A kinetic decomposition for singular limits of non-local conservation laws

Journal of Differential Equations

Volumn 247, Issue 12, 2009, Pages 3338-3356

  Kissling, Frederike ^a   LeFloch, Philippe G ^b   Rohde, Christian ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

^b UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Conservation law; Entropy solution; Kinetic decomposition; Nonclassical shock wave; Singular limit

Indexed keywords

EID: 70349322676     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.05.006     Document Type: Article

References (33)

1. Abeyaratne R., and Knowles J.K. Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114 (1991) 119-154
2. Anderson D.M., McFadden G.B., and Wheeler A.A. Diffuse-interface methods in fluid mechanics. Annual Reviews. Palo Alto, CA. Annu. Rev. Fluid Mech. vol. 30 (1998) 139-165
3. Bedjaoui N., and LeFloch P.G. Diffusive-dispersive traveling waves and kinetic relations. I. Nonconvex hyperbolic conservation laws. J. Differential Equations 178 (2002) 574-607
4. Bertozzi A.L., Münch A., and Shearer M. Undercompressive shocks in thin film flows. Phys. D 134 (1999) 431-464
5. Coclite G.M., and Karlsen K.H. A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation. Comm. Partial Differential Equations 31 (2006) 1253-1272
6. Correia J.M., and LeFloch P.G. Nonlinear diffusive-dispersive limits for multidimensional conservation laws. Advances in Nonlinear Partial Differential Equations and Related Areas. Beijing, 1997 (1998), World Sci. Publ., River Edge, NJ 103-123. http://arxiv.org/abs/0810.1880 available at:
7. Correia J.M., and LeFloch P.G. Nonlinear hyperbolic conservation laws. Nonlinear Evolution Equations and Their Applications. Macau, 1998 (1999), World Sci. Publ., River Edge, NJ 21-44
8. DiPerna R.J. Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88 (1985) 223-270
9. Fan H., and Slemrod M. Dynamic flows with liquid-vapor phase transitions. Handbook of Mathematical Fluid Dynamics, vol. I (2002), North-Holland, Amsterdam 373-420
10. Haink J., and Rohde C. Local discontinuous Galerkin schemes for model problems in phase transition theory. Commun. Comput. Phys. 4 (2008) 860-893
11. Hayes B.T., and LeFloch P.G. Nonclassical shocks and kinetic relations: Scalar conservation laws. Arch. Ration. Mech. Anal. 139 (1997) 1-56
12. Hayes B.T., and LeFloch P.G. Nonclassical shocks and kinetic relations: Strictly hyperbolic systems. SIAM J. Math. Anal. 31 (2000) 941-991
13. Hwang S. Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws. J. Differential Equations 225 (2006) 90-102
14. Hwang S. Singular limit problem of the Camassa-Holm type equation. J. Differential Equations 235 (2007) 74-84
15. Hwang S., and Tzavaras A.E. Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations. Comm. Partial Differential Equations 27 (2002) 1229-1254
16. Jacobs D., McKinney B., and Shearer M. Travelling wave solutions of the modified Korteweg de Vries-Burgers equation. J. Differential Equations 116 (1995) 448-467
17. Kondo C.I., and LeFloch P.G. Zero diffusion-dispersion limits for scalar conservation laws. SIAM J. Math. Anal. 33 (2002) 1320-1329
18. Kruzkov S. First-order quasilinear equations with several space variables. Math. USSR Sb. 10 (1970) 217-243
19. LeFloch P.G. Propagating phase boundaries: Formulation of the problem and existence via the Glimm scheme. Arch. Ration. Mech. Anal. 123 (1993) 153-197
20. LeFloch P.G. An introduction to nonclassical shocks of systems of conservation laws. In: Kröner D., Ohlberger M., and Rohde C. (Eds). Proc. International School on Hyperbolic Problems. Freiburg, Germany, Oct. 97. Lect. Notes Comput. Sci. Eng. vol. 5 (1999), Springer-Verlag 28-72
21. LeFloch P.G. Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves. Lectures Math. ETH Zürich (2002), Birkhäuser Verlag, Basel
22. LeFloch P.G., and Mohamadian M. Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations. J. Comput. Phys. 227 (2008) 4162-4189
23. LeFloch P.G., and Natalini R. Conservation laws with vanishing nonlinear diffusion and dispersion. Nonlinear Anal. 36 (1999) 213-230
24. Lions P.-L., Perthame B., and Tadmor E.T. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994) 169-191
25. Perthame B. Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl. 77 (1998) 1055-1064
26. Perthame B., and Souganidis P.E. A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. 31 (1998) 591-598
27. Presutti E. Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics, Theoretical and Mathematical Physics (2008), Springer-Verlag
28. Rohde C. Scalar conservation laws with mixed local and non-local diffusion-dispersion terms. SIAM J. Math. Anal. 37 (2005) 103-129
29. Rohde C. On local and non-local Navier-Stokes-Korteweg systems for liquid-vapour phase transitions. ZAMM Z. Angew. Math. Mech. 85 (2005) 839-857
30. Schonbek M.E. Convergence of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 7 (1982) 959-1000
31. Slemrod M. Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Anal. 81 (1983) 301-315
32. Truskinovsky L. Kinks versus shocks. In: Fosdick R., Dunn E., and Slemrod M. (Eds). Shock Induced Transitions and Phase Structures in General Media. IMA Vol. Math. Appl. vol. 52 (1993), Springer-Verlag, New York 185-229
33. Van Duijn C.J., Peletier L.A., and Pop I.S. A new class of entropy solutions of the Buckley-Leverett equation. SIAM J. Math. Anal. 39 (2007) 507-536