General class of nonlinear bifurcation problems from a point in the essential spectrum: application to shock wave solutions of kinetic equations

Abstract

An abstract class of bifurcation problems is investigated from the essential spectrum of the associated Frechet derivative. This class is a very general framework for the theory of one-dimensional, steady-profile traveling- shock-wave solutions to a wide family of kinetic integro-differential equations from nonequilibrium statistical mechanics. Such integro-differential equations usually admit the Navier--Stokes system of compressible gas dynamics or the MHD systems in plasma dynamics as a singular limit, and exhibit similar viscous shock layer solutions. The mathematical methods associated with systems of partial differential equations must, however, be replaced by the considerably more complex Bifurcation Theory setting. A hierarchy of bifurcation problems is considered, starting with a simple bifurcation problem from a simple eigenvalue. Sections are entitled as follows: introduction and background from mechanics; the mathematical problem: principal results; a generalized operational calculus, and the derivation of the generalized Lyapunov--Schmidt equations; and methods of solution for the Lyapunov--Schmidt and the functional differential equations. 1 figure. (RWR