Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations
[Solutions globales pour des perturbations nonlinéaires à longue portée de l’équation de Schrödinger en dimension 2]
Mémoires de la Société Mathématique de France, no. 91 (2002) , 100 p.

Soient Q 1 ,Q 2 deux formes quadratiques et u solution locale de l’équation de Schrödinger en dimension 2 d’espace (i t +Δ)u=Q 1 (u, x u)+Q 2 (u ¯, x u ¯). Nous prouvons que si Q 1 et Q 2 dépendent effectivement des dérivées de u, et si la donnée de Cauchy est assez petite et assez décroissante à l’infini, la solution existe globalement en temps. La difficulté du problème réside dans le fait que la perturbation nonlinéaire est à longue portée, en ce sens qu’elle s’écrit comme un produit (d’une dérivée) de u par un potentiel dont la norme L en espace n’est pas intégrable lorsque t+.

Let Q 1 ,Q 2 be two quadratic forms, and u a local solution of the two dimensional Schrödinger equation (i t +Δ)u=Q 1 (u, x u)+Q 2 (u ¯, x u ¯). We prove that if Q 1 and Q 2 do depend on the derivatives of u, and if the Cauchy datum is small enough and decaying enough at infinity, the solution exists for all times. The difficulty of the problem originates in the fact that the nonlinear perturbation is a long range one: by this, we mean that it can be written as the product of (a derivative of) u and of a potential whose L space-norm is not time integrable at infinity.

DOI : 10.24033/msmf.404
Classification : 35Q55, 35S50
Keywords: Global existence, Nonlinear Schrödinger equation
Mot clés : Existence globale, équation de Schrödinger nonlinéaire
@book{MSMF_2002_2_91__1_0,
     author = {Delort, Jean-Marc},
     title = {Global solutions for~small~nonlinear long~range~perturbations of~two~dimensional {Schr\"odinger~equations}},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {91},
     year = {2002},
     doi = {10.24033/msmf.404},
     mrnumber = {1942854},
     zbl = {1008.35072},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2002_2_91__1_0/}
}
TY  - BOOK
AU  - Delort, Jean-Marc
TI  - Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations
T3  - Mémoires de la Société Mathématique de France
PY  - 2002
IS  - 91
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2002_2_91__1_0/
DO  - 10.24033/msmf.404
LA  - en
ID  - MSMF_2002_2_91__1_0
ER  - 
%0 Book
%A Delort, Jean-Marc
%T Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations
%S Mémoires de la Société Mathématique de France
%D 2002
%N 91
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2002_2_91__1_0/
%R 10.24033/msmf.404
%G en
%F MSMF_2002_2_91__1_0
Delort, Jean-Marc. Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations. Mémoires de la Société Mathématique de France, Série 2, no. 91 (2002), 100 p. doi : 10.24033/msmf.404. http://numdam.org/item/MSMF_2002_2_91__1_0/

[1] J.-M. BonyCalcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires, Ann. Scient. Éc. Norm. Sup. 14, (1981) 209–256. | MR | EuDML

[2] A. P. Calderón & R. VaillancourtOn the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23, (1971), 374–378. | MR | Zbl

[3] J.-Y. CheminFluides parfaits incompressibles, Astérisque 230, (1995). | MR

[4] H. ChiharaLocal existence for semi-linear Schrödinger equations, Math. Japonica 42, (1995), 35–52. | MR

[5] —, Global existence of small solutions to semi-linear Schrödinger equations, Comm. Partial Differential Equations 21, (1996), 63–78. | MR | Zbl

[6] —, The initial value problem for cubic semi-linear Schrödinger equations, Publ. RIMS, Kyoto Univ. 32, (1996), 445–471. | MR | Zbl

[7] S. CohnGlobal existence for the nonresonant Schrödinger equation in two space dimensions, Canad. Appl. Math. Quart. 2, (1994), 247-282. | MR

[8] J.-M. DelortExistence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Scient. Éc. Norm. Sup. 34, (2001), 1–61. | MR | EuDML

[9] S. DoiOn the Cauchy problem for Schrödinger type equations and the regularity of the solutions, J. Math. Kyoto Univ. 34, (1994), 319–328. | MR | Zbl

[10] N. Hayashi & H. HirataGlobal existence of small solutions to nonlinear Schrödinger equations, Nonlinear Analysis, Theory, Methods and Applications 31, (1998), 671–685. | MR | Zbl

[11] N. Hayashi & K. KatoGlobal existence of small analytic solutions to Schrödinger equations with quadratic nonlinearities, Comm. Partial Differential Equations 22, (1997), 773–798. | MR | Zbl

[12] N. Hayashi, C. Miao & P.I. NaumkinGlobal existence of small solutions to the generalized derivative nonlinear Schrödinger equation, Asymptotic Analysis 21, (1999), 133–147. | MR | Zbl

[13] N. Hayashi & P.I. NaumkinOn the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices (2000), 115–132. | MR | Zbl

[14] —, Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, preprint, 12 pp.

[15] —, Global existence of small solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, SIAM J. Math. Anal. 32 (2001), no. 6, 1390–1409.

[16] —, Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equation in two space dimensions, preprint, 15 pp.

[17] —, A quadratic nonlinear Schrödinger equation in one space dimension, preprint, 18 pp.

[18] N. Hayashi & T. OzawaRemarks on nonlinear Schrödinger equations in one space dimension, Diff. Integral Eqs 7, (1994), 453–461. | MR | Zbl

[19] C. Kenig, G. Ponce & L. VegaSmall solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 10, (1993), 255–288. | MR | EuDML | Zbl | Numdam

[20] —, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998), no. 3, 489–545. | MR | Zbl

Cité par Sources :