## Differential and Integral Equations

### The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension

Nikolaos Tzirakis

#### Abstract

We show that the Cauchy problem for the quintic NLS on $\mathbf{R}$ is globally well posed in $H^s$ for $4/9<s\leq 1/2$. Since we work below the energy space we cannot immediately use the energy. Instead we use the "I-method" introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and 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.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 8 (2005), 947-960.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060152

Mathematical Reviews number (MathSciNet)
MR2150447

Zentralblatt MATH identifier
1212.35459

#### Citation

Tzirakis, Nikolaos. The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension. Differential Integral Equations 18 (2005), no. 8, 947--960. https://projecteuclid.org/euclid.die/1356060152