Almost Global Existence for 2-D Incompressible Isotropic Elastodynamics

We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/{\epsilon}]$, for initial data of the form $\epsilon U_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/{\epsilon}]$, provided that $\epsilon$ 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 $L^2$ estimates and the ghost weight introduced by Alinhac.