Let be two quadratic forms, and a local solution of the two dimensional Schrödinger equation . We prove that if and do depend on the derivatives of , 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) and of a potential whose space-norm is not time integrable at infinity.
Soient deux formes quadratiques et solution locale de l’équation de Schrödinger en dimension d’espace . Nous prouvons que si et dépendent effectivement des dérivées de , 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 par un potentiel dont la norme en espace n’est pas intégrable lorsque .
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://archive.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://archive.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://archive.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, Serie 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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