Conditional stability and symmetry in hydrodinamics and mathematical biology
Convection with temperature dependent viscosity in a porous medium: nonlinear stability and the Brinkman effect.
We establish a nonlinear energy stability theory for the problem of convection in a porous medium when the viscosity depends on the temperature. This is, in fact, the situation which is true in real life and has many applications to geophysics. The nonlinear analysis presented here would appear to require the presence of a Brinkman term in the momentum equation, rather than just the normal form of Darcy's law.
Unconditional nonlinear stability in a polarized dielectric liquid
We derive a very sharp nonlinear stability result for the problem of thermal convection in a layer of dielectric fluid subject to an alternating current (AC). It is particularly important to note that the size of the initial energy in which we establish global nonlinear stability is not restricted whatsoever, and the Rayleigh-Roberts number boundary coincides with that found by a formal linear instability analysis.
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