The system of equations governing isothermal flow of an inviscid and compressible fluid in Lagrangian coordinates is given by (1.1) \(v_ t- u_ x=0\), \(u_ t+p(v)_ x=0\), where u, v, and p are the velocity, the specific volume, and the pressure of the fluid, respectively. For the isothermal flow of an ideal gas, \(p(v)=RT/v\), where R is a universal constant, and T is the constant absolute temperature. In this case, the system (1.1) is hyperbolic. On the other hand, the equation of state of a van der Waals fluid is (1.3) \(p(v)=(RT/v-b)-(a/v^ 2)\), where a and b are the characteristic constants of the fluid. Thus when the temperature is sufficiently low, \(p'(v)\) is positive, and thus the system (1.1) is elliptic on a certain interval (\(\alpha\),\(\beta)\), while \(p'(v)\) is negative on (b,\(\alpha)\) (liquid phase) as well as on (\(\beta\),\(\infty)\) (vapor phase) on which (1.1) is hyperbolic.
The Riemann problem for (1.1) is a special initial-value problem in which the initial data are of the form \[ (1.3)\quad (u(x,0),v(x,0))=(u_ 0,v_ 0),\quad x<0,\quad (u_ 1,v_ 1),\quad x>0. \] Since both the system (1.1) and the data (1.3) are invariant under the transformation (x,t)\(\to (\gamma x,\gamma t)\), \(\gamma >0\), the solution of (1.1), (1.3) is a function of x/t, i.e., it is a fan of waves that emanate from the origin and propagate with individual speeds. For hyperbolic problems the structure of solutions of the Riemann problem has been investigated thoroughly [T. P. Liu, Trans. Amer. Math. Soc. 199, 89-112 (1974; Zbl 0289.35063) and L. Leibovich, J. Math. Anal. Appl. 45, 81-90 (1974; Zbl 0273.35052)]. In particular, it is known that the problem specified by (1.1), and (1.3) generally admits several solutions. Admissibility criteria, such as the LAX entropy condition, the viscosity criterion, the entropy rate admissibility criterion, and the extended entropy condition, have been employed in order to single out a physically admissible solution.
Failure of uniqueness in the problem set by (1.1), and (1.3) arises also when the system is not hyperbolic, for instance for the van der Waals fluid (1.2), and so admissibility criteria have to be postulated there, too.
This paper has five sections. In Section 2 the author formulates the problem; in Section 3 he discusses the fundamental properties of the energy rate along the shock curve and the phase boundary curve. In Section 4 the possible solutions which join two constant states lying in different phases by backward waves, phase boundaries, and forward waves is treated. Also, he compares the energy rate for different solutions. Finally, in Section 5 a special Riemann problem is discussed and it is shown that the well known Maxwell construction is admissible according to the energy rate criterion.