Regularity Of Attractor Research Articles

Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damping part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove the well-posedness of the corresponding system in energy space and establish the existence of a regular global attractor under certain conditions on the nonlinearities and coefficients of the damped part only. Besides, we study the singular limits of the problem under consideration when curvature tends to zero, or curvature tends to zero, and simultaneously shear moduli tend to infinity and perform numerical modeling for these processes.

Stability of regular attractors for non-autonomous random dynamical systems and applications to stochastic Newton–Boussinesq equations with delays

In this paper, we establish theoretical results on the stability of random regular attractors. First, we introduce a backward regular attractor, which is a new type of attractor defined by a minimal backward pullback attracting set. We then establish an existence theorem for such an attractor, and prove it is long time stable. Eventually, we prove the long time stability of regular pullback random attractors. As an application, we consider stochastic non-autonomous Newton–Boussinesq equations with variable and distributed delays. Since solutions of the equations have no higher regularity, we prove their regular asymptotic compactness via the spectrum decomposition technique.

Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of Nge 2 components occupying a bounded region Omega subset mathbb {R}^{d},dle 3. Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if dle 2, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d=3, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d=3, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.

Regular attractor of the β-evolution equation with fractional damping on Rn

We study the well-posedness and longtime dynamics of the β-evolution equation with fractional damping: ∂t2u+(−Δ)βu+γ(1−Δ)α∂tu+f(u)=g(x) on the whole space Rn, with β > 2α > 0. First, we find a critical exponent p*=n+4αn−2β for the well-posedness of energy solutions. In fact, if the nonlinear term grows with the order p ∈ [1, p*) and satisfies some dissipative conditions, then the equation is globally well-posed in the energy space. Moreover, both u and ∂tu have a smoothing effect as a parabolic equation. Finally, we show that the solution semigroup has a global attractor A in the energy space. The main difficulties come from the non-compactness of the Sobolev embedding on Rn and the nonlocal characteristic of the equation. We overcome them by establishing some new commutator estimates.

Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations

<p style='text-indent:20px;'>In the current issue, we consider a general class of two coupled weakly dissipative fractional Schrödinger-type equations. We will prove that the asymptotic dynamics of the solutions for such NLS system will be described by the existence of a regular compact global attractor in the phase space that has finite fractal dimension.</p>

Finite Dimensional Global Attractor for a Fractional Schrödinger Type Equation with Mixed Anisotropic Dispersion

We study the Cauchy problem for a class of nonlinear damped fractional Schrodinger type equation in a two dimensional unbounded domain. Then, we focus on long-time behaviour of the solutions proving that this behaviour is described by the existence of regular finite-dimensional global attractor in the energy space.

Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $

<p style='text-indent:20px;'>We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> that reads</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.</p>

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Regular random attractors for non-autonomous stochastic evolution equations with time-varying delays on thin domains

This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction–diffusion equations with time-varying delays on thin domains. First, we prove the existence and uniqueness of the regular random attractor. Then, we prove the upper semicontinuity of the regular random attractors for the equations on a family of (n + 1)-dimensional thin domains that collapses onto an n-dimensional domain.

Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains

<p style='text-indent:20px;'>This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of <inline-formula><tex-math id="M1">$ (n+1) $</tex-math></inline-formula>-dimensional thin domains collapses onto an <inline-formula><tex-math id="M2">$ n $</tex-math></inline-formula>-dimensional domain.</p>

Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.

Asymptotic dynamics of the solutions for a system of N-coupled fractional nonlinear Schrödinger equations

In the current issue, we consider a system of N-coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities. We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor with finite fractal dimension.

Finite dimensional global attractor for a dissipative anisotropic fourth order Schrödinger equation

We study the long-time behavior of the solutions to a nonlinear damped anisotropic fourth order Schrödinger type equation in R2. We prove that this behavior is described by the existence of regular finite-dimensional global attractor in the energy space.

Regularity of uniform attractor for 3D non-autonomous Navier–Stokes–Voigt equation

In this paper, we study the large time behavior for 3D viscoelastic incompressible fluid flow subject to Kelvin–Voigt damping and a time varying external force. The evolution of the dynamic is governed by a 3D non-autonomous Navier–Stokes–Voigt (NSV) equation. We assume that the external force is in the space of translation bounded functions (or its sub-spaces) that requires less compactness than the space of translation compact functions. We formulate the system in the framework of skew product flow. By defining an appropriate energy space that incorporates the Kelvin–Voigt damping, we established the existence of regular strong uniform attractor for the NSV equation in energy space W˜ and provide a description of the general structure of the uniform attractor. Our result improves the H01 regularity of global attractor of the system under study.

Global attractors for the 2D hyperbolic Cahn–Hilliard equations

We consider the initial boundary value problem for the hyperbolic relaxation of the 2D Cahn–Hilliard equation with sub-cubic nonlinearity. Under mild regularity conditions on the nonlinearity, we prove the uniform (with respect to the initial data) boundedness of the weak solutions without assuming lower bound condition on the first derivative of the nonlinear term. Then, we prove the existence of the regular global attractor for the weak solutions.

Navier–Stokes–Voigt Equations with Memory in 3D Lacking Instantaneous Kinematic Viscosity

We consider a Navier-Stokes-Voigt fluid model where the instantaneous kinematic viscosity has been completely replaced by a memory term incorporating hereditary effects, in presence of Ekman damping. The dissipative character of our model is weaker than the one where hereditary and instantaneous viscosity coexist, previously studied by Gal and Tachim-Medjo. Nevertheless, we prove the existence of a regular exponential attractor of finite fractal dimension under rather sharp assumptions on the memory kernel.

Regular attractor for damped KdV‐Burgers equations on R

We study the well‐posedness and dynamic behavior for the KdV‐Burgers equation with a force on R. We establish Lp−Lq estimates of the evolution , as an application we obtain the local well‐posedness. Then the global well‐posedness follows from a uniform estimate for solutions as tgoes to infinity. Next, we prove the asymptotical regularity of solutions in space and by the smoothing effect of . The regularity and the asymptotical compactness in L2 yields the asymptotical compactness in by an interpolation arguement. Finally, we conclude the existence of an globalattractor.

Buckling and nonlinear dynamics of elastically coupled double-beam systems

This paper deals with damped transverse vibrations of elastically coupled double-beam system under even compressive axial loading. Each beam is assumed to be elastic, extensible and supported at the ends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is involved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends on structural parameters and applied axial loads. The occurrence of a so complex structure of the steady states motivates a global analysis of the longtime dynamics. In this regard, we are able to prove the existence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, it consists of the unstable manifolds connecting them.

On the Existence of Regular Global Attractor for $$p$$ p -Laplacian Evolution Equation

In this study, we consider the nonlinear evolution equation of parabolic type $$\begin{aligned} u_t-\mathrm{div}\big (|\nabla u|^{p-2}\nabla u\big )+f(u)=g. \end{aligned}$$ut-div(| u|p-2 u)+f(u)=g....

Well-posedness and longtime behaviour of a coupled nonlinear system modeling a suspension bridge

In this paper we discuss the well-posedness and the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a Kelvin–Voigt string–beam system which models a suspension bridge. For this model we obtain the existence and uniqueness of solutions and the exponential stability of the homogeneous system, provided that the constant axial force \(p\) is smaller than a critical value. For a general \(p,\) the existence of the regular global attractor is proved when the external loads are independent of time.