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 Zbl | 1 citation dans Numdam

DOI : 10.24033/msmf.404

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

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     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://archive.numdam.org/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. http://numdam.org/item/MSMF_2002_2_91__1_0/

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