Georgia Institute of Technology College of Sciences

We will give an overview of results on the global existence of solutions to the initial value problem for nonlinear elastic and viscoelastic materials in 3d without boundary. Materials will be assumed to be isotropic, but both compressible and incompressible cases will be discussed. In the compressible case, a key null condition must be imposed to control nonlinear interactions of pressure waves. This necessary assumption is consistent with the physical model. Initial conditions are small perturbations of a stress free reference state. Existence is proven using a fixed point argument which combines energy estimates and with some new dispersive estimates.