some suitable assumptions, we firstly prove the global well‐posedness of the problem by using the Faedo–Galerkin approximations together with some energy estimates and then obtain the general decay results of the energy via suitable Lyapunov functionals. Finally, we prove the existence of uniform attractors. Copyright © 2013 John Wiley & Sons, Ltd.     

Hasegawa-Mima方程的整体吸引子

考虑了带有耗散项的Hasegawa-Mima方程解的长时间性态, 研究了具有初值周期边值条件的Hasegawa-Mima方程的整体吸引子问题．运用关于时间的一致先验估计,证明了该问题整体吸引子的存在性,并获得了整体吸引子的维数估计．