Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations

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]

Delort, Jean-Marc

Mémoires de la Société Mathématique de France, no. 91 (2002) , 100 p.

Résumé
Abstract

Soient $Q_{1}, Q_{2}$ deux formes quadratiques et $u$ solution locale de l’équation de Schrödinger en dimension $2$ d’espace $(i \partial_{t} + Δ) u = Q_{1} (u, \nabla_{x} u) + Q_{2} (\bar{u}, \nabla_{x} \bar{u})$ . 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 $L^{\infty}$ en espace n’est pas intégrable lorsque $t \to + \infty$ .

Let $Q_{1}, Q_{2}$ be two quadratic forms, and $u$ a local solution of the two dimensional Schrödinger equation $(i \partial_{t} + Δ) u = Q_{1} (u, \nabla_{x} u) + Q_{2} (\bar{u}, \nabla_{x} \bar{u})$ . 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 $L^{\infty}$ space-norm is not time integrable at infinity.

MR 1942854 | Zbl 1008.35072 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/msmf.404
Classification : 35Q55, 35S50
Mots clés : Existence globale, équation de Schrödinger nonlinéaire

BibTeX
RIS

@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},
     zbl = {1008.35072},
     mrnumber = {1942854},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2002_2_91__1_0/}
}

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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
DA  - 2002///
IS  - 91
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2002_2_91__1_0/
UR  - https://zbmath.org/?q=an%3A1008.35072
UR  - https://www.ams.org/mathscinet-getitem?mr=1942854
UR  - https://doi.org/10.24033/msmf.404
DO  - 10.24033/msmf.404
LA  - en
ID  - MSMF_2002_2_91__1_0
ER  -

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/

Bibliographie
Cité par

[1] J.-M. Bony – Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires, Ann. Scient. Éc. Norm. Sup. 14, (1981) 209–256. | EuDML 82073 | MR 631751

[2] A. P. Calderón & R. Vaillancourt – On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23, (1971), 374–378. | MR 284872 | Zbl 0203.45903

[3] J.-Y. Chemin – Fluides parfaits incompressibles, Astérisque 230, (1995). | MR 1340046

[4] H. Chihara – Local existence for semi-linear Schrödinger equations, Math. Japonica 42, (1995), 35–52. | MR 1344627

[5] —, Global existence of small solutions to semi-linear Schrödinger equations, Comm. Partial Differential Equations 21, (1996), 63–78. | MR 1373765 | Zbl 0843.35111

[6] —, The initial value problem for cubic semi-linear Schrödinger equations, Publ. RIMS, Kyoto Univ. 32, (1996), 445–471. | MR 1409797 | Zbl 0870.35095

[7] S. Cohn – Global existence for the nonresonant Schrödinger equation in two space dimensions, Canad. Appl. Math. Quart. 2, (1994), 247-282. | MR 1318545

[8] J.-M. Delort – Existence 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. | EuDML 82538 | MR 1833089

[9] S. Doi – On the Cauchy problem for Schrödinger type equations and the regularity of the solutions, J. Math. Kyoto Univ. 34, (1994), 319–328. | MR 1284428 | Zbl 0807.35026

[10] N. Hayashi & H. Hirata – Global existence of small solutions to nonlinear Schrödinger equations, Nonlinear Analysis, Theory, Methods and Applications 31, (1998), 671–685. | MR 1487854 | Zbl 0886.35141

[11] N. Hayashi & K. Kato – Global existence of small analytic solutions to Schrödinger equations with quadratic nonlinearities, Comm. Partial Differential Equations 22, (1997), 773–798. | MR 1452167 | Zbl 0881.35106

[12] N. Hayashi, C. Miao & P.I. Naumkin – Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation, Asymptotic Analysis 21, (1999), 133–147. | MR 1723555 | Zbl 0937.35171

[13] N. Hayashi & P.I. Naumkin – On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices (2000), 115–132. | MR 1741610 | Zbl 1004.35112

[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. Ozawa – Remarks on nonlinear Schrödinger equations in one space dimension, Diff. Integral Eqs 7, (1994), 453–461. | MR 1255899 | Zbl 0803.35137

[19] C. Kenig, G. Ponce & L. Vega – Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 10, (1993), 255–288. | EuDML 78303 | Numdam | MR 1230709 | Zbl 0786.35121

[20] —, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998), no. 3, 489–545. | MR 1660933 | Zbl 0928.35158

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