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

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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.

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Abstract (VO)
Résumé

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.

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 $t \to + \infty$ .

MR Zbl

DOI : 10.24033/msmf.404

Classification : 35Q55, 35S50
Keywords: Global existence, Nonlinear Schrödinger equation
Mots-clés : Existence globale, équation de Schrödinger nonlinéaire

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     author = {Delort, Jean-Marc},
     title = {Global solutions for~small~nonlinear long~range~perturbations of~two~dimensional {Schr\"odinger~equations}},
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     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 = {https://geodesic-test.mathdoc.fr/item/MSMF_2002_2_91__1_0/}
}

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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. https://geodesic-test.mathdoc.fr/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. | MR | EuDML

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

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

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

[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. | MR | EuDML

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[13] N. Hayashi & P.I. Naumkin – On 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. Ozawa – Remarks on nonlinear Schrödinger equations in one space dimension, Diff. Integral Eqs 7, (1994), 453–461. | MR | Zbl

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

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

Huang, Jiaxi; Zhao, Lifeng Asymptotic behavior of incompressible Schrödinger flow for small data in three dimensions, Journal of Differential Equations, Volume 386 (2024), pp. 519-556 | DOI:10.1016/j.jde.2024.01.002 | Zbl:1532.35423
Sakoda, Daisuke; Sunagawa, Hideaki Small data global existence for a class of quadratic derivative nonlinear Schrödinger systems in two space dimensions, Journal of Differential Equations, Volume 268 (2020) no. 4, pp. 1722-1749 | DOI:10.1016/j.jde.2019.09.032 | Zbl:1431.35181
Bai, Ruobing; Wu, Yifei; Xue, Jun Optimal small data scattering for the generalized derivative nonlinear Schrödinger equations, Journal of Differential Equations, Volume 269 (2020) no. 9, pp. 6422-6447 | DOI:10.1016/j.jde.2020.05.001 | Zbl:1440.35304
Ikeda, Masahiro; Katayama, Soichiro; Sunagawa, Hideaki Null structure in a system of quadratic derivative nonlinear Schrödinger equations, Annales Henri Poincaré, Volume 16 (2015) no. 2, pp. 535-567 | DOI:10.1007/s00023-014-0316-6 | Zbl:1315.35203
Ionescu, Alexandru D.; Pausader, Benoit Global solutions of quasilinear systems of Klein-Gordon equations in 3D, Journal of the European Mathematical Society (JEMS), Volume 16 (2014) no. 11, pp. 2355-2431 | DOI:10.4171/jems/489 | Zbl:1316.35180
Hayashi, Nakao; Naumkin, Pavel I. Global existence for two dimensional quadratic derivative nonlinear Schrödinger equations, Communications in Partial Differential Equations, Volume 37 (2012) no. 4-6, pp. 732-752 | DOI:10.1080/03605302.2011.626099 | Zbl:1247.35145
Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 2D quadratic Schrödinger equations, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 97 (2012) no. 5, pp. 505-543 | DOI:10.1016/j.matpur.2011.09.008 | Zbl:1244.35134
Bernal-Vílchis, Fernando; Hayashi, Nakao; Naumkin, Pavel I. Quadratic derivative nonlinear Schrödinger equations in two space dimensions, NoDEA. Nonlinear Differential Equations and Applications, Volume 18 (2011) no. 3, pp. 329-355 | DOI:10.1007/s00030-011-0098-1 | Zbl:1228.35217
Hayashi, Nakao; Naumkin, Pavel I. Remark on the global existence and large time asymptotics of solutions for the quadratic NLS, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 74 (2011) no. 18, pp. 6950-6964 | DOI:10.1016/j.na.2011.07.016 | Zbl:1229.35262

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