Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions
type: Item
instance of: scholarly article
title: Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions
zbMATH Open document ID: 1304.35643
author: Gaku Hoshino
author: Tohru Ozawa
published in: Osaka Journal of Mathematics
publication date: 2014-11-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1414090794
review text: The unknown function u is a complex-valued function that depends on the time variable and a spatial variable which belongs to a two-dimensional Euclidean space. Nonlinearity is given the function ``right member'' \(f\) which is a complex-valued function satisfying a gauge condition. Also, this function behaves like a cubic power at the origin. The authors state some typical examples of the nonlinearities \(f\). Using a Cauchy condition of the form \(u(0)\) associated to equation (1.1), is obtained a Cauchy problem and the purpose of the present paper is to prove the global existence of solutions for this Cauchy problem. Also, the authors study the analyticity of mentioned solutions. The global existence of solutions and the analyticity of solutions are demonstrated for sufficiently small Cauchy data with exponential decay at infinity. Should be noted that the authors describe the analytic smoothing effects for nonlinear Schrödinger equation by using some Lebesgue spaces of a \(p\)-th integrable function on \(\mathbb R^2\), the usual Sobolev space (or the space of Bessel potentials) and the homogeneous Sobolev space (or the space of Riesz potentials). Also, it is used the Galilei transform.
reviewed by: Marin I. Marin
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35Q41
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 6368440
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: smoothing effect
zbMATH Keywords: analytic solutions
zbMATH Keywords: cubic power
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6368440