UC Santa Barbara

The first result in this dissertation concerns wave equations in three space dimensions with small O(v) viscous dissipation and O(d) non-null quadratic nonlinearities. Small O(e) solutions are shown to exist globally provided that ed/v is sufficiently small. When this condition is not met, small solutions exist almost globally, and in certain parameter ranges, the addition of dissipation enhances the lifespan. We study next a system of nonlinear partial differential equations modeling the motion of incompressible Hookean isotropic viscoelastic materials. The nonlinearity inherently satisfies a null condition and our second result establishes global solutions with small initial data independent of viscosity. In the proofs we use vector fields, energy estimates, and weighted decay estimates.