Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data

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July 2015, 14(4): 1563-1580. doi: 10.3934/cpaa.2015.14.1563

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

Received July 2014 Revised December 2014 Published April 2014

Abstract
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We consider the Cauchy problem for the defocusing nonlinear Schrödinger equations (NLS) on the real line with a special subclass of almost periodic functions as initial data. In particular, we prove global existence of solutions to NLS with limit periodic functions as initial data under some regularity assumption.

Keywords: global existence, well-posedness, limit periodic functions., almost periodic functions, Nonlinear Schrödinger equation.

Mathematics Subject Classification: Primanry: 35Q55; Secondary: 11K70, 42A7.

Citation: Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563

References:

[1]	A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar
[7]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). doi: 10.1017/CBO9781139165372. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253. doi: 10.1137/140973384. Google Scholar
[17]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar
[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. Google Scholar
[19]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

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References:

[1]	A. Besicovitch, Almost Periodic Functions,, Dover Publications, (1955). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext. Springer, (2011). Google Scholar
[7]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	Y. Katznelson, An Introduction to Harmonic Analysis,, Third edition. Cambridge Mathematical Library. Cambridge University Press, (2004). doi: 10.1017/CBO9781139165372. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	T. Oh, On nonlinear Schrödinger equations with almost periodic initial data,, \emph{SIAM J. Math. Anal.}, 47 (2015), 1253. doi: 10.1137/140973384. Google Scholar
[17]	T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis,, CBMS Regional Conference Series in Mathematics, (2006). Google Scholar
[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. Google Scholar
[19]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups,, \emph{ Funkcial. Ekvac.}, 30 (1987), 115. Google Scholar

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