Thermodynamics and stability of non-equilibrium steady states in open systems -- incompressible heat conducting viscous fluid subject to a temperature gradient

Thermodynamical arguments are known to be useful in the construction of physically motivated Lyapunov functionals for nonlinear stability analysis of spatially homogeneous equilibrium states in thermodynamically isolated systems. Unfortunately, the limitation to isolated systems is essential, and standard arguments are not applicable even for some very simple thermodynamically open systems. On the other hand, the nonlinear stability of thermodynamically open systems is usually investigated using the so-called energy method. The mathematical quantity that is referred to as the “energy” is, however, in most cases not linked to the energy in the physical sense of the word. Consequently, it would seem that genuine thermo-dynamical concepts are of no use in the nonlinear stability analysis of thermodynamically open systems. We show that this is not the case. In particular, we propose a construction that in the case of a simple heat conduction problem leads to a physically well-motivated Lyapunov type functional, which effectively replaces the artificial Lyapunov functional used in the standard energy method. The proposed construction seems to be general enough to be applied in complex thermomechanical settings.