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Remarks on the full dispersion Davey-Stewartson systems
Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data
1. | School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom |
References:
[1] |
A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955).
|
[2] |
H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen,, \emph{Acta Math.}, 45 (1925), 29.
doi: 10.1007/BF02395468. |
[3] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, \emph{Geom. Funct. Anal.}, 3 (1993), 107.
doi: 10.1007/BF01896020. |
[4] |
J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS,, \emph{J. Anal. Math.}, 94 (2004), 125.
doi: 10.1007/BF02789044. |
[5] |
A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum,, \emph{J. Anal. Math.}, 72 (1997), 1.
doi: 10.1007/BF02843151. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011).
|
[7] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705.
doi: 10.1090/S0894-0347-03-00421-1. |
[9] |
C. Corduneanu, Almost Periodic Functions,, With the collaboration of N. Gheorghiu and V. Barbu. Translated from the Romanian by Gitta Bernstein and Eugene Tomer. Interscience Tracts in Pure and Applied Mathematics, (1968).
|
[10] |
D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data,, preprint, (). Google Scholar |
[11] |
I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense,, in \emph{Spectral Operator Theory and Related Topics, (1994), 181.
|
[12] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3,, \emph{Ann. Inst. H. Poincar\'e Sect. A (N.S.)}, 28 (1978), 287.
|
[13] |
Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
doi: 10.1017/CBO9781139165372. |
[14] |
T. Ogawa and Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition,, \emph{Functional-analytic Methods for Partial Differential Equations} (Tokyo, (1989), 236.
doi: 10.1007/BFb0084910. |
[15] |
T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 487.
doi: 10.2307/2048340. |
[16] |
T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253.
doi: 10.1137/140973384. |
[17] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[18] |
K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data,, \emph{SIAM J. Math. Anal.}, 44 (2012), 3412.
doi: 10.1137/110849973. |
[19] |
Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115.
|
show all references
References:
[1] |
A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955).
|
[2] |
H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen,, \emph{Acta Math.}, 45 (1925), 29.
doi: 10.1007/BF02395468. |
[3] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, \emph{Geom. Funct. Anal.}, 3 (1993), 107.
doi: 10.1007/BF01896020. |
[4] |
J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS,, \emph{J. Anal. Math.}, 94 (2004), 125.
doi: 10.1007/BF02789044. |
[5] |
A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum,, \emph{J. Anal. Math.}, 72 (1997), 1.
doi: 10.1007/BF02843151. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011).
|
[7] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$,, \emph{J. Amer. Math. Soc.}, 16 (2003), 705.
doi: 10.1090/S0894-0347-03-00421-1. |
[9] |
C. Corduneanu, Almost Periodic Functions,, With the collaboration of N. Gheorghiu and V. Barbu. Translated from the Romanian by Gitta Bernstein and Eugene Tomer. Interscience Tracts in Pure and Applied Mathematics, (1968).
|
[10] |
D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data,, preprint, (). Google Scholar |
[11] |
I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense,, in \emph{Spectral Operator Theory and Related Topics, (1994), 181.
|
[12] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3,, \emph{Ann. Inst. H. Poincar\'e Sect. A (N.S.)}, 28 (1978), 287.
|
[13] |
Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004).
doi: 10.1017/CBO9781139165372. |
[14] |
T. Ogawa and Y. Tsutsumi, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary condition,, \emph{Functional-analytic Methods for Partial Differential Equations} (Tokyo, (1989), 236.
doi: 10.1007/BFb0084910. |
[15] |
T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity,, \emph{Proc. Amer. Math. Soc.}, 111 (1991), 487.
doi: 10.2307/2048340. |
[16] |
T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253.
doi: 10.1137/140973384. |
[17] |
T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006).
|
[18] |
K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data,, \emph{SIAM J. Math. Anal.}, 44 (2012), 3412.
doi: 10.1137/110849973. |
[19] |
Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115.
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