GLOBAL EXISTENCE OF INHOMOGENEOUS INCOMPRESSIBLE ISOTROPIC ELASTODYNAMICS IN THREE DIMENSIONS

This paper concerns the Cauchy problem of the incompressible isotropic elastodynamics in three dimensions. The homogeneous case was studied by Sideris and Thomases [Comm. Pure Appi. Math., 58 (2005), pp. 750-788; Comm. Pure Appl. Math., 60 (2007), pp. 1707-1730] in Euler coordinates. Here we generalize their result to the inhomogeneous case and prove the global existence of classical solution in Lagrangian coordinates under the assumption that the density is a small perturbation around a constant state. The generalized energy method is utilized to derive the energy estimates in certain weighted Sobolev spaces.