Attractors of dissipative hyperbolic equations with singularly oscillating external forces

We study a uniform attractor $\mathcal{A}^\varepsilon $ for a dissipative wave equation in a bounded domain Ω ? ?ⁿ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g ₀(x, t)+ ε^?α g ₁ (x, t/ε), x ∈ Ω, t ∈ ?, where α > 0, 0 < ε ≤ 1. In E = H ₀ ¹ × L ₂, this equation has an absorbing set B ^ε estimated as ‖B ^ε‖ _E ≤C ₁+C ₂ε^?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g ₁(x, z), x ∈ Ω, z ∈ ?, we prove that, for 0 < α ≤ α ₀, the global attractors $\mathcal{A}^\varepsilon $ of such an equation are bounded in E, i.e., $\parallel \mathcal{A}^\varepsilon \parallel _E \leqslant C_3 $ , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g ₀(x, t) that also has a global attractor $\mathcal{A}^0 $ . For the case in which g ₀(x, t) = g ₀(x) and the global attractor $\mathcal{A}^0 $ of the limiting equation is exponential, it is established that, for 0 < α ≤ α ₀, the Hausdorff distance satisfies the estimate $dist_E (\mathcal{A}^\varepsilon ,\mathcal{A}^0 ) \leqslant C\varepsilon ^{\eta (\alpha )} $ , where η(α) > 0. For η(α) and α ₀, explicit formulas are given. We also study the nonautonomous case in which g ₀ = g ₀(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $\mathcal{A}^\varepsilon $ from $\mathcal{A}^0 $ , similar to those given above.     

Averaging of a 3D primitive equations with oscillating external forces

In this article, we consider a non-autonomous three-dimensional primitive equations of the ocean with a singularly oscillating external force g ^ε?=?g ₀(t)?+?ε^?ρ g ₁(t/ε) depending on a small parameter ε?>?0 and ρ?∈?[0,?1) together with the averaged system with the external force g ₀(t), formally corresponding to the case ε?=?0. Under suitable assumptions on the external force, we prove as in [V.V. Chepyzhov, V. Pata, and M.M.I. Vishik, Averaging of 2D Navier–Stokes equations with singularly oscillating forces, Nonlinearity, 22 (2009), pp. 351–370] the boundness of the uniform global attractor 𝒜^ε as well as the convergence of the attractors 𝒜^ε of the singular systems to the attractor 𝒜⁰ of the averaged system as ε?→?0⁺. When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by Kε^(1?ρ). Let us note that the main difference between this work and that of Chepyzhov et al. (2009) is that the non-linearity involved in the three-dimensional primitive equation is stronger than the one in the two-dimensional Navier–Stokes equations considered in Chepyzhov et al. (2009), which makes the analysis of the problem studied in this article more involved.     

Kolmogorov ε-entropy of attractor for a non-autonomous strongly damped wave equation

In this paper, we study the long-time behavior of solutions for a non-autonomous strongly damped wave equation. We first prove the existence of a uniform attractor for the equation with a translation compact driving force and then obtain an upper estimate for the Kolmogorov ε-entropy of the uniform attractor. Finally we obtain an upper bound of the fractal dimension of the uniform attractor with quasiperiodic force.     

Convergence of the wave equation damped on the interior to the one damped on the boundary

In this paper, we study the convergence of the wave equation with variable internal damping term γn(x)ut to the wave equation with boundary damping γ(x)⊗δx∈∂Ωut when (γn(x)) converges to γ(x)⊗δx∈∂Ω in the sense of distributions. When the domain Ω in which these equations are defined is an interval in R, we show that, under natural hypotheses, the compact global attractor of the wave equation damped on the interior converges in X=H¹(Ω)×L²(Ω) to the one of the wave equation damped on the boundary, and that the dynamics on these attractors are equivalent. We also prove, in the higher-dimensional case, that the attractors are lower-semicontinuous in X and upper-semicontinuous in H1−ε(Ω)×H−ε(Ω).     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.     

Polynomially bounded solutions to the Loewner differential equation in several complex variables

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z,t)=etAz+?, where A∈L(Cn,Cn) has the property m(A)>0. Here m(A)=min{R〈A(z),z〉:‖z‖=1}. We also give sufficient conditions for g(z,t)=L(f(z,t)) to be polynomially bounded, where f(z,t) is an A-normalized polynomially bounded Loewner chain solution to the Loewner differential equation.