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

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 L1-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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