Skip to main content
eScholarship
Open Access Publications from the University of California

UC Santa Barbara

UC Santa Barbara Electronic Theses and Dissertations bannerUC Santa Barbara

Longtime behavior of small solutions to viscous perturbations of nonlinear hyperbolic systems in 3D

Abstract

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.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View