High Frequency Analysis of Families of Solutions to the Equation of Viscoelasticity of Kelvin–voigt

Abstract

In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.

Microlocal defect measures for a degenerate thermoelasticity system

“…They crucially use a result of Henry et al [18] which shows that the semigroup associated to (1.1) is equal up to a compact operator to the semi-group of a system consisting of a damped wave equation coupled with a heat equation on the temperature θ (see the Appendix for details). The method is based on the use of microlocal defect measures, and similar works have been achieved for the Lamé system in [7], for the equations of magnetoelasticity in [10], and for the equation of viscoelastic waves by the authors [2]. One obtains that the energy propagates along the rays of the geometric optic associated with the wave operator ∂ 2 t − ∆ with a damping depending simultaneously on the position and the speed of the trajectory.…”

“…Then, one associates with the sequence u n (t, x) its H 1 space-time microlocal defect measure µ by the following: up to extraction of a subsequence, for all q = q b + q i ∈ A 2 admitting a principal symbol, (2) q ∞ (z, ζ )dµ(z, ζ ).…”

“…Heuristically, following a bicharacteristic which reaches ∂Ω in a hyperbolic point, the measure is totally restituted (since µ = 0 above H) and the restitution is described by 2 1 H , which means propagation along the reflected curve. Heuristically, following a bicharacteristic which reaches ∂Ω in a hyperbolic point, the measure is totally restituted (since µ = 0 above H) and the restitution is described by 2 1 H , which means propagation along the reflected curve.…”

Non-structural controllability of linear elastic systems with structural damping

“…The wave equation with square root damping (α = 1/2) is just out of reach and seems to us an interesting open problem (the appendix in [20] proves that it is not controllable by a one-dimensional input). It results from [1] that the wave equation (γ = 1) with Kelvin-Voigt damping (α = 1) is not controllable from any Ω = M. It results from [5] that the damped wave equation (γ = 1, α = 0), is controllable from Ω or not depending on whether the control time is greater or lower than L Ω (defined before Theorem 2).…”

“…The structural vibration modes of the conservative system represented by (1) with B = D = 0 are prescribed by the positive self-adjoint operator S. This ideal system is perturbed by a dissipative mechanism prescribed by the positive self-adjoint operator D. The system is actuated through a control mechanism prescribed by the operator B (possibly unbounded to take into account trace operators prescribing the boundary value of distributed states). Throughout this paper, controllability will always mean the ability of steering any initial state (z(0),ż(0)) to zero over a finite time by some appropriate input function u (i.e.…”