The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension , the defocusing case
نویسنده
چکیده
We show that the Cauchy problem for the quintic NLS on R is globally well-posed in Hs for 4/9 < s ≤ 1/2. Since we work below the energy space we can not immediately use the energy.Instead we use the “I-method” introduced by J.Colliander,M.Keel,G.Staffilani, H.Takaoka,T.Tao.This method allows us to define a modification of the energy functional that is “almost conserved” and thus can be used to iterate the local result.
منابع مشابه
2 00 6 Global Existence and Scattering for Rough Solutions to Generalized Nonlinear Schrödinger Equations On
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori Lt,x spacetime control on solutions.
متن کاملGlobal Existence and Scattering for Rough Solutions to Generalized Nonlinear Schrödinger Equations on R
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori Lt,x spacetime control on solutions.
متن کاملIll-Posedness for Semilinear Wave Equations with Very Low Regularity
In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on u and ∂tu. We prove a ill-posedness result for the “defocusing” case, and give an alternative proof for the supercritical “focusing” case, which improves the result in [4].
متن کامل20 07 Invariant Measures for the Defocusing Nls
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane R 2. We also prove an estimate giving some intuition to what may happen in 3 dimensions. Résumé. On démontre l'existence et l'invariance d'une mesure de Gibbs par le flot de l'´ equation de Schrödinger non linéaire posée sur le disque du ...
متن کاملA Refined Global Well-Posedness Result for Schrödinger Equations with Derivative
In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in H s , for s > 1 2 for data small in L 2. To understand the strength of this result one should recall that for s < 1 2 the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost cons...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009