Electron. J. Diff. Eqns., Vol. 2000(2000), No. 72, pp. 1-19.
### Nonclassical Riemann solvers and kinetic relations III:
A nonconvex hyperbolic model for Van der Waals fluids

Philippe G. LeFloch & Mai Duc Thanh

**Abstract:**

This paper deals with the so-called p-system describing
the dynamics of isothermal and compressible fluids.
The constitutive equation is assumed to have the typical
convexity/concavity properties of the van der Waals equation.
We search for discontinuous solutions constrained by the
associated mathematical entropy inequality.
First, following a strategy proposed by Abeyaratne and Knowles
and by Hayes and LeFloch, we describe here the whole family of
nonclassical Riemann solutions for this model.
Second, we supplement the set of equations with a kinetic relation
for the propagation of nonclassical undercompressive shocks,
and we arrive at a uniquely defined solution of the Riemann problem.
We also prove that the solutions depend
L^{1}-continuously upon
their data.
The main novelty of the present paper is the presence of
two inflection points in the constitutive equation.
The Riemann solver constructed here is relevant for fluids
in which viscosity and capillarity effects are kept in balance.
Submitted June 7, 2000. Published December 1, 2000.

Math Subject Classifications: 35L65, 76N10, 76L05.

Key Words: compressible fluid dynamics, phase transitions, Van der Waals,
entropy inequality, hyperbolic conservation law, kinetic relation,
nonclassical solutions, Riemann solver.

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Philippe G. LeFloch

Centre de Mathematiques Appliquees

Centre National de la Recherche Scientifique, U.M.R. 7641,

Ecole Polytechnique, 91128 Palaiseau Cedex, France.

e-mail: lefloch@cmapx.polytechnique.fr
Mai Duc Thanh

Hanoi Institute of Mathematics, P.O. Box 631 BoHo,

10.000 Hanoi, Vietnam

e-mail: mdthanh@hanimath.ac.vn
thanh@cmapx.polytechnique.fr

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