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

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

doi:10.1016/j.anihpc.2013.11.001

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

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

Résumé

This paper is concerned with the cubic Szegő equation

i \partial_{t} u = Π ({| u |}^{2} u),

defined on the

L^{2}

Hardy space on the one-dimensional torus

𝕋

, where

Π : L^{2} (𝕋) \to L_{+}^{2} (𝕋)

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

ℓ^{1}

norm of Fourier transforms (the Wiener algebra).

MR Zbl | 2 citations dans Numdam

DOI : 10.1016/j.anihpc.2013.11.001

Classification : 35B10, 35B65, 47B35
Mots-clés : Cubic Szegő equation, Gevrey class regularity, Analytic solutions, Hankel operators

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     title = {On the radius of analyticity of solutions to the cubic {Szeg\H{o}} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {97--108},
     publisher = {Elsevier},
     volume = {32},
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     doi = {10.1016/j.anihpc.2013.11.001},
     mrnumber = {3303943},
     zbl = {1332.35058},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.001/}
}

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Gérard, Patrick; Guo, Yanqiu; Titi, Edriss S. On the radius of analyticity of solutions to the cubic Szegő equation. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 97-108. doi : 10.1016/j.anihpc.2013.11.001. http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.001/

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