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GLOBAL CONTINUOUS SOLUTIONS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

Laboratoire de Mathématiques Appliquées de Compiègne, Université de Technologie de Compiègne, 60205 Compiègne Cedex, France

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RÉGIS MONNEAU

CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France

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https://doi.org/10.1142/S0219891610002050Cited by:12 (Source: Crossref)

Abstract

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic system appears naturally in the modeling of the dynamics of dislocation densities.

Keywords:

AMSC: 35L45, 35Q35, 35Q72, 74H25

References

R. A. Adams , Sobolev Spaces , Pure and Applied Mathematics 65 ( Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] , New York, London , 1975 ) . Google Scholar
L. Ambrosio, Invent. Math. 158, 227 (2004), DOI: 10.1007/s00222-004-0367-2. Web of Science, Google Scholar
S. Bianchini and A. Bressan, Ann. Math. (2) 161, 223 (2005), DOI: 10.4007/annals.2005.161.223. Web of Science, Google Scholar
H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree] (Masson, Paris, 1983). Théorie et applications. [Theory and applications] . Google Scholar
M. Cannoneet al., Arch. Ration. Mech. Anal. (2007). Google Scholar
R. J. DiPerna, Arch. Ration. Mech. Anal. 82, 27 (1983), DOI: 10.1007/BF00251724. Web of Science, Google Scholar
R. J. DiPerna, Trans. Amer. Math. Soc. 292, 383 (1985). Web of Science, Google Scholar
R. J. DiPerna and P.-L. Lions, Invent. Math. 98, 511 (1989), DOI: 10.1007/BF01393835. Web of Science, Google Scholar
A. El Hajj, SIAM J. Math. Anal. 39, 965 (2007), DOI: 10.1137/060672170. Web of Science, Google Scholar
A. El Hajj and N. Forcadel, Math. Comp. 77, 789 (2008), DOI: 10.1090/S0025-5718-07-02038-8. Web of Science, Google Scholar
A. El Hajj and R. Monneau, Some uniqueness results for diagonal hyperbolic systems with large and monotone data, in preparation (2009) . Google Scholar
R. Eymard, T. Gallouët and R. Herbin, Chinese Ann. Math. Ser. B 16, 1 (1995). Web of Science, Google Scholar
L. Gȧrding, Les Équations aux Dérivées Partielles (Éditions du Centre National de la Recherche Scientifique, Paris, 1962) pp. 33–40. Google Scholar
J. Glimm, Commun. Pure Appl. Math. 18, 697 (1965), DOI: 10.1002/cpa.3160180408. Web of Science, Google Scholar
I. Groma and P. Balogh, Acta Mater 47, 3647 (1999), DOI: 10.1016/S1359-6454(99)00215-3. Web of Science, Google Scholar
J. P. Hirth and J. Lothe , Theory of Dislocations , 2nd edn. ( Krieger , Malabar, Florida , 1992 ) . Google Scholar
S. N. Kružkov, Mat. Sb. (N.S.) 81(123), 228 (1970). Google Scholar
P. D. Lax , Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 11 ( Society for Industrial and Applied Mathematics , Philadelphia, Pa , 1973 ) . Google Scholar
P. LeFloch, Commun. Partial Differential Equations 13, 669 (1988). Web of Science, Google Scholar
P. LeFloch and T.-P. Liu, Forum Math. 5, 261 (1993). Web of Science, Google Scholar
P. LeFloch, J. Hyperbolic Differential Equations 1, 643 (2004), DOI: 10.1142/S0219891604000287. Link, Web of Science, Google Scholar
G. M. Lieberman , Second Order Parabolic Differential Equations ( World Scientific Publishing Co. Inc. , River Edge, NJ , 1996 ) . Link, Google Scholar
P.-L. Lions, B. Perthame and E. Tadmor, J. Amer. Math. Soc. 7, 169 (1994), DOI: 10.2307/2152725. Web of Science, Google Scholar
O. A. Oleinik, Amer. Math. Soc. Transl. (2) 26, 95 (1963). Web of Science, Google Scholar
F. Poupaud, Ann. Fac. Sci. Toulouse Math. (6) 8, 649 (1999). Google Scholar
D. Serre , Systems of Conservation Laws. I, II , Geometric structures, oscillations, and initial-boundary value problems ( Cambridge University Press , Cambridge , 2000 ) . Google Scholar
J. Simon, Ann. Mat. Pura Appl. (4) 146, 65 (1987), DOI: 10.1007/BF01762360. Web of Science, Google Scholar
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Research Notes in Mathematics IV (Pitman, Boston, MA, 1979) pp. 136–212. Google Scholar
S. Yefimov , Discrete Dislocation and Nonlocal Crystal Plasticity Modelling ( Netherlands Institute for Metals Research, University of Groningen , 2004 ) . Google Scholar
S. Yefimov and E. Van der Giessen, Int. J. Solids Structure 42, 3375 (2005), DOI: 10.1016/j.ijsolstr.2004.10.025. Web of Science, Google Scholar