[Existence globale de solutions pour un système couplé parabolique/Hamilton–Jacobi singulier avec condition de Dirichlet]
Nous étudions l'existence de solutions mixtes (distribution/viscosité) pour un système couplé parabolique/Hamilton–Jacobi posé sur un interval. Notre motivation vient de l'étude de la dynamique de densités de dislocations dans un cristal de taille finie. L'idée de la preuve consiste à considérer une régularisation parabolique appropriée, et ensuite à passer à la limite en utilisant en particulier une estimation entropique pour les densités.
We study the existence of (distribution/viscosity) solutions of a singular parabolic/Hamilton–Jacobi coupled system. Our motivation stems from the study of the dynamics of dislocation densities in a crystal of finite size. The method of the proof consists in considering a parabolic regularization of the system, and then passing to the limit after obtaining some uniform bounds using in particular an entropy estimate for the densities.
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@article{CRMATH_2008__346_17-18_945_0,
author = {Ibrahim, Hassan and Jazar, Mustapha and Monneau, R\'egis},
title = {Global existence of solutions to a singular {parabolic/Hamilton{\textendash}Jacobi} coupled system with {Dirichlet} conditions},
journal = {Comptes Rendus. Math\'ematique},
pages = {945--950},
publisher = {Elsevier},
volume = {346},
number = {17-18},
year = {2008},
doi = {10.1016/j.crma.2008.07.031},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.031/}
}
TY - JOUR AU - Ibrahim, Hassan AU - Jazar, Mustapha AU - Monneau, Régis TI - Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions JO - Comptes Rendus. Mathématique PY - 2008 SP - 945 EP - 950 VL - 346 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.07.031/ DO - 10.1016/j.crma.2008.07.031 LA - en ID - CRMATH_2008__346_17-18_945_0 ER -
%0 Journal Article %A Ibrahim, Hassan %A Jazar, Mustapha %A Monneau, Régis %T Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions %J Comptes Rendus. Mathématique %D 2008 %P 945-950 %V 346 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.07.031/ %R 10.1016/j.crma.2008.07.031 %G en %F CRMATH_2008__346_17-18_945_0
Ibrahim, Hassan; Jazar, Mustapha; Monneau, Régis. Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 945-950. doi : 10.1016/j.crma.2008.07.031. http://www.numdam.org/articles/10.1016/j.crma.2008.07.031/
[1] Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications (Berlin), Mathematics & Applications, vol. 17, Springer-Verlag, Paris, 1994
[2] Nonlinear Schrödinger evolution equations, Nonlinear Anal., Volume 4 (1980), pp. 677-681
[3] A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, Volume 5 (1980), pp. 773-789
[4] Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics, Acta Mater., Volume 51 (2003), pp. 1271-1281
[5] Theory of Dislocations, Kreiger Publishing Company, Florida 32950, 1982
[6] H. Ibrahim, M. Jazar, R. Monneau, Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system, preprint, hal-00281487
[7] H. Ibrahim, M. Jazar, R. Monneau, Dynamics of dislocation densities in a bounded channel. Part II: existence of weak solutions to a singular Hamilton–Jacobi/parabolic strongly coupled system, preprint, hal-00281859
[8] Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Commun. Math. Phys., Volume 214 (2000), pp. 191-200
[9] Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967 (Translated from the Russian by S. Smith)
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