Volume (2002) no. 91

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

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

Abstract
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 1942854 | Zbl 1008.35072 | 1 citation in Numdam

DOI : https://doi.org/10.24033/msmf.404
Classification: 35Q55, 35S50
Keywords: Global existence, Nonlinear Schrödinger equation

@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}
}

Delort, Jean-Marc. Global solutions for small nonlinear long range perturbations of two dimensional Schrödinger equations. Mémoires de la Société Mathématique de France, Serie 2, , no. 91 (2002), 100 p. doi : 10.24033/msmf.404. http://www.numdam.org/item/MSMF_2002_2_91__1_0/

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