Global Solutions of the Equations of Elastodynamics of Incompressible Neo-Hookean Materials

We prove that the initial-value problem for the motion of a certain type of elastic body has a solution for all time if the initial data are sufficiently small. The body must fill all of three space, obey a ``neo-Hookean'' stress-strain law, and be incompressible. The proof takes advantage of the delayed singularity formation which occurs for solutions of quasi-linear hyperbolic equations in more than one space dimension. It turns out that the curl of the displacement of the body obeys such an equation. Thus using Klainerman's inequality, one derives the necessary estimates to gaurantee that solutions persist for all time.