Stability of a Hot Smoluchowski Fluid

Details Stability of a Hot Smoluchowski Fluid Stability of a Hot Smoluchowski Fluid

2001-03-21

arxiv.org/abs/math-ph/0103026v1

Physics

Mathematical Physics

9 pages, to appear in "Open Systems and Information Dynamics", Kluwer

Scientific paper

We study coupled non-linear parabolic equations for a fluid described by a material density and a temperature, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearised equations for small perturbations about a stationary solution are derived; they are subject to the boundary conditions, Dirichlet for the temperature and no-flow conditions for the material. The spectrum of the generator L of time-evolution, regarded as an operator on the Hilbert space of square-integrable functions on [0,1], 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; its numerical range is the whole of the complex plane.

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