Stability of a Hot Smoluchowski Fluid

TL;DR

This paper analyzes the stability of stationary solutions in a coupled fluid model involving density and temperature, demonstrating spectral properties of the linearized operator that imply stability in a one-dimensional setting.

Contribution

It derives the spectral properties of the linearized operator for a coupled density-temperature fluid model, establishing conditions for stability of stationary solutions.

Findings

Spectrum of the linearized operator is real, discrete, and non-positive.

Stationary solutions are stable under small perturbations.

The linearized operator is not sectorial, with a numerical range covering the entire complex plane.

Abstract

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.