GLOBAL WELL-POSEDNESS OF INCOMPRESSIBLE ELASTODYNAMICS IN THREE-DIMENSIONAL THIN DOMAIN

In this article, we prove global existence of classical solutions to the incompressible isotropic Hookean elastodynamics in three-dimensional thin domain Omega(delta) = R-2 x [0, delta] with periodic boundary condition. This system essentially consists of two-dimensional quasilinear wave-type equations. Following the classical vector field theory and the generalized energy method of Klainerman, we introduce the anisotropic weighted Sobolev-type inequalities, the anisotropic generalized energy, and the anisotropic weighted L-2 norm adapted to the thin domain. The main issue in the anisotropic generalized energy estimate is that the pressure gradient brings an extra singular (-1)(delta) factor arising from the partial derivative(2)(3) derivative. Based on the inherent cancellation structure, we introduce an auxiliary anisotropic generalized energy to overcome this difficulty. The ghost weight technique introduced by Alinhac [Invent. Math., 145 (2001), pp. 597-618] and the strong null condition introduced by Lei [Comm. Pure Appl. Math., 69 (2016), pp. 2072-2106] play important roles for temporal decay estimates.