Traveling waves of an elliptic–hyperbolic model of phase transitions via varying viscosity–capillarity

We consider an elliptic–hyperbolic model of phase transitions and we show that any Lax shock can be approximated by a traveling wave with a suitable choice of viscosity and capillarity. By varying viscosity and capillarity coefficients, we can cover any Lax shock which either remains in the same phase, or admits a phase transition. The argument used in this paper extends the one in our earlier works. The method relies on LaSalleʼs invariance principle and on estimating attraction region of the asymptotically stable of the associated autonomous system of differential equations. We will show that the saddle point of this system of differential equations lies on the boundary of the attraction region and that there is a trajectory leaving the saddle point and entering the attraction region. This gives us a traveling wave connecting the two states of the Lax shock. We also present numerical illustrations of traveling waves.