On Traveling Waves in Compressible Euler Equations with Thermal Conductivity

Heat conduction plays an important role in fluid dynamics. However, the modeling of thermal conductivity involves higher order derivatives which causes a tough obstacle for the study of traveling waves. In this work, we propose a modified term for the thermal conductivity coefficient in viscous–capillary compressible Euler equations. By approximation, which is crucial in any mathematical modeling, the heat conduction may be assumed to depend only on the specific volume. Then, we can derive a 2×2 system of first-order differential equations for traveling waves of the given model, whose equilibria can be shown to admit a stable–saddle connection for 1-shocks and a saddle–stable connection for 3-shocks. This establishes the existence of a traveling wave of the viscous–capillary Euler equations with the presence of a modified thermal conductivity effect.