Geodesic
Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for Van der Waals fluids
Electronic journal of differential equations, Tome 2000 (2000)
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.
Classification : 35L65, 76N10, 76L05
Keywords: compressible fluid dynamics, phase transitions, van der Waals, entropy inequality, hyperbolic conservation law, kinetic relation, nonclassical solutions, Riemann solver 
@article{EJDE_2000__2000__a44,
     author = {LeFloch,  Philippe G. and Mai Duc Thanh},
     title = {Nonclassical {Riemann} solvers and kinetic relations {III:} {A} nonconvex hyperbolic model for {Van} der {Waals} fluids},
     journal = {Electronic journal of differential equations},
     year = {2000},
     volume = {2000},
     zbl = {0962.35121},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a44/}
}
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AU  - LeFloch,  Philippe G.
AU  - Mai Duc Thanh
TI  - Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for Van der Waals fluids
JO  - Electronic journal of differential equations
PY  - 2000
VL  - 2000
UR  - http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a44/
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%A LeFloch,  Philippe G.
%A Mai Duc Thanh
%T Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for Van der Waals fluids
%J Electronic journal of differential equations
%D 2000
%V 2000
%U http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a44/
%G en
%F EJDE_2000__2000__a44
LeFloch,  Philippe G.; Mai Duc Thanh. Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for Van der Waals fluids. Electronic journal of differential equations, Tome 2000 (2000). http://geodesic.mathdoc.fr/item/EJDE_2000__2000__a44/