Existence and Uniqueness for a Nonlinear Parabolic/Hamilton-Jacobi Coupled System Describing the Dynamics of Dislocation Densities
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 2, pp. 415-435.
@article{AIHPC_2009__26_2_415_0,
     author = {Ibrahim, Hassan},
     title = {Existence and {Uniqueness} for a {Nonlinear} {Parabolic/Hamilton-Jacobi} {Coupled} {System} {Describing} the {Dynamics} of {Dislocation} {Densities}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {415--435},
     publisher = {Elsevier},
     volume = {26},
     number = {2},
     year = {2009},
     doi = {10.1016/j.anihpc.2007.09.005},
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     zbl = {1159.74010},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2007.09.005/}
}
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Ibrahim, Hassan. Existence and Uniqueness for a Nonlinear Parabolic/Hamilton-Jacobi Coupled System Describing the Dynamics of Dislocation Densities. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 2, pp. 415-435. doi : 10.1016/j.anihpc.2007.09.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2007.09.005/

[1] Barles G., Solutions De Viscosité Des Équations De Hamilton-Jacobi, Springer-Verlag, Paris, 1994. | MR | Zbl

[2] Brézis H., Analyse Fonctionelle. Théorie Et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983, xiv+234 pp. | MR | Zbl

[3] M. Cannone, A. El Hajj, R. Monneau, F. Ribaud, Global existence of a system of non-linear transport equations describing the dynamics of dislocation densities, 2007, submitted for publication.

[4] Caselles V., Scalar Conservation Laws and Hamilton-Jacobi Equations in One Space Variables, Nonlinear Anal. 18 (5) (1992) 461-469. | MR | Zbl

[5] Cleveringa H. H.M., Van Der Giessen E., Needleman A., Acta Materialia 54 (1997) 3164.

[6] Corrias L., Falcone M., Natalini R., Numerical Schemes for Conservation Laws Via Hamilton-Jacobi Equations, Math. Comp. 64 (210) (1995) 555-580, S13-S18. | MR | Zbl

[7] Crandall M. G., Lions P. L., On Existence and Uniqueness of Solutions of Hamilton-Jacobi Equations, Nonlinear Anal. Methods Appl. 10 (4) (1986) 353-370. | MR | Zbl

[8] El Hajj A., Well-Posedness Theory for a Nonconservative Burgers-Type System Arising in Dislocation Dynamics, SIAM J. Math. Anal. 39 (3) (2007) 965-986. | MR | Zbl

[9] A. El Hajj, N. Forcadel, A convergent scheme for a non-local coupled system modelling dislocations densities dynamics, Math. Comp. (2006), in press. | Zbl

[10] Evans L. C., Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[11] Eymard R., Gallouët T., Herbin R., Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation, Chin. Ann. of Math. Ser. B 16 (1) (1995) 1-14. | MR | Zbl

[12] Gimse T., Risebro N. H., A Note on Reservoir Simulation for Heterogeneous Porous Media, Transport Porous Media 10 (1993) 257-270.

[13] Groma I., Balogh P., Investigation of Dislocation Pattern Formation in a Two-Dimensional Self-Consistent Field Approximation, Acta Materialia 47 (1999) 3647-3654.

[14] Groma I., Czikor F. F., Zaiser M., Spatial Correlations and Higher-Order Gradient Terms in a Continuum Description of Dislocation Dynamics, Acta Materialia 51 (2003) 1271-1281.

[15] Hirth J. R., Lothe L., Theory of Dislocations, second ed., Krieger, Malabar, FL, 1992.

[16] Ishii H., Existence and Uniqueness of Solutions of Hamilton-Jacobi Equations, Funkcial. Ekvac. 29 (1986) 167-188. | MR | Zbl

[17] H. Ibrahim, Rapport de recherche du CERMICS 2007-338.

[18] Karlsen K. H., Risebro N. H., A Note on Front Tracking and the Equivalence Between Viscosity Solutions of Hamilton-Jacobi Equations and Entropy Solutions of Scalar Conservation Laws, Nonlinear Anal. 50 (2002) 455-469. | MR | Zbl

[19] Kruškov S. N., First Order Quasilinear Equations With Several Space Variables, Math. USSR Sb. 10 (1970) 217-243. | Zbl

[20] Kruškov S. N., The Cauchy Problem in the Large for Non-Linear Equations and for Certain First-Order Quasilinear Systems With Several Variables, Dokl. Akad. Nauk SSSR 155 (1964) 743-746. | MR | Zbl

[21] Lax P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, PA, 1973, v+48 pp. | MR | Zbl

[22] Ley O., Lower-Bound Gradient Estimates for First-Order Hamilton-Jacobi Equations and Applications to the Regularity of Propagating Fronts, Adv. Differential Equations 6 (5) (2001) 547-576. | MR | Zbl

[23] Lions P. L., Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, MA, 1982, Advanced Publishing Program. | MR | Zbl

[24] Nabarro F. R.N., Theory of Crystal Dislocations, Oxford, Clarendon Press, 1969.

[25] Ostrov D., Solutions of Hamilton-Jacobi Equations and Scalar Conservation Laws With Discontinuous Space-Time Dependence, J. Differential Equations 182 (2002) 51-77. | MR | Zbl

[26] Whitham G. B., Linear and Nonlinear Waves, Wiley, New York, 1974. | MR | Zbl

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