Zero diffusion-dispersion limits for scalar conservation laws

SIAM Journal on Mathematical Analysis

Volumn 33, Issue 6, 2002, Pages 1320-1329

  Kondo, Cezar I ^a   LeFloch, Philippe G ^a  

^a FEDERAL UNIVERSITY OF SÃO CARLOS   (Brazil)

Author keywords

A priori estimate; Conservation law; Diffusion; Dispersion; Entropy solution; Measure valued solution; Shock wave; Singular limit

Indexed keywords

EID: 0036434955     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141000374269     Document Type: Article

References (24)

1. P. Baiti, P.G. LeFloch, and B. Piccoli, Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems, J. Differential Equations, 172 (2001), pp. 59-82.
2. N. Bedjaoui and P.G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations I. Nonconvex hyperbolic conservation laws, J. Differential Equations, 178 (2002), pp. 574-607.
3. J. Correia and P.G. LeFloch, Nonlinear diffusive-dispersive limits for multidimensional conservation laws, in Advances in Nonlinear P.D.E.'s and Related Areas, G.Q. Chen et al., eds., World Scientific, River Edge, NJ, 1999, pp. 103-123.
4. R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), pp. 223-270.
5. B.T. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations: Scalar conservation laws, Arch. Ration. Mech. Anal., 139 (1997), pp. 1-56.
6. B.T. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations: Strictly hyperbolic systems, SIAM J. Math. Anal., 31 (2000), pp. 941-991.
7. B.T. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations: Finite difference schemes, SIAM J. Numer. Anal., 35 (1998), pp. 2169-2194.
8. D. Jacobs, W.R. McKinney, and M. Shearer, Traveling wave solutions of the modified Korteweg-deVries Burgers equation, J. Differential Equations, 116 (1995), pp. 448-467.
9. C. Kondo and P.G. LeFloch, Measure-valued solutions and well-posedness of multi-dimensional conservation laws in a bounded domain, Portugal. Math., 58 (2001), pp. 171-194.
10. S.N. Kružkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), pp. 285-355 (in Russian); Math. USSR-Sb., 10 (1970), pp. 217-243 (in English).
11. S.N. Kružkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), pp. 285-355 (in Russian); Math. USSR-Sb., 10 (1970), pp. 217-243 (in English).
12. P.D. LAX, The zero dispersion limit, a deterministic analogue of turbulence, Comm. Pure Appl. Math., 44 (1991), pp. 1047-1056.
13. P.D. Lax and C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. I, Comm. Pure Appl. Math., 36 (1983), pp. 253-290.
14. P.D. Lax and C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. II, Comm. Pure Appl. Math., 36 (1983), pp. 571-593.
15. P.D. Lax and C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation. III, Comm. Pure Appl. Math., 36 (1983), pp. 809-829.
16. P.G. LeFloch, An introduction to nonclassical shocks of systems of conservation laws, Proc. International School on Theory and Numerics for Conservation Laws, Freiburg, Germany, 20-24 Oct. 97, Lectures Notes Comput. Sci. Eng., D. Kröner, M. Ohlberger and C. Rohde, eds., Springer-Verlag, New York, 1999, pp. 28-73.
17. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Non-classical Shock Waves, E.T.H. Lecture Notes Series: Birkhäuser, 2002.
18. P.G. LeFloch and R. Natalini, Conservation laws with vanishing nonlinear diffusion and dispersion, Nonlinear Anal., 36 (1999), pp. 213-230.
19. P.G. LeFloch and C. Rohde, High-order schemes, entropy inequalities, and nonclassical shocks, SIAM J. Numer. Anal., 37 (2000), pp. 2023-2060.
20. P.G. LeFloch and M.D. Thanh, Nonclassical Riemann solvers and kinetic relations III. A non-convex hyperbolic model for van der Waals fluids, Electron. J. Differential Equations, 72 (2000), 19 pp.
21. F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), pp. 489-507.
22. M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations, 7 (1982), pp. 959-1000.
23. A. Szepessy, An existence result for scalar conservation laws using measure-valued solutions, Comm. Partial Differential Equations, 14 (1989), pp. 1329-1350.
24. L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. III, J.M. Ball, ed., Reidel, Dordrecht, The Netherlands, 1983, pp. 263-285.