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.
Citation
Nikolaos Tzirakis. "The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension." Differential Integral Equations 18 (8) 947 - 960, 2005. https://doi.org/10.57262/die/1356060152
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