Mathematical Sciences: Stability of Solutions of Nonlinear P.D.E.

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The primary concern of this mathematical research is the asymptotic stability of solutions of nonlinear partial differential equations. Recently developed geometric and topological tools will be exploited in the analysis of the spectrum of differential operators. The central idea is that of linearizing about a distinguished solution such as a steady state, a travelling wave or a radially symmetric solution. Such solutions play a central role in formulating a qualitative theory in that they can be thought of as asymptotic states for an associated evolution problem. In this regard it is of fundamental importance to develop methods for assessing their stability. This work based on abstract semigroup theory, uses information about the spectrum of an operator, obtained by linearizing about the solution in question, to predict its stability or instability. The particular obstacle to be overcome in any particular application is in accumulating sufficiently accurate estimates for the location and multiplicity of the potentially unstable eigenvalues of the operator. The stability analysis of several particular systems will be initiated. Applications to be considered are the stability of travelling wave solutions of higher order phase field models, and the stability of shock wave profiles on nonlinear conservation laws. The latter problem raises new issues, namely the role played by the continuous spectrum, but it is focused mainly on circumventing, rather than understanding it. Much of this work derives from concrete problems in the study of viscous profiles of shocks arising in isentropic gas dynamics and nonadiabatic tubular reactors.