Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

Exponential attractor for the wave equation with structural damping and supercritical exponent

The paper studies the existence of an exponential attractor for the wave equation with structural damping and supercritical nonlinearity [Formula: see text]. By constructing a bounded absorbing set with higher global regularity (rather than the long-standing partial regularity) and by using the weak quasi-stability estimates (rather than the strong ones as usual), we establish the existence of an exponential attractor in the natural energy space.

Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary conditionutt-Δu+Δ2u-Δut-Δg(u)=f(x). The main result is concerned with nonlinearitiesg(u)with supercritical growth. In that case, we construct a bounded absorbing set with further regularity and obtain quasi-stability estimates. Then the exponential attractor is established in natural energy spaceV2×H.

FINITE-DIMENSIONAL GLOBAL ATTRACTOR FOR A NONLOCAL PHASE-FIELD SYSTEM

We analyze a phase-field system where the energy balance equation is linearly coupled with a nonlinear and nonlocal ODE for the order parameter . The latter equation is characterized by a space convolution term which models long-range interactions and a singular configuration potential that forces to take values in the interval (􀀀1; 1). We prove that the corresponding dynamical system is dissipative, i.e., it has a bounded absorbing set in a suitable phase space. Then we establish the existence of a finite-dimensional global attractor.

Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.

Flattening, squeezing and the existence of random attractors

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

Amorphous molecular beam epitaxy: global solutions and absorbing sets

The parabolic equation \[u_t + u_{xxxx} + u_{xx} = - (|u_x|^\alpha)_{xx}, \qquad \alpha>1\], is studied under the boundary conditions $u_x|_{\partial\Omega}=u_{xxx}|_{\partial\Omega}=0$ in a bounded real interval $\Omega$. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any $\alpha>1$ and initial data $u_0 \in W^{1,q}(\Omega)$ ($q>\alpha$), and that they are global if $\alpha \le \frac{5}{3}$. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for $\alpha < \frac{10}{3}$, and for suitable transforms of such solutions the existence of a bounded absorbing set in $L^1(\Omega)$ is proved for $\alpha \in [2,\frac{10}{3})$. The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets.