A survey of the Szeg? equation Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday
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摘要
We survey the main properties of the cubic Szeg? equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping.
We survey the main properties of the cubic Szeg? equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping.
We survey the main properties of the cubic Szeg? equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of high Sobolev norms and the effects of weak damping.
引文
1 Bahouri H,Chemin J-Y.Equations de transport relatives`a des champs de vecteurs non-lipschitziens et m′ecanique des fluides.Arch Ration Mech Anal,1994,127:159-181
2 Bahouri H,Chemin J-Y,Danchin R.Fourier Analysis and Nonlinear Partial Differential Equations.Comprehensive Studies in Mathematics,vol.243.Berlin-Heidelberg:Springer,2011
3 Bahouri H,G′erard P,Xu C-J.Espaces de Besov et estimations de Strichartz g′en′eralis′ees sur le groupe de Heisenberg.J Anal Math,2000,82:93-118
4 Brezis H,Gallouet T.Nonlinear Schr¨odinger evolution equations.Nonlinear Anal,1980,4:677-681
5 Burq N,G′erard P,Tzvetkov N.Strichartz inequalities and the nonlinear Schr¨odinger equation on compact manifolds.Amer J Math,2004,126:569-605
6 Chemin J-Y.Fluides Parfaits Incompressibles.Ast′erisque,vol.230.Paris:Soc Math France,1995
7 Chemin J-Y,Lerner N.Flot de champs de vecteurs non Lipschitziens et′equations de Navier-Stokes.J Differential Equations,1995,121:314-328
8 Fefferman C.Characterizations of bounded mean oscillation.Bull Amer Math Soc(NS),1971,77:587-588
9 G′erard P.Nonlinear Schr¨odinger equations in inhomogeneous media:Wellposedness and illposedness of the Cauchy problem.In:Proceedings of the International Congress of Mathematicians,vol.3.Z¨urich:Eur Math Soc,2006,157-182
10 G′erard P,Grellier S.The cubic Szeg?o equation.Ann Sci′Ec Norm Sup′er(4),2010,43:761-810
11 G′erard P,Grellier S.Effective integrable dynamics for a certain nonlinear wave equation.Anal PDE,2012,5:1139-1155
12 G′erard P,Grellier S.The Cubic Szeg?o Equation and Hankel Operators.Ast′erisque,vol.389.Paris:Soc Math France,2017
13 G′erard P,Grellier S.Generic colourful tori and inverse spectral transform for Hankel operators.Tunisian J Math,2019,1:347-372
14 G′erard P,Guo Y,Titi E.On the radius of analyticity of solutions to the cubic Szeg?o equation.Ann Inst H Poincar′e Anal Non Lin′eaire,2015,32:97-108
15 G′erard P,Koch H.The cubic Szeg?o flow at low regularity.S′eminaire Laurent Schwartz-EDP et Applications Expos′e No.14,2016
16 G′erard P,Pushnitski A.Weighted model spaces and Schmidt subspaces of Hankel operators.ArXiv:1803.04295,2018
17 G′erard P,Pushnitski A.Inverse spectral theory for a class of non compact Hankel operators.Mathematika,2019,65:132-156
18 John F,Nirenberg F L.On functions of bounded mean oscillation.Comm Pure Appl Math,1961,14:415-426
19 Nehari Z.On bounded bilinear forms.Ann of Math(2),1957,65:153-162
20 Ozawa T,Visciglia N.An improvement on the Br′ezis-Gallou¨et technique for 2D NLS and 1D half-wave equation.Ann Inst H Poincar′e Anal Non Lin′eaire,2016,33:1069-1079
21 Peller V.Hankel Operators and Their Applications.Springer Monographs in Mathematics.New York:Springer-Verlag,2003
22 Pocovnicu O.Explicit formula for the solution of the cubic Szeg?o equation on the real line and applications.Discrete Contin Dyn Syst,2011,31:607-649
23 Thirouin J.Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szeg?o equation.Trans Amer Math Soc,2019,371:3673-3690
24 Xu H.Large-time blowup for a perturbation of the cubic Szeg?o equation.Anal PDE,2014,7:717-731
25 Xu H.Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schr¨odinger equation.Math Z,2017,286:443-489
1)G′erard P, Grellier S. On a damped Szeg?o equation. In preparation
2 Bahouri H,Chemin J-Y,Danchin R.Fourier Analysis and Nonlinear Partial Differential Equations.Comprehensive Studies in Mathematics,vol.243.Berlin-Heidelberg:Springer,2011
3 Bahouri H,G′erard P,Xu C-J.Espaces de Besov et estimations de Strichartz g′en′eralis′ees sur le groupe de Heisenberg.J Anal Math,2000,82:93-118
4 Brezis H,Gallouet T.Nonlinear Schr¨odinger evolution equations.Nonlinear Anal,1980,4:677-681
5 Burq N,G′erard P,Tzvetkov N.Strichartz inequalities and the nonlinear Schr¨odinger equation on compact manifolds.Amer J Math,2004,126:569-605
6 Chemin J-Y.Fluides Parfaits Incompressibles.Ast′erisque,vol.230.Paris:Soc Math France,1995
7 Chemin J-Y,Lerner N.Flot de champs de vecteurs non Lipschitziens et′equations de Navier-Stokes.J Differential Equations,1995,121:314-328
8 Fefferman C.Characterizations of bounded mean oscillation.Bull Amer Math Soc(NS),1971,77:587-588
9 G′erard P.Nonlinear Schr¨odinger equations in inhomogeneous media:Wellposedness and illposedness of the Cauchy problem.In:Proceedings of the International Congress of Mathematicians,vol.3.Z¨urich:Eur Math Soc,2006,157-182
10 G′erard P,Grellier S.The cubic Szeg?o equation.Ann Sci′Ec Norm Sup′er(4),2010,43:761-810
11 G′erard P,Grellier S.Effective integrable dynamics for a certain nonlinear wave equation.Anal PDE,2012,5:1139-1155
12 G′erard P,Grellier S.The Cubic Szeg?o Equation and Hankel Operators.Ast′erisque,vol.389.Paris:Soc Math France,2017
13 G′erard P,Grellier S.Generic colourful tori and inverse spectral transform for Hankel operators.Tunisian J Math,2019,1:347-372
14 G′erard P,Guo Y,Titi E.On the radius of analyticity of solutions to the cubic Szeg?o equation.Ann Inst H Poincar′e Anal Non Lin′eaire,2015,32:97-108
15 G′erard P,Koch H.The cubic Szeg?o flow at low regularity.S′eminaire Laurent Schwartz-EDP et Applications Expos′e No.14,2016
16 G′erard P,Pushnitski A.Weighted model spaces and Schmidt subspaces of Hankel operators.ArXiv:1803.04295,2018
17 G′erard P,Pushnitski A.Inverse spectral theory for a class of non compact Hankel operators.Mathematika,2019,65:132-156
18 John F,Nirenberg F L.On functions of bounded mean oscillation.Comm Pure Appl Math,1961,14:415-426
19 Nehari Z.On bounded bilinear forms.Ann of Math(2),1957,65:153-162
20 Ozawa T,Visciglia N.An improvement on the Br′ezis-Gallou¨et technique for 2D NLS and 1D half-wave equation.Ann Inst H Poincar′e Anal Non Lin′eaire,2016,33:1069-1079
21 Peller V.Hankel Operators and Their Applications.Springer Monographs in Mathematics.New York:Springer-Verlag,2003
22 Pocovnicu O.Explicit formula for the solution of the cubic Szeg?o equation on the real line and applications.Discrete Contin Dyn Syst,2011,31:607-649
23 Thirouin J.Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szeg?o equation.Trans Amer Math Soc,2019,371:3673-3690
24 Xu H.Large-time blowup for a perturbation of the cubic Szeg?o equation.Anal PDE,2014,7:717-731
25 Xu H.Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schr¨odinger equation.Math Z,2017,286:443-489
1)G′erard P, Grellier S. On a damped Szeg?o equation. In preparation