On the rate of decay of solutions to linear viscoelastic equation

Remark on the rate of decay of solutions to linearized compressible Navier–Stokes equations

“…But, the Green function of the system (1.1) and the linear viscoelastic equation is not written exactly. Tai-P. Liu and W. Wang [4] gave the pointwise estimate for the solutions to the system (1.1) and the nonlinear problem in odd multi-dimension case, and Y. Shibata [6] gave the L p estimate for the solution to the linear viscoelastic equations by directly using Fourier transform method. The main difference of the structure to the solutions between (1.1) or effective artificial viscosity system and linear viscoelastic equation is the Riesz kernel R j (x) = F −1 [ξ j /|ξ|] (x), where F −1 denotes the Fourier inverse transform.…”

“…As a result, the solution may grow without bound in L p for p < 2. This result was investigated by D. Hoff and K. Zumbrun [2,3] in the case of the Navier-Stokes system describing the compressible fluid flow, and Y. Shibata [6] in the case of the linear viscoelastic equation. D. Hoff and K. Zumbrun [2,3] considered the linear effective artificial viscosity system as the first approximation of the compressible Navier-Stokes equation in several space dimension.…”

“…D. Hoff and K. Zumbrun [2] overcame this difficulty by applying the weak version of the Paley-Wiener theorem to the general, symmetrizable, hyperbolic-strictly parabolic systems. In this paper, we shall estimate directly using Fourier transform method in [6]. In particular, we shall detect the cancellation in the Green function.…”

High-order energy decay for structural damped systems in the electromagnetical field

“…As a matter of fact, when we apply a certain viscoelastic effect to the damped term, the situation will change. This is demonstrated in [5].…”

Diffusion and wave behaviour in linear Voigt model

“…[10]- [18] ). In particular, the behaviour of solutions of (1.1) when ε → 0 has been analized in various applications of artificial viscosity method to non linear second order wave equation [19], [20].…”