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

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.
Citation: Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563
References:
[1]

A. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.

[2]

H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen, Acta Math., 45 (1925), 29-127. doi: 10.1007/BF02395468.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS, J. Anal. Math., 94 (2004), 125-157. doi: 10.1007/BF02789044.

[5]

A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum, J. Anal. Math., 72 (1997), 1-20. doi: 10.1007/BF02843151.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. 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, No. 22. Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 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, arXiv:1212.2674v2.

[11]

I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, in Spectral Operator Theory and Related Topics, 181-208, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994.

[12]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3, Ann. Inst. H. Poincaré Sect. A (N.S.), 28 (1978), 287-316.

[13]

Y. Katznelson, An Introduction to Harmonic Analysis, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 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, Functional-analytic Methods for Partial Differential Equations (Tokyo, 1989), 236-251, Lecture Notes in Math., 1450, Springer, Berlin, 1990. 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, Proc. Amer. Math. Soc., 111 (1991), 487-496. doi: 10.2307/2048340.

[16]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data, SIAM J. Math. Anal., 47 (2015), 1253-1270. doi: 10.1137/140973384.

[17]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.

[18]

K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data, SIAM J. Math. Anal., 44 (2012), 3412-3428. doi: 10.1137/110849973.

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

show all references

References:
[1]

A. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.

[2]

H. Bohr, Zur theorie der fast periodischen funktionen. I. Eine verallgemeinerung der theorie der fourierreihen, Acta Math., 45 (1925), 29-127. doi: 10.1007/BF02395468.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, A remark on normal forms and the "$I$-method'' for periodic NLS, J. Anal. Math., 94 (2004), 125-157. doi: 10.1007/BF02789044.

[5]

A. Boutet de Monvel and I. Egorova, On solutions of nonlinear Schrödinger equations with Cantor-type spectrum, J. Anal. Math., 72 (1997), 1-20. doi: 10.1007/BF02843151.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749. 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, No. 22. Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 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, arXiv:1212.2674v2.

[11]

I. Egorova, The Cauchy problem for the KdV equation with almost periodic initial data whose spectrum is nowhere dense, in Spectral Operator Theory and Related Topics, 181-208, Adv. Soviet Math., 19, Amer. Math. Soc., Providence, RI, 1994.

[12]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. III. Special theories in dimensions 1, 2 and 3, Ann. Inst. H. Poincaré Sect. A (N.S.), 28 (1978), 287-316.

[13]

Y. Katznelson, An Introduction to Harmonic Analysis, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 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, Functional-analytic Methods for Partial Differential Equations (Tokyo, 1989), 236-251, Lecture Notes in Math., 1450, Springer, Berlin, 1990. 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, Proc. Amer. Math. Soc., 111 (1991), 487-496. doi: 10.2307/2048340.

[16]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data, SIAM J. Math. Anal., 47 (2015), 1253-1270. doi: 10.1137/140973384.

[17]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.

[18]

K. Tsugawa, Local well-posedness of the KdV equation with quasi-periodic initial data, SIAM J. Math. Anal., 44 (2012), 3412-3428. doi: 10.1137/110849973.

[19]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

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