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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.
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O. Pocovnicu, Explicit formula for the solution of the Szegő equation on the real line and applications, Discrete Contin. Dyn. Syst. 31 (2011), no.3, 607-649
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A. Larios, E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete and Continuous Dynamical Systems Series B 14 (2010), 603-627
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