Abstract
We give a necessary and sufficient condition, of geometrical type, for the uniform decay of energy of solutions of the linear system of magnetoelasticity in a bounded domain with smooth boundary. A Dirichlet-type boundary condition is assumed. Our strategy is to use microlocal defect measures to show suitable observability inequalities on high-frequency solutions of the Lamé system.
Highlights
Let Ω be a bounded, connected domain of R3, with a smooth boundary
A B-resistant ray, whose direction of propagation is orthogonal to B in the transversal case and parallel to B in the longitudinal case carries essentially the component of v which is parallel to B, cancelling the dissipative term R(v) := curl (∂tv ∧ B)
We shall introduce a measure describing, from a microlocal point of view, the defect of compactness in H1 of the sequence. This description is of fundamental importance to show the observability inequalities of the preceding section, since the Lame system decomposes into two waves equation
Summary
Let Ω be a bounded, connected domain of R3, with a smooth boundary. Let us consider the following system, modelling the displacement of an elastic solid in a magnetic field:. Lebeau [3] have given (under a spectral assumption) a necessary and sufficient condition on Ω, of geometrical nature, for the uniform decay in dimension 2 or 3 This decay is equivalent to the non-existence of rays, called “transversal polarization rays”, carrying the transversal component of v (the divergence-free component), which resists to the dissipation. A B-resistant ray, whose direction of propagation is orthogonal to B in the transversal case and parallel to B in the longitudinal case carries essentially the component of v which is parallel to B, cancelling the dissipative term R(v) := curl (∂tv ∧ B) From this point of view, Theorem 1.2 is very natural.
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