Stability of a hot Smoluchowski fluid

Open Systems and Information Dynamics, 7, 1-9, 2000. Archived as math-ph/0103026. The local preprint number is KCL-MTH-01-27.

ABSTRACT

We study coupled nonlinear parabolic equations for a fluid described by a material density rho and a temperature Theta, both functions of space and time. In one dimension, the linearised equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for Theta and no-flow for the material.

The spectrum of the generator L of time-evolution is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is C.

Further analysis of the linearised equation, and generalisations, has been carried out by Lyonell Boulton, in a paper "Spectral behaviour of a simple non-self-adjoint operator", King's College preprint, 2001, found at arXiv:math.SP/0102170v2.