Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

Exponential stability of an elastic string with local Kelvin–Voigt damping

This paper is devoted to analyzing an elastic string with local Kelvin–Voigt damping. We prove the exponential stability of the system when the material coefficient function near the interface is smooth enough. Our method is based on the frequency method and semigroup theory.     

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The damping is localized in a suitable open subset of the domain under consideration. The exponential stability result proposed by Liu and Rao for that system assumes that the damping is localized in a neighborhood of the whole boundary, and the damping coefficient is continuously differentiable with a bounded Laplacian. We propose a new solution to the exponential stability problem based on the introduction of a new variable, and a constructive frequency domain approach. The main features of our method are: (i) the damping region need not be a neighborhood of the whole boundary; (ii) the damping coefficient is assumed to be bounded measurable with bounded measurable gradient only; (iii) the introduction of a new variable. These features enable us to improve on the damping coefficient smoothness and more especially on the feedback control region. Further, when combined with a recent result of Borichev and Tomilov on the polynomial decay of bounded semigroups, the new method enables us to prove a polynomial decay estimate of the energy when the damping coefficient is bounded measurable only.     

Two-grid stabilized algorithms for the steady Navier–Stokes equations with damping

This paper presents and studies three two-grid stabilized quadratic equal-order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two-grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two-grid algorithms save a large amount of CPU time than the one-grid stabilized algorithm.     

Lack of exponential stability to Timoshenko system with viscoelastic Kelvin–Voigt type

We study the Timoshenko systems with a viscoelastic dissipative mechanism of Kelvin–Voigt type. We prove that the model is analytical if and only if the viscoelastic damping is present in both the shear stress and the bending moment. Otherwise, the corresponding semigroup is not exponentially stable no matter the choice of the coefficients. This result is different to all others related to Timoshenko model with partial dissipation, which establish that the system is exponentially stable if and only if the wave speeds are equal. Finally, we show that the solution decays polynomially to zero as \({t^{-1/2}}\) , no matter where the viscoelastic mechanism is effective and that the rate is optimal whenever the initial data are taken on the domain of the infinitesimal operator.     

On a system of nonlinear wave equations with Balakrishnan–Taylor damping

In this paper, we study the initial-boundary value problem for a coupled system of nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan–Taylor damping terms. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Also, we show that nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of stronger damping.     

Mixed initial–boundary value problem for equations of motion of Kelvin–Voigt fluids

The initial–boundary value problem for equations of motion of Kelvin–Voigt fluids with mixed boundary conditions is studied. The no-slip condition is used on some portion of the boundary, while the impermeability condition and the tangential component of the surface force field are specified on the rest of the boundary. The global-in-time existence of a weak solution is proved. It is shown that the solution is unique and depends continuously on the field of external forces, the field of surface forces, and initial data.     

Existence of weak solutions for the generalized Navier–Stokes equations with damping

In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any ${q > \frac{2N}{N+2}}$ and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.     

General decay rate estimates for viscoelastic wave equation with Balakrishnan–Taylor damping

In this paper, we consider the viscoelastic wave equation with Balakrishnan–Taylor damping. This work is devoted to prove uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on behavior of the relaxation function at infinity.     

Trajectory and Global Attractors for a Modified Kelvin–Voigt Model

Journal of Applied and Industrial Mathematics - We study the qualitative behavior of weak solutions to an autonomous modified Kelvin–Voigt model on the base of the theory of attractors for...     

Global stability of viscous contact wave for 1-D compressible Navier–Stokes equations

The viscous contact waves for one-dimensional compressible Navier–Stokes equations has recently been shown to be asymptotically stable. The stability results are called local stability or global stability depending on whether the norms of initial perturbations are small or not. Up to now, local stability results toward viscous contact waves of compressible Navier–Stokes equations have been well established (see Huang et al., 2006, 2008, 2009 [9], [10], [7]), but there are few results for the global stability in the case of Cauchy problem which is the purpose of this paper. The proof is based on an elementary energy method using an inequality concerning the heat kernel (see Lemma 1 of Huang et al., 2010 [7]).