Uniform Bound of the Highest Energy for the Three Dimensional Incompressible Elastodynamics

Abstract

This article concerns the time growth of Sobolev norms of classical solutions to the three dimensional incompressible isotropic elastodynamics with small initial displacements. Given initial data in ${H^k_\Lambda}$ for a fixed big integer k, the global well-posedness of this Cauchy problem has been established by Sideris and Thomases in (Commun Pure Appl Math 58(6):750–788, 2005) and (J Hyperbolic Differ Equ 3(4):673–690, 2006, Commun Pure Appl Math 60(12):1707–1730, 2007), where the highest-order generalized energy E _k (t) may have a certain growth in time. Alinhac conjectured that such a growth in time may be a true phenomenon, in (Geometric analysis of hyperbolic differential equations: an introduction, lecture note series: 374. Mathematical Society, London) he proved that E _k (t) is still uniformly bounded in time only for the three dimensional scalar quasilinear wave equation under a null condition. In this paper, we show that the highest-order generalized energy E _k (t) is still uniformly bounded for the three dimensional incompressible isotropic elastodynamics. The equations of incompressible elastodynamics can be viewed as nonlocal systems of wave type and are inherently linearly degenerate in the isotropic case. There are three ingredients in our proof: the first is that we still have a decay rate of ${t^{-\frac{3}{2}}}$ when we do the highest energy estimate away from the light cone even though in this case the Lorentz invariance is not available. The second one is that the ${L^\infty}$ norm of the good unknowns, in particular ${\nabla(v + G\omega)}$, is shown to have a decay rate of ${t^{-\frac{3}{2}}}$ near the light cone. The third one is that the pressure is estimated in a novel way as a nonlocal nonlinear term with null structure, as has been recently observed in [16]. The proof employs the generalized energy method of Klainerman, enhanced by weighted L ² estimates and the ghost weight introduced by Alinhac.