DIFFUSIVE-DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV. COMPRESSIBLE EULER EQUATIONS

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumptions and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals’ equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The authors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.