Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion

This work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity effects proposed in our earlier work Thanh (2010) [26]. Precisely, we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients. The shock wave can be classical or nonclassical. Interestingly, we in particular show the existence of a traveling wave for a given Lax shock but rather nonclassical when the straight line connecting the two left-hand and right-hand states crosses the graph of the pressure function two more times in the middle region. Furthermore, we also discuss all the possibilities of saddle–stable, saddle–saddle, or stable–stable connections.