Almost global existence for 2-D incompressible isotropic elastodynamics

Abstract

We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval [ 0 , T / ϵ ] [0,{T}/{\epsilon }] for initial data of the form ϵ U 0 \epsilon U_0 , where T T depends only on some Sobolev norm of U 0 U_0 . We show that for such data there exists a unique solution on a time interval [ 0 , exp ⁡ T / ϵ ] [0, \exp {T}/{\epsilon }] , provided that ϵ \epsilon is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted L 2 L^2 estimates and the ghost weight introduced by Alinhac.