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 | 1 citation 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

@article{AIHPC_2015__32_1_97_0,
     author = {G\'erard, Patrick and Guo, Yanqiu and Titi, Edriss S.},
     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},
     number = {1},
     year = {2015},
     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/}
}

TY  - JOUR
AU  - Gérard, Patrick
AU  - Guo, Yanqiu
AU  - Titi, Edriss S.
TI  - On the radius of analyticity of solutions to the cubic Szegő equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 97
EP  - 108
VL  - 32
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.001/
DO  - 10.1016/j.anihpc.2013.11.001
LA  - en
ID  - AIHPC_2015__32_1_97_0
ER  -

%0 Journal Article
%A Gérard, Patrick
%A Guo, Yanqiu
%A Titi, Edriss S.
%T On the radius of analyticity of solutions to the cubic Szegő equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 97-108
%V 32
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.001/
%R 10.1016/j.anihpc.2013.11.001
%G en
%F AIHPC_2015__32_1_97_0

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/

Bibliographie
Cité par

[1] N. Burq, P. Gérard, N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^{d}$ , Math. Res. Lett. 9 no. 2–3 (2002), 323 -335 | MR | Zbl

[2] N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 no. 1 (2005), 187 -223 | MR | Zbl

[3] J.B. Conway, A Course in Operator Theory, Grad. Stud. Math. vol. 21 , American Mathematical Society, Providence, RI (2000) | MR

[4] A.B. Ferrari, E.S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Commun. Partial Differ. Equ. 23 (1998), 1 -16 | Zbl

[5] C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier–Stokes equations, J. Funct. Anal. 87 (1989), 359 -369 | MR | Zbl

[6] P. Gérard, S. Grellier, The cubic Szegő equation, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 761 -810 | EuDML | Numdam | MR | Zbl

[7] P. Gérard, S. Grellier, Invariant tori for the cubic Szegő equation, Invent. Math. 187 no. 3 (2012), 707 -754 | MR | Zbl

[8] P. Gérard, S. Grellier, An explicit formula for the cubic Szegő equation, arXiv:1304.2619 (2013) | MR

[9] Y. Guo, E.S. Titi, Persistency of analyticity for nonlinear wave equations: an energy-like approach, Bull. Inst. Math. Acad. Sin. (N. S.) 8 no. 4 (2013), 445 -479 | MR | Zbl

[10] I. Kukavica, V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Am. Math. Soc. 137 (2009), 669 -677 | MR | Zbl

[11] S.B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal. 2 (1997), 338 -363 | MR | Zbl

[12] A. Larios, E.S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst., Ser. B 14 (2010), 603 -627 | MR | Zbl

[13] C.D. Levermore, M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differ. Equ. 133 (1997), 321 -339 | MR | Zbl

[14] T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation, Funkc. Ekvacioj 54 no. 3 (2011), 335 -365 | MR | Zbl

[15] M. Oliver, E.S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier–Stokes equations in $ℝ^{n}$ , J. Funct. Anal. 172 (2000), 1 -18 | MR | Zbl

[16] M. Oliver, E.S. Titi, On the domain of analyticity for solutions of second order analytic nonlinear differential equations, J. Differ. Equ. 174 (2001), 55 -74 | MR | Zbl

[17] V.V. Peller, Hankel operators of class $𝔖_{p}$ and their applications (rational approximation, Gaussian processes, the problem of majorization of operators), Math. USSR Sb. 41 (1982), 443 -479 | MR | Zbl

[18] V.V. Peller, Hankel Operators and Their Applications, Springer Monogr. Math. , Springer-Verlag, New York (2003) | MR | Zbl

[19] O. Pocovnicu, Explicit formula for the solution of the Szegő equation on the real line and applications, Discrete Contin. Dyn. Syst. 31 no. 3 (2011), 607 -649 | MR | Zbl

[20] O. Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE 4 no. 3 (2011), 379 -404 | MR | Zbl

[21] H. Xu, Large time blow up for a perturbation of the cubic Szegő equation, arXiv:1307.5284 (2013) | MR

Cité par Sources :