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On the Radius of Analyticity of Solutions to the Cubic Szeg\"o Equation

Authors
  • Gerard, Patrick
  • Guo, Yanqiu
  • Titi, Edriss S.
Type
Preprint
Publication Date
Aug 06, 2013
Submission Date
Mar 25, 2013
Identifiers
arXiv ID: 1303.6148
Source
arXiv
License
Yellow
External links

Abstract

This paper is concerned with the cubic Szeg\H{o} equation $$ i\partial_t u=\Pi(|u|^2 u), $$ defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb T$, where $\Pi: L^2(\mathbb T)\rightarrow L^2_+(\mathbb T)$ is the Szeg\H{o} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time $t\in (-\infty,\infty)$. In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the $\ell^1$ norm of Fourier transforms (the Wiener algebra).

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