Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions
type: Item
instance of: scholarly article
title: Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions
zbMATH Open document ID: 0824.35026
DOI: 10.1007/BF02572354
author: Nakao Hayashi
author: Jean Ginibre
published in: Mathematische Zeitschrift
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/174762
review text: This paper is devoted to the study of the Cauchy problem for the nonlinear Schrödinger equation \(i \partial_ tu = - (1/2) \Delta u + F(u, \overline u, \nabla u, \nabla \overline u)\) in the borderline case of space dimension 3, with \(F\) a quadratic function of its arguments and \(\partial F/ \partial (\nabla u)\) pure imaginary. The authors prove the existence of small almost global solutions, namely of solutions with a lifespan \(T = \exp [C (\varphi)/ \varepsilon]\) for initial data \(u(0) = \varepsilon \varphi\) small of order \(\varepsilon\) in a suitable norm. This paper continues previous work by the second author proving the existence of small global solutions of the same problem, either in dimension 4, or in dimension 3 with \(F\) at least linear in \(\nabla u\), \(\nabla \overline u\). The method relies on energy estimates and Sobolev inequalities associated with a Lie algebra of vector fields which commute with the linear part of the equation, in the same spirit as in Klainerman's work on the nonlinear wave equation. In contrast with the previous works, however, it requires the use of half integer derivatives.
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 784104
zbMATH Keywords: quadratic nonlinear Schrödinger equations
zbMATH Keywords: small almost global solutions
reviewed by: Jean Ginibre
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784104