Remarks on traveling waves and equilibria in fluid dynamics with viscosity, capillarity, and heat conduction

Heat conduction causes a tough obstacle in studying traveling waves in fluid dynamics. In this note we consider the fluid dynamics equations where viscosity, capillarity and heat conduction coefficients are present. First we transform the model into the one with an equation for the entropy as the conservation of energy. Then, given any traveling wave of the viscous–capillary–heat conductive model connecting two given states, we derive a corresponding system of differential equations. Then we show that this system of differential equations possesses the equilibria which correspond to the two states of the given traveling wave. This work may therefore motivate future study to solve challenging open questions on the stability of these equilibria and the existence of the traveling waves in fluid dynamics with heat conduction.