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

Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, UMR 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France

INSSET, Université de Picardie, 48 rue Raspail, 02109 Saint-Quentin, France

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and

P. G. LEFLOCH

Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, UMR 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France

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https://doi.org/10.1142/S0252959903000037Cited by:12 (Source: Crossref)

Abstract

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 assumption 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 autors 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.

Document Code: A, Article ID: 0252-9599(2003)01-0017-18.

Keywords:

References

R. Abeyaratne and J. K. Knowles, Arch. Rational Mech. Anal. 114, 119 (1991), DOI: 10.1007/BF00375400. Crossref, Google Scholar
R. Abeyaratne and J. K. Knowles, SIAM J. Appl. Math. 51, 1205 (1991), DOI: 10.1137/0151061. Crossref, Google Scholar
N. Bedjaoui and P. G. LeFloch, Ann. Univ. Ferrara Sc. Mat. 47, 117 (2001). Crossref, Google Scholar
N. Bedjaoui and P. G. LeFloch, J. Differential Equations 178, 574 (2002), DOI: 10.1006/jdeq.2000.4009. Crossref, Google Scholar
N. Bedjaoui and P. G. LeFloch, Proc. Royal Soc. Edinburgh 133A, (2002). Google Scholar
D. Gilbard, Amer. J. Math. 7, 256 (1951). Google Scholar
B. T. Hayes and P. G. LeFloch, Arch. Rational Mech. Anal. 139, 1 (1997), DOI: 10.1007/s002050050046. Crossref, Google Scholar
B. T. Hayes and P. G. LeFloch, SIAM J. Math. Anal. 31, 941 (2000), DOI: 10.1137/S0036141097319826. Crossref, Google Scholar
D. Jacobs, W. R. McKinney and M. Shearer, J. Differential Equations 116, 448 (1995), DOI: 10.1006/jdeq.1995.1043. Crossref, Google Scholar
P. G. LeFloch, Arch. Rational Mech. Anal. 123, 153 (1993). Crossref, Google Scholar
P. G. LeFloch, An introduction to nonclassical shocks of systems of conservation laws, Proceedings of the "International School on Theory and Numerics for Conservation Laws", Lecture Notes in Computational Science and Engineering, eds. D. Kröner, M. Ohlberger and C. Rohde (1998) pp. 28–72. Google Scholar
P. G. LeFloch , ETH Lecture Notes , Hyperbolic systems of conservation laws : Classical and nonclassical shock waves and kinetic relations ( Birkhäuser , 2002 ) . Google Scholar
R. L. Pego, Arch. Rational Mech. Anal. 94, 165 (1986), DOI: 10.1007/BF00280432. Crossref, Google Scholar
S. Schulze and M. Shearer, J. Math. Anal. Appl. 229, 344 (1999), DOI: 10.1006/jmaa.1998.6186. Crossref, Google Scholar
M. Shearer and Y. Yang, Proc. Royal Soc. Edinburgh. 125A, 675 (1995). Google Scholar
M. Slemrod, Arch. Rational Mech. Anal. 81, 301 (1983), DOI: 10.1007/BF00250857. Crossref, Google Scholar
M. Slemrod, Arch. Rational. Mech. Anal. 83, 333 (1983). Google Scholar
M. Slemrod, Non-classical continuum mechanics, Vanishing viscosity-capillarity approach to the Riemann problem for a van der Waals fluid 122 (Cambridge Univ. Press, Cambridge, 1987) pp. 325–335. Crossref, Google Scholar
L. Truskinovsky, J. Appl. Math. and Mech. (PMM) 51, 777 (1987). Crossref, Google Scholar
L. Truskinovsky, Shock induced transitions and phase structures in general media, Kinks versus shocks, eds. R. Fosdick, E. Dunn and H. Slemrod (Springer-Verlag, 1993) pp. 185–229. Crossref, Google Scholar
H. Weyl, Comm. Pure Appl. Math. 2, 103 (1949), DOI: 10.1002/cpa.3160020201. Crossref, Google Scholar