Nonlinear evolution and stability of the heat flow in nanosystems: Beyond linear phonon hydrodynamics

A heat-transport equation incorporating nonlocal and nonlinear contributions of the heat flux is derived in the framework of weakly nonlocal nonequilibrium thermodynamics. The motivation for these terms arises from applications to nanosystems, where strong gradients are found, due to the small distance over which changes in temperature and heat flux take place. This equation generalizes to the nonlinear domain previous equations used in the context of phonon hydrodynamics. Compatibility with second law of thermodynamics is investigated and a comparison with the thermomass model of heat transport is carried out. The analogy between the equations describing the heat flow problem and the hydrodynamic equations is shown and the stability of the heat flow is analyzed in a special case.