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An infinite, horizontal layer of viscous fluid is considered, the fluid being everywhere homogeneous and incompressible, except that (i) as usual, the Boussinesq approximation leads to temperature-induced changes in density solely with a view to simulating buoyancy forces, (ii) the diffusivity depends in an arbitrary manner on the temperature, but is bounded below by a given positive constant. The upper and lower faces of the layer are each subject to constant temperatures, that on the bottom being higher than that on the top. The steady, motionless equilibrium state is identified, and its nonlinear stability to spatially periodic perturbations in the plane of the layer is investigated. A Liapunov functional, or measure, is identified, and the perturbations are shown to be asymptotically, exponentially stable in this measure, provided that a certain parameter (called the Rayleigh number), descriptive of the thermal gradient, is sufficiently small. Both fixed-fixed and free-free boundary conditions for velocity are considered

A novel Liapunov functional was used in previous work to establish nonlinear stability of certain nontrivial equilibrium states; essentially the context was that of pure nonlinear diffusion. This paper uses the same Liapunov functional to derive a nonlinear stability criterion in the context of a highly nonlinear system of p.d.e.'s involving nonlinear diffusion as an element. The context is that of convection of a thermofluid (i) conforming to Darcy's law and the Boussinesq approximation, (ii) with temperature dependent thermal diffusivity and viscosity, in an infinite vertical slab of porous material. The vertical faces are held at different constant temperatures, a steady state is identified, and is shown to be nonlinearly stable provided that the Rayleigh number does not exceed a quantity which reflects the temperature dependence of the pertinent physical properties. It may be that the versatility of the Liapunov functional thus exhibited may extend to other nonlinear systems involving nonlinear diffusion.

A nonlinear double diffusive system is the subject of this article: we consider two simultaneous p.d.e.'s in two dependent variables, first order in time and second order in the spatial variables. Dirichlet boundary conditions, independent of time, are assumed. The principal concern of the article is the stability of the steady states together with the convergence of the unsteady states thereto. Novel Liapunov functionals are used to this end. Both linear and nonlinear stability are discussed together with some aspects of (linear) instability. Prior to these considerations, some relevant remarks are made concerning uniqueness and nonuniqueness of the steady states

A novel, very effective Liapunov functional was used in previous papers to derive decay and asymptotic stability estimates (with respect to time) in a variety of thermal and thermo-mechanical contexts. The purpose of this note is to show that the versatility of this functional extends to certain non-linear elliptic boundary value problems in a right cylinder, the axial co-ordinate in this context replacing the time variable in the previous one. A steady-state temperature problem is considered with Dirichlet boundary conditions, the condition on the boundary being independent of the axial co-ordinate. The functional is used to obtain an estimate of the error committed in approximating the temperature field by the two-dimensional field induced by the boundary condition on the lateral surface. Copyright © 2002 John Wiley & Sons, Ltd.