On the global attractor of 2D incompressible turbulence with random forcing

This study revisits bounds on the projection of the global attractor in the energy–enstrophy plane for 2D incompressible turbulence [Dascaliuc, Foias, and Jolly, 2005, 2010]. In addition to providing more elegant proofs of some of the required nonlinear identities, the treatment is extended from the case of constant forcing to the more realistic case of random forcing. Numerical simulations in particular often use a stochastic white-noise forcing to achieve a prescribed mean energy injection rate. The analytical bounds are demonstrated numerically for the case of white-noise forcing.     

Sharp Estimates for the Number of Degrees of Freedom for the Damped-Driven 2-D Navier-Stokes Equations

We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier-Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier-Stokes system with damping, our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2-D Navier-Stokes system is inspired by the Stommel-Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the viscous Stommel-Charney model.     

非线性光学中薛定谔型方程的整体吸引子

研究了一类非线性薛定谔型方程,描述了光波在光折射晶体中的传播.首先构造了该模型整体弱的吸引子,然后通过能量方程的精确分析,证明整体弱吸引子实际为系统整体强吸引子.最后给出了整体吸引子的分形维数和Hausdorff维数的上界估计.     

Structure of the global attractor for a second order strongly damped lattice system

In this paper, we consider the dynamical behavior of a second order strongly damped lattice system where the coupled operator is nonnegative definite symmetric. Firstly, we prove the existence of a global attractor, and give an upper bound of Hausdorff dimension of the global attractor, which keeps bounded for large strongly damping. Then we show that when the damping term is linear and the damping is suitable large, the system has an unbounded one-dimensional global attractor, which is a restricted horizontal curve.     

Distributional Enstrophy Dissipation Via the Collapse of Three Point Vortices

Dissipation of enstrophy in 2D incompressible flows in the zero viscous limit is considered to play a significant role in the emergence of the inertial range corresponding to the forward enstrophy cascade in the energy spectrum of 2D turbulent flows. However, since smooth solutions of the 2D incompressible Euler equations conserve the enstrophy, we need to consider non-smooth inviscid and incompressible flows so that the enstrophy dissipates. Moreover, it is physically uncertain what kind of a flow evolution gives rise to such an anomalous enstrophy dissipation. In this paper, in order to acquire an insight about the singular phenomenon mathematically as well as physically, we consider a dispersive regularization of the 2D Euler equations, known as the Euler-\(\alpha \) equations, for the initial vorticity distributions whose support consists of three points, i.e., three \(\alpha \)-point vortices, and take the \(\alpha \rightarrow 0\) limit of its global solutions. We prove with mathematical rigor that, under a certain condition on their vortex strengths, the limit solution becomes a self-similar evolution collapsing to a point followed by the expansion from the collapse point to infinity for a wide range of initial configurations of point vortices. We also find that the enstrophy always dissipates in the sense of distributions at the collapse time. This indicates that the triple collapse is a mechanism for the anomalous enstrophy dissipation in non-smooth inviscid and incompressible flows. Furthermore, it is an interesting example elucidating the emergence of the irreversibility of time in a Hamiltonian dynamical system.     

Long time data series and diffuculties with the characterization of chaotic attractors: a case with intermittency III

We show that experimental holes appearing in attractors may not be a consequence of repeller action of fixed points (a local property) but rather a consequence of global properties of the attractor where regions have practically vanishing probability of being visited.     

Maximum Rate of Growth of Enstrophy in Solutions of the Fractional Burgers Equation

This investigation is a part of a research program aiming to characterize the extreme behavior possible in hydrodynamic models by analyzing the maximum growth of certain fundamental quantities. We consider here the rate of growth of the classical and fractional enstrophy in the fractional Burgers equation in the subcritical and supercritical regimes. Since solutions to this equation exhibit, respectively, globally well-posed behavior and finite-time blowup in these two regimes, this makes it a useful model to study the maximum instantaneous growth of enstrophy possible in these two distinct situations. First, we obtain estimates on the rates of growth and then show that these estimates are sharp up to numerical prefactors. This is done by numerically solving suitably defined constrained maximization problems and then demonstrating that for different values of the fractional dissipation exponent the obtained maximizers saturate the upper bounds in the estimates as the enstrophy increases. We conclude that the power-law dependence of the enstrophy rate of growth on the fractional dissipation exponent has the same global form in the subcritical, critical and parts of the supercritical regime. This indicates that the maximum enstrophy rate of growth changes smoothly as global well-posedness is lost when the fractional dissipation exponent attains supercritical values. In addition, nontrivial behavior is revealed for the maximum rate of growth of the fractional enstrophy obtained for small values of the fractional dissipation exponents. We also characterize the structure of the maximizers in different cases.     

Bubbling,riddling, blowout and critical curves

This paper provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaoti1c trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.     

Asymptotic dynamics of the one‐dimensional attraction–repulsion Keller–Segel model

The asymptotic behavior of the attraction–repulsion Keller–Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction–repulsion Keller–Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time‐dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform‐in‐time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. Copyright © 2014 John Wiley & Sons, Ltd.     

Global Attractor of One Nonlinear Parabolic Equation

We apply the theory of multivalued semiflows to a nonlinear parabolic equation of the reaction–diffusion type in the case where it is impossible to prove the uniqueness of its solution. A multivalued semiflow is generated by solutions satisfying a certain estimate global in time. We establish the existence of a global compact attractor in the phase space for the multivalued semiflow generated by a nonlinear parabolic equation. We prove that this attractor is an upper-semicontinuous function of a parameter.     

Well-posedness in Sobolev spaces of the full water wave problem in 3-D

We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion of the interface of the water wave is not subject to Taylor instability, as long as the interface separates the whole 3-D space into two simply connected regions. We prove further the existence and uniqueness of solutions of the full 3-D water wave problem, locally in time, for any initial interface that separates the whole 3-D space into two simply connected regions.

     

常微分方程的區域分析理論(Ⅲ)

<正> 本文是繼續[Ⅰ],[Ⅱ]的發展,並初步總結其中若干部分,定名為分區線性方程.總結其特性及簡易的定性處理法,使這類的非線性方程的規律性非常容易掌握;正如我們掌握常係數線性方程一樣.     

On global attractors of the 3D Navier-Stokes equations

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.     

Random Sequential Adsorption of Discs on Surfaces of Constant Curvature: Plane,Sphere, Hyperboloid,and Projective Plane

We present an algorithm to simulate random sequential adsorption (random “parking”) of discs on constant curvature surfaces: the plane, sphere, hyperboloid, and projective plane, all embedded in three-dimensional space. We simulate complete parkings efficiently by explicitly calculating the boundary of the available area in which discs can park and concentrating new points in this area. We use our algorithm to study the number distribution and density of discs parked in each space, where for the plane and hyperboloid we consider two different periodic tilings each. We make several notable observations: (1) on the sphere, there is a critical disc radius such the number of discs parked is always exactly four: the random parking is deterministic. We prove this statement and also show that random parking on the surface of a d-dimensional sphere would have deterministic behaviour at the same critical radius. (2) The average number of parked discs does not always monotonically increase as the disc radius decreases: on the plane (square with periodic boundary conditions), there is an interval of decreasing radius over which the average decreases. We give a heuristic explanation for this counterintuitive finding. (3) As the disc radius shrinks to zero, the density (average fraction of area covered by parked discs) appears to converge to the same constant for all spaces, though it is always slightly larger for a sphere and slightly smaller for a hyperboloid. Therefore, for parkings on a general curved surface we would expect higher local densities in regions of positive curvature and lower local densities in regions of negative curvature.     

Regularity of the attractor for 3-D complex Ginzburg-Landau equation

In this paper,the existence of global attractor for 3-D complex Ginzburg Landau equation is considered.By a decomposition of solution operator,it is shown that the global attractor A_i in H~i(Ω) is actually equal to a global attractor Aj in H~j(Ω)(i≠j,i,j = 1,2,…m).     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.     

有界多连通区域具线性阻尼Navier-Stokes方程的有限维行为

考虑二维有界多连通区域上具线性阻尼Navier-Stokes方程,在适当的边界条件下证明了解的存在唯一性及整体吸引子A的存在性,并给出了A的Hausdorff维数与Fractal维数.     

Global attractor for Hirota equation

The long time behavior of solution for Hirota equation with zero order dissipation is studied. The global weak attractor for this system in Hper^k is constructed. And then by exact analysis of the energy equation, it is shown that the global weak attractor is actually the global strong attractor in Hper^k.     

Global attractors for the discrete Klein–Gordon–Schrödinger type equations

The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice. We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution is continuous and possesses an absorbing set. Using truncation technique, we show that there exists a global attractor for the semi-group. Finally, we extend the criteria of Zhou et al. [S. Zhou, C. Zhao, and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Discrete Contin. Dyn. Syst. A 21 (2008), pp. 1259–1277.] for finite fractal dimension of a family of compact subsets in a Hilbert space to obtain an upper bound of fractal dimension for the global attractor.     

Backward behavior of solutions of the Kuramoto-Sivashinsky equation

We prove that any solution of the Kuramoto-Sivashinsky equation either belongs to the global attractor or it cannot be continued to a solution defined for all negative times. This extends a previous result of the first author who proved that solutions which do not belong to the global attractor have superexponential backward growth. A particular consequence of the result is that the global attractor can be characterized as the maximal invariant set.