On traveling waves in viscous-capillary Euler equations with thermal conductivity

This work establishes the stability of the equilibrium states corresponding to traveling waves in viscous-capillary Euler equations when a standard thermal conductivity coefficient is present. Due to the presence of the heat conduction, the associated system of ordinary differential equations is much more involved and complicated. Given a shock wave, we can obtain exactly the sign of the real part of each eigenvalue of the Jacobian matrix of the corresponding system of ODEs at the two equilibria associated with the left-hand and right-hand states of the shock. It turns out that one equilibrium point is asymptotically stable, why the other point is unstable and admits eigenvalues whose real parts have opposite signs. Suitable approximate connections between the unstable and stable equilibria are obtained by numerical tests for various ranges of thermal conductivity: low, medium, and high values. Moreover, numerical tests also suggest that a trajectory could leave the unstable equilibrium point and enters the attraction domain of the asymptotically stable equilibrium point. This work therefore may motivate for further study on the existence of the traveling waves of the viscous-capillary gas dynamics equations with the presence of heat conduction.