Hodge-Carleman estimates and global attractors for heterogeneous elastic waves with localized damping on Riemannian manifolds

Abstract

We study the long-time dynamics of the semilinear Lamé system of isotropic elasticity on a compact Riemannian manifold with boundary, for a heterogeneous medium whose density and Lamé moduli vary in space, under a frictional dissipation which acts only on an open set $ω$. The elastic operator is the divergence of the Hooke stress tensor, and the starting point is a structural identity which shows that the dilatation and the vorticity of the displacement satisfy wave equations with the pressure and the shear speeds, respectively, coupled to the displacement only through first-order terms. A single weight function, free of critical points and strictly convex with respect to the two ray metrics of the medium, then serves for the displacement, its dilatation and its vorticity alike, and this Hodge structure yields Carleman estimates for the Lamé operator with variable coefficients at two energy levels, provided that $ω$ contains a neighbourhood of the boundary and that such a weight exists on the undamped region. The hypothesis is fulfilled on every compact manifold with boundary by damping regions of arbitrarily small volume. From these estimates we derive unique continuation for elastic waves with bounded, merely measurable, time-dependent potentials and an internal observability inequality, which in turn yield, for subcritical restoring forces in dimension three, a compact global attractor which coincides with the unstable manifold of the equilibria, has finite fractal dimension and is bounded in $(H^2\cap H^1_0)\times H^1_0$, together with a generalized exponential attractor.