[Théorèmes de structure pour les équations de Schrödinger linéaire et non linéaire en dimension deux d'espace]
On étudie le comportement des solutions de l'équation de Schrödinger non linéaire à croissance exponentielle, où la norme d'Orlicz joue un rôle crucial. L'approche qu'on a adoptée dans ce travail consiste à comparer des suites de solutions des équations de Schrödinger linéaires et non linéaires issues de la même suite de données de Cauchy, moyennant un terme de reste petit à la fois en normes de Strichartz et d'Orlicz. Cette analyse, qui est basée sur les décompositions en profils, met en lumière le rôle distingué de la composante 1-oscillante de la suite des données initiales. Ce phénomène est complètement différent de ceux obtenus dans le cadre des équations semi-linéaires dispersives critiques, comme dans [2,13], où toutes les composantes oscillantes créent le même effet non linéaire, à un changement d'échelle près.
We investigate the behavior of solutions to 2D nonlinear Schrödinger equations with exponential growth, where the Orlicz norm plays a crucial role. The approach we adopted in this paper consists in comparing the evolution of oscillations and concentration effects displayed by sequences of solutions to linear and nonlinear Schrödinger equations associated with the same sequence of Cauchy data, up to small remainder terms both in Strichartz and Orlicz norms. The analysis we conducted in this work emphasizes that the nonlinear effect highlighted in this framework only arises from the 1-oscillating component of the sequence of the Cauchy data. This phenomenon is strikingly different from those obtained for critical semilinear dispersive equations, such as for instance in [2,13], where all the oscillating components induce the same nonlinear effect, up to a change of scale.
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@article{CRMATH_2015__353_3_235_0,
author = {Bahouri, Hajer},
title = {Structure theorems for {2D} linear and nonlinear {Schr\"odinger} equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {235--240},
publisher = {Elsevier},
volume = {353},
number = {3},
year = {2015},
doi = {10.1016/j.crma.2015.01.009},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.01.009/}
}
TY - JOUR AU - Bahouri, Hajer TI - Structure theorems for 2D linear and nonlinear Schrödinger equations JO - Comptes Rendus. Mathématique PY - 2015 SP - 235 EP - 240 VL - 353 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.01.009/ DO - 10.1016/j.crma.2015.01.009 LA - en ID - CRMATH_2015__353_3_235_0 ER -
%0 Journal Article %A Bahouri, Hajer %T Structure theorems for 2D linear and nonlinear Schrödinger equations %J Comptes Rendus. Mathématique %D 2015 %P 235-240 %V 353 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.01.009/ %R 10.1016/j.crma.2015.01.009 %G en %F CRMATH_2015__353_3_235_0
Bahouri, Hajer. Structure theorems for 2D linear and nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 235-240. doi : 10.1016/j.crma.2015.01.009. http://archive.numdam.org/articles/10.1016/j.crma.2015.01.009/
[1] Structure theorems for 2D linear and nonlinear Schrödinger equations | HAL
[2] High-frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., Volume 121 (1999), pp. 131-175
[3] On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal., Volume 260 (2011), pp. 208-252
[4] Scattering for the critical 2-D NLS with exponential growth, Differ. Integral Equ., Volume 27 (2014), pp. 233-268
[5] A Fourier approach to the profile decomposition in Orlicz spaces, Math. Res. Lett., Volume 21 (2014), pp. 33-54
[6] Convergence of solutions of H-Systems or how to blow bubbles, Arch. Ration. Mech. Anal., Volume 89 (1985), pp. 21-86
[7] Some new estimates on oscillatoryon integrals, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton Math. Ser., vol. 42, 1995, pp. 83-112
[8] Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 213-233
[9] Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal., Volume 133 (1996), pp. 50-68
[10] Energy critical NLS in two space dimension, J. Hyperbolic Differ. Equ., Volume 6 (2009), pp. 549-575
[11] Scattering for the two dimensional NLS with exponential nonlinearity, Nonlinearity, Volume 25 (2012), pp. 1843-1849
[12] Self trapped laser beams in plasma, Phys. Fluids, Volume 20 (1977), pp. 1176-1179
[13] Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not., Volume 8 (1998), pp. 399-425
[14] A sharp form of an inequality of N. Trudinger, Indiana Univ. Math. J., Volume 20 (1971), pp. 1077-1092
[15] Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002
[16] A global compactness result for boundary value problems involving limiting nonlinearities, Math. Z., Volume 187 (1984), pp. 511-517
[17] On imbedding into Orlicz spaces and some applications, J. Math. Mech., Volume 17 (1967), pp. 473-484
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