Pointwise estimates for the stability of a scalar conservation law

iscous conservation laws are a fundamental class of partial differential equations that arise in such important areas as gas and fluid dynamics, the behavior of materials and magnetohydrodynamics (MHD). A viscous profile solution to such an equation is a solution whose form is preserved as it translates in a fixed direction.;One question that arises in the study of viscous conservation laws is whether a given viscous profile solution will persist in time, or be asymptotically stable. The stability of viscous profiles is of considerable interest, as it is closely related to such fundamental issues as the convergence of numerical schemes, the validity of matched asymptotic expansion, and convergence in the inviscid limit. In this thesis we obtain two results pertaining to the stability of scalar viscous conservation laws.;Result 1. First, pointwise estimates are obtained on the Green's function for the scalar linear convection-diffusion equations that arise when a scalar conservation law with nonconstant diffusion is linearized about a viscous shock profile of arbitrary strength. These estimates take the form of Gaussian kernels centered around paths determined by the (typically different) asymptotic states of the convection function. The analysis extends the spectral transform method to the nonconstant coefficient case.;Result 1 yields numerous applications. For one, these estimates lead to tight pointwise estimates on perturbations of viscous profiles, as discussed below under Result 2. Further, the methods are suitable for extension to the case with systems and to higher order scalar equations such as those involving dispersion.;Result 2. For Result 2, we apply the pointwise estimates of Result 1 to the study of pointwise behavior of the perturbation of viscous shock solutions to a scalar conservation law, obtaining a result independent of shock strength. We find that for...