The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces

Tosinobu Muramatu, Shifu Taoka

Type: Article

Publication Date: 2004-07-01

Citations: 22

DOI: https://doi.org/10.2969/jmsj/1191334089

Abstract

We define a class of Besov type spaces which is a generalization of that defined by Kenig-Ponce-Vega ([4], [5]) in their study on KdV equation and nonlinear Schrödinger equation. Using these spaces, we prove the following results: the 1-dimendional semilinear Schrödinger equation with the nonlinear term c 1u 2+ c2 u ¯2 has a unique local-in-time solution for the initial data ∊B2, 1-3/ 4, and that with cuu ¯ has a unique local-in-time solution for the initial data $\in B_{2,1}^{-1/4, \sharp}$. Note that $B_{2,1}^{-1/4, \sharp}(\mathbf{R}) \supset B_{2,1}^{-1/4}(\mathbf{R}) \supset H^5(\mathbf{R})$ for any $s > -1/4$.

Locations

Journal of the Mathematical Society of Japan - View - PDF

Similar Works

Action	Title	Year	Authors
+ PDF Chat	Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces	2005	Shifu Taoka
+	The initial value problem for the semilinear Schrodinger equation in Besov spaces	2004	志婦田岡
+	Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations	1999	Fabrice Planchon
+	The existence of the solution of the initial value problem for the semilinear Schrodinger equation in Besov spaces〔和文〕 (調和解析学と非線形偏微分方程式研究集会報告集)	2001	志婦田岡
+	THE CAUCHY PROBLEM OF NONLINEAR SCHROEDINGER-BOUSSINESQ EQUATIONS IN H^3（R^d）	2005	HanYongqian
+ PDF Chat	Local well-posedness for the derivative nonlinear Schrödinger Equation in Besov Spaces	2019	Cai Constantin Cloos
+	Besov Spaces and Unconditional Well-Posedness for the Nonlinear Schrödinger Equation in $\dot{\rm H}^s({\mathbb R}^n)$	2003	Giulia Furioli Elide Terraneo
+	Local Well-Posedness for the Derivative Nonlinear Schrödinger Equation in Besov spaces	2016	Cai Constantin Cloos
+	ON THE CAUCHY PROBLEM IN BESOV SPACES FOR A NON-LINEAR SCHRÖDINGER EQUATION	2000	Fabrice Planchon
+ PDF Chat	Vlasov–Poisson Equation in Besov Space	2022	Cong He Jingchun Chen
+	Local Well-Posedness for the Derivative Nonlinear Schr\"odinger Equation in Besov spaces	2016	Cai Constantin Cloos
+	Nonhomogeneous Boundary Value Problems of Nonlinear Schrödinger Equations in a Half Plane	2016	Yu Ran Shu-Ming Sun Bing‐Yu Zhang
+	The existence of the solution of the initial value problem for the semilinear Schrodinger equation in Besov spaces (Harmonic Analysis and Nonlinear Partial Differential Equations)	2001	志婦田岡
+ PDF Chat	Local well-posedness for the Schr\"{o}dinger-KdV system in $H^{s_1}\times H^{s_2}$, II	2024	Yingzhe Ban Jie Chen Ying Zhang
+	Global well-posedness results for the 2D-Schrödinger–Debye system	2018	Raphael Santos
+	Cauchy problem of nonlinear Schrödinger equation with Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$	2007	Yi Zhou
+	Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$	2008	Nobu Kishimoto
+	Local Well-Posedness for the Derivative Nonlinear Schrödinger Equation in Besov spaces	2016	Cai Constantin Cloos
+ PDF Chat	Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation	2017	Yuanyuan Ren Yongsheng Li Wei Yan
+	Sharp bilinear estimates and well-posedness for the 1-D Schrödinger-Debye system	2006	Adán J. Corcho Carlos Matheus

Works That Cite This (22)

Action	Title	Year	Authors
+	Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation	2005	Ioan Bejenaru Terence Tao
+ PDF Chat	Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations	2003	Michael Christ J. Colliander T. Tao
+ PDF Chat	Well-Posedness for the Fifth Order KdV Equation	2012	Takamori Kato
+	Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation	2005	Ioan Bejenaru Terence Tao
+	Global well-posedness of the Benjamin-Ono equation in H^1(R)	2003	Terence Tao
+	Sharp local well-posedness for the "good" Boussinesq equation	2012	Nobu Kishimoto
+	The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity	2015	Yongsheng Li Jianhua Huang Wei Yan
+ PDF Chat	Sharp local well-posedness for the “good” Boussinesq equation	2012	Nobu Kishimoto
+ PDF Chat	The Cauchy problem for higher-order modified Camassa–Holm equations on the circle	2019	Wei Yan Yongsheng Li Xiaoping Zhai Yimin Zhang
+ PDF Chat	The Cauchy problem for the Ostrovsky equation with positive dispersion	2018	Wei Yan Yongsheng Li Jianhua Huang Jinqiao Duan

Works Cited by This (8)

Action	Title	Year	Authors
+ PDF Chat	Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrödinger equation	2001	Kenji Nakanishi Hideo Takaoka Yoshio Tsutsumi
+ PDF Chat	Quadratic forms for the 1-D semilinear Schrödinger equation	1996	Carlos E. Kenig Gustavo Ponce Luis Vega
+	Quadratic forms for a 2-D semilinear Schrödinger equation	1997	Gigliola Staffilani
+ PDF Chat	On Besov Spaces and Sobolev Spaces of Generalized Functions Defined on a General Region	1973	Tosinobu Muramatu
+	Interaction Equations for Short and Long Dispersive Waves	1998	Daniella Bekiranov Takayoshi Ogawa Gustavo Ponce
+ PDF Chat	On spaces of potentials connected with $L^p$ classes	1963	N. Aronszajn P. Szeptycki
+ PDF Chat	A bilinear estimate with applications to the KdV equation	1996	Carlos E. Kenig Gustavo Ponce Luis Vega
+	Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations	1993	Jean Bourgain