Strichartz and uniform Sobolev inequalities for the elastic wave equation

Abstract

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.

Pointwise convergence for the elastic wave equation

“…Recently, the three of the authors and Lee [6] diagonalized the Lamé operator so that ∆ * = P ∆P −1 with a certain invertible matrix P to study the elastic wave equation. We utilize the diagonalization to prove Theorem 1.1 in the following sections.…”

“…Let us recall from [6, Section 2] the diagonalization process and notations to make this article self-contained (the readers are encouraged to refer to [6] for details; also see Figure 2). For the unit vector e 1 = (1, 0, .…”

Spectral enclosures for the damped elastic wave equation

“…From these results Strichartz estimates for the evolution equation followed (in the same manner as for classical wave equation, see [11,12]). These Strichartz estimates were then generalized in [23,24]. In particular in [23] the endpoint case is deduced.…”

“…These Strichartz estimates were then generalized in [23,24]. In particular in [23] the endpoint case is deduced.…”

“…Surprisingly, differently from the Laplacian, in [26] the authors also showed the failure of uniform Sobolev and Carleman inequalities for the Lamé operator. In [23] it was proved that if spacial lower-order perturbations are replaced by temporal ones, i.e. if one considers the damped equation, then those estimates become available again.…”