Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Global existence to the Cauchy problem for hyperbolic conservation laws with an isolated umbilic point


Authors: Elisabetta Felaco, Bruno Rubino and Rosella Sampalmieri
Journal: Quart. Appl. Math. 71 (2013), 629-659
MSC (2010): Primary 35L65, 35L80
DOI: https://doi.org/10.1090/S0033-569X-2013-01328-6
Published electronically: August 28, 2013
MathSciNet review: 3136988
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Abstract: In this paper the existence of global weak solutions for a $ 2 \times 2$ system of non-strictly hyperbolic non-linear conservation laws is established for data in $ L^{\infty }$.

The result is proven by means of viscous approximation and application of the compensated compactness method.

The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain an existence result. For this purpose we combine the classical techniques referring to a singular Euler-Poisson-Darboux equation with the compensated compactness method.


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  • 1. G.-Q. Chen and P.T. Kan, Hyperbolic conservation laws with umbilic degeneracy. I, Arch. Rational Mech. Anal. 130 (1995), no. 3, 231-276. MR 1337115 (96c:35111)
  • 2. -, Hyperbolic conservation laws with umbilic degeneracy, Arch. Ration. Mech. Anal. 160 (2001), no. 4, 325-354. MR 1869669 (2003b:35130)
  • 3. K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), no. 2, 373-392. MR 0430536 (55:541)
  • 4. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1989, Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication. MR 1013360 (90k:35001)
  • 5. R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27-70. MR 684413 (84k:35091)
  • 6. -, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1-30. MR 719807 (85i:35118)
  • 7. -, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), no. 2, 383-420. MR 808729 (87g:35148)
  • 8. L.C. Evans, Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
  • 9. H. Frid and M.M. Santos, Nonstrictly hyperbolic systems of conservation laws of the conjugate type, Communications in partial differential equations 19 (1994), no. 1-2, 27-59. MR 1256997 (95d:35100)
  • 10. X. Garaizar, The small anisotropy formulation of elastic deformation, Acta Appl. Math. 14 (1989), no. 3, 259-268. MR 995288 (90j:73021)
  • 11. J. Glimm, The interaction of nonlinear hyperbolic waves, Comm. Pure Appl. Math. 41 (1988), no. 5, 569-590. MR 948072 (89h:35198b)
  • 12. E.L. Isaacson, D. Marchesin, C.F. Palmeira, and B.J. Plohr, A global formalism for nonlinear waves in conservation laws, Comm. Math. Phys. 146 (1992), no. 3, 505-552. MR 1167301 (93h:35124)
  • 13. P.T. Kan, On the Cauchy problem of a $ 2\times 2$ system of non-strictly hyperbolic conservation laws, Courant Institute of Math. Sciences, N.Y. University (1989), (Ph.D. Thesis, Advisor: J. Glimm).
  • 14. -, Non-strictly hyperbolic conservation laws, Appl. Math. Lett. 4 (1991), no. 4, 83-87. MR 1117777 (92c:35077)
  • 15. B.L. Keyfitz and H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219-241. MR 549642 (80k:35050)
  • 16. P.D. Lax, The multiplicity of eigenvalues, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 213-214. MR 640948 (83a:15009)
  • 17. Y. G. Lu, Convergence of the viscosity method for a non-strictly hyperbolic conservation law, Comm. Math. Phys. 150 (1992), no. 1, 59-64. MR 1188496 (94k:35190)
  • 18. P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differential Equations 84 (1990), no. 1, 129-147. MR 1042662 (91i:35156)
  • 19. P. Marcati and B. Rubino, Entropy methods for nonstrictly hyperbolic systems, J. Partial Differential Equations 10 (1997), no. 4, 333-346. MR 1486714 (99f:35135)
  • 20. F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 3, 489-507. MR 506997 (80h:46043a)
  • 21. B. Rubino, Approximate solutions to the Cauchy problem for a class of $ 2\times 2$ nonstrictly hyperbolic systems of conservation laws, Boll. Un. Mat. Ital. B (7) 8 (1994), no. 3, 583-614. MR 1294451 (95h:35135)
  • 22. -, Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a $ 2\times 2$ nonstrictly hyperbolic system of conservation laws, Quart. Appl. Math. 53 (1995), no. 3, 401-421. MR 1343459 (96h:35122)
  • 23. D.G. Schaeffer and M. Shearer, The classification of $ 2\times 2$ systems of nonstrictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. 40 (1987), no. 2, 141-178. MR 872382 (88a:35155)
  • 24. -, Riemann problems for nonstrictly hyperbolic $ 2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc. 304 (1987), no. 1, 267-306. MR 906816 (88m:35101)
  • 25. D. Serre, La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à une dimension d'espace, J. Math. Pures Appl. (9) 65 (1986), no. 4, 423-468. MR 881690 (88d:35123)
  • 26. -, Systems of conservation laws. 2, Cambridge University Press, Cambridge, 2000, Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon. MR 1775057 (2001c:35146)
  • 27. M. Shearer, Loss of strict hyperbolicity of the Buckley-Leverett equations for three-phase flow in a porous medium, Numerical simulation in oil recovery (Minneapolis, Minn., 1986), IMA Vol. Math. Appl., vol. 11, Springer, New York, 1988, pp. 263-283. MR 922970
  • 28. M. Shearer, D. G. Schaeffer, D. Marchesin, and P. L. Paes-Leme, Solution of the Riemann problem for a prototype $ 2\times 2$ system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97 (1987), no. 4, 299-320. MR 865843 (88a:35156)
  • 29. J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York, 1983. MR 688146 (84d:35002)
  • 30. Z. J. Tang and T. C. T. Ting, Wave curves for the Riemann problem of plane waves in isotropic elastic solids, Internat. J. Eng. Sci. 25 (1987), no. 11-12, 1343-1381. MR 921358 (88m:73011)
  • 31. L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass., 1979, pp. 136-212. MR 584398 (81m:35014)

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Additional Information

Elisabetta Felaco
Affiliation: Department of Mathematics, University of Hamburg, Bundesstrasse 55 Hamburg, 20146, Germany
Email: elisabetta.felaco@math.uni-hamburg.de

Bruno Rubino
Affiliation: Department of Mathematics, University of L’Aquila, via Vetoio 1, L’Aquila, 67100, Italy
Email: rubino@ing.univaq.it

Rosella Sampalmieri
Affiliation: Department of Mathematics, University of L’Aquila, via Vetoio 1, L’Aquila, 67100, Italy
Email: sampalm@ing.univaq.it

DOI: https://doi.org/10.1090/S0033-569X-2013-01328-6
Received by editor(s): October 4, 2011
Received by editor(s) in revised form: March 27, 2012
Published electronically: August 28, 2013
Article copyright: © Copyright 2013 Brown University

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