DCDS-B
Global attractors for the Gray-Scott equations in locally uniform spaces
Gaocheng Yue Chengkui Zhong
In this paper, we prove the existence of a $(L_{lu}^2(\mathbb{R}^N)\times L_{lu}^2(\mathbb{R}^N),L_{\rho}^2(\mathbb{R}^N)\times L_{\rho}^2(\mathbb{R}^N))$-global attractor for the solution semigroup generated by the Gray-Scott equations on unbounded domains of space dimension $N\leq3.$
keywords: asymptotic compactness Gray-Scott equations well-posedness global attractor. locally uniform spaces
DCDS
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models
Yonghai Wang Chengkui Zhong
In this paper, we consider the upper semicontinuity of pullback attractors for a nonautonomous Kirchhoff wave model with strong damping. For this purpose, some necessary abstract results are established.
keywords: Pullback attractor upper semicontinuity Kirchhoff type.
DCDS
On the global well-posedness to the 3-D Navier-Stokes-Maxwell system
Gaocheng Yue Chengkui Zhong
The present paper is devoted to the well-posedness issue of solutions of a full system of the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations. By means of Littlewood-Paley analysis we prove the global well-posedness of solutions in the Besov spaces $\dot{B}_{2,1}^\frac1{2}\times B_{2,1}^\frac3{2}\times B_{2,1}^\frac3{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} \big(\|u_0^h\|_{\dot{B}_{2,1}^\frac1{2}} +\|E_0\|_{B_{2,1}^\frac{3}{2}}+\|B_0\|_{B_{2,1}^\frac{3}{2}}\big)\exp \Big\{\frac{C_0}{\nu^2}\|u_0^3\|_{\dot{B}_{2,1}^\frac1{2}}^2\Big\}\leq c_0, \end{align*} for some sufficiently small constant $c_0.$
keywords: Navier-Stokes-Maxwell system Besov spaces. Littlewood-Paley theory Well-posedness
DCDS-B
Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations
Gaocheng Yue Chengkui Zhong
In this paper we study the long time behavior of the three dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Navier-Stokes-Voight equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to $V'.$ For the nonautonomous case, we only assume that $f(x,t)$ is translation bounded instead of translation compact, where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal projection. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Navier-Stokes-Voight equations and also obtain the existence of the uniform attractor. Finally, we describe the structure of the uniform attractor and its regularity.
keywords: the uniform attractor 3D Navier-Stokes-Voight equations global attractor asymptotic regularity.
DCDS
The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces
Shan Ma Chengkui Zhong
For weakly damped non-autonomous hyperbolic equations, we introduce a new concept Condition (C*), denote the set of all functions satisfying Condition (C*) by L2 C* $(R;X)$ which are translation bounded but not translation compact in $L^2$ loc$(R;X)$, and show that there are many functions satisfying Condition (C*); then we study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time dependent external forces $g(x,t)\in $ L2 C* $(R;X)$ and prove the existence of the uniform attractors for the family of processes corresponding to the equation in $H^1_0\times L^2$ and $D(A)\times H^1_0$.
keywords: uniform attractor; translation compact; uniformly ω-limit compact; uniform Condition (C); Condition (C*); damped hyperbolic equations.
DCDS-B
Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent
Fengjuan Meng Chengkui Zhong
In this paper, we are concerned with some properties of the global attractor of weakly damped wave equations. We get the existence of multiple stationary solutions for wave equations with weakly damping. Furthermore, we provide some approaches to verify the small neighborhood of the origin is an attracting domain which is important to obtain the multiple equilibrium points in global attractor.
keywords: wave equations. global attractor Lyapunov functional $Z_2$ index equilibrium points
DCDS
On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations
Gaocheng Yue Chengkui Zhong
The present paper is devoted to the well-posedness issue of solutions to the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations with horizontal dissipation and horizontal magnetic diffusion. By means of anisotropic Littlewood-Paley analysis we prove the global well-posedness of solutions in the anisotropic Sobolev spaces of type $H^{0,s_0}(\mathbb{R}^3)$ with $s_0>\frac1{2}$ provided the norm of initial data is small enough in the sense that \begin{align*} (\|u_n^h(0)\|_{H^{0,s_0}}^2+\|B_n^h(0)\|_{H^{0,s_0}}^2)\exp \Big\{C_1(\|u_0^3\|_{H^{0,s_0}}^4+\|B_0^3\|_{H^{0,s_0}}^4)\Big\}\leq\varepsilon_0, \end{align*} for some sufficiently small constant $\varepsilon_0.$
keywords: anisotropic Littlewood-Paley theory MHD equations Well-posedness Besov spaces.
DCDS
Global exponential κ-dissipative semigroups and exponential attraction
Jin Zhang Peter E. Kloeden Meihua Yang Chengkui Zhong

Globally exponential $κ-$dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup $\{S(t)\}_{t≥q 0}$ has a bounded absorbing set, then $\{S(t)\}_{t≥q 0}$ is globally exponentially $κ-$dissipative if and only if there exists a compact set $\mathcal{A}^*$ that is positive invariant and attracts any bounded subset exponentially. The set $\mathcal{A}^*$ need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

keywords: Exponential dissipativity global attractor measures of noncompactness
DCDS
Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces
Songsong Lu Hongqing Wu Chengkui Zhong
The existence and structure of uniform attractors in $V$ is proved for nonautonomous 2D Navier-stokes equations on bounded domain with a new class of external forces, termed normal in $L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are translation bounded but not translation compact in $L_{l o c}^2(\mathbb R; H)$. To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.
keywords: nonautonomous equation uniform Condition (C) uniformly $\omega$-limit compact process normal function. Uniform attractor Navier-Stokes equations
DCDS-B
Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation
Bo You Chengkui Zhong Fang Li
This paper is devoted to the existence of pullback attractors for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. We first prove the existence of pullback absorbing sets in $H$ and $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8), and then we prove the existence of a pullback attractor in $H$ by the Sobolev compactness embedding theorem. Finally, we obtain the existence of a pullback attractor in $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8) by verifying the pullback $\mathcal{D}$ condition $(PDC)$.
keywords: planetary geostrophic viscous equations Sobolev compactness embedding theory pullback $\mathcal{D}$ condition $(PDC)$. Pullback attractor

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