On the radius of analyticity of solutions to the cubic Szegő equation

Gérard, Patrick; Guo, Yanqiu; Titi, Edriss S.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 97-108.

This paper is concerned with the cubic Szegő equation

$$i{\partial }_{t}u=\Pi \left({\left|u\right|}^{2}u\right),$$

defined on the $L^2$ Hardy space on the one-dimensional torus $𝕋$, where $\Pi :L^2\left(𝕋\right)\to L_{+}^2\left(𝕋\right)$ is the Szegő 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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