Global well-posedness of critical nonlinear Schrödinger equations below L^2

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April 2013, 33(4): 1389-1405. doi: 10.3934/dcds.2013.33.1389

Global well-posedness of critical nonlinear Schrödinger equations below $L^2$

Yonggeun Cho ^1, , Gyeongha Hwang ^2, and Tohru Ozawa ^3,

1.	Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.	Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
3.	Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received May 2011 Revised August 2012 Published October 2012

Abstract
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The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below $L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

Keywords: critical nonlinearity below $L^2$, global well-posedness, weighted Strichartz estimate, Hartree equations, angular regularity..

Mathematics Subject Classification: Primary: 35Q55; Secondary: 42B3.

Citation: Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389

References:

[1]	C. Ahn and Y. Cho, Lorentz space extension of Strichartz estimate, Proc. Amer. Math. Soc., 133 (2005), 3497-3503.
[2]	L. Bergé, Soliton stability versus collapse, Phys. Rev. E., 62 (2000), 3071-3074.
[3]	A. Bouard and R. Fukuizumi, Stability of standing waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities, Annales de l'IHP., 6 (2005), 1-21.
[4]	N. L. Carothers, "A Short Course on Banach Space Theory," London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005.
[5]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003.
[6]	M. Chae, Y. Cho and S. Lee, Mixed norm estimates of Schrodinger waves and their applications, Commun. Partial Differential Equations, 35 (2010), 906-943.
[7]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., ().
[8]	Y. Cho, S. Lee and T. Ozawa, On Hartree equations with derivatives, Nonlinear Analysis TMA, 74 (2011), 2098-2108.
[9]	Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations, RIMS Kokyuroku Bessatsu, B22 (2010), 145-166.
[10]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contem. Math., 11 (2009), 355-365.
[11]	Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations, DCDS-A, 23 (2009), 1273-1290.
[12]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Func. Anal., 179 (2001), 409-425.
[13]	J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation in 2d, Int. Math. Res. Not. 23 (2007), Art. ID rnm090, 30 pp.
[14]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
[15]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
[16]	J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1080/15332969.1993.9985061.
[17]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[18]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations,, in preprint, ().
[19]	N. Hayashi and T. Ozawa, Smoothing effect for Schrödinger equations, J. Fuctional. Anal., 85 (1989), 307-348.
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2} - Ze^2/r$, Commun. Math. Physics, 53 (1977), 285-294.
[21]	K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.
[22]	J. Kato, M. Nakamura and T. Ozawa, A generalization of the weighted Strichartz estimates for wave equations and an application to self-similar solutions, Comm. Pure Appl. Math., 60 (2007), 164-186. doi: 10.1002/cpa.20133.
[23]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[24]	S. Machihara, M. Nakamura, K. Nakanashi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Func. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.
[25]	F. Merle, Nonexistence of minimal blow-up solutions of equations $iu_t=-\Delta u - k(x)\|u\|^{4/N}u \text{ in } \mathbbR^n$, Ann. Inst. H. Poincaré Phys. Théor, 64 (1996), 33-85.
[26]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Func. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.
[27]	_______, The Cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.
[28]	_______, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pure Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003.
[29]	_______, Global well-posedness and scattering for the defocusing $H^{1/2}$-subcritical Hartree equation in $R^d$, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1831-1852.
[30]	_______, Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation in $\mathbbR^{1+n}$, Commun. Partial Differential Equations, 36 (2011), 729-776.
[31]	K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space. II, Ann. Henri Poincaré, 3 (2002), 503-535.
[32]	M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann., 335 (2006), 645-673. doi: 10.1007/s00208-006-0757-4.
[33]	T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Commun. Partial Differential Equations, 25 (2000), 1471-1485.
[34]	_______, "Nonlinear Dispersive Equations," Local and global analysis, CBMS 106, eds: AMS, 2006.
[35]	I. Towers and B. A. Malomed, Stable, $(2 + 1)$dimensional solutions in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Am. B, 19 (2002), 537-543.
[36]	Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups, Funkcial. Ekvac., 30 (1987), 115-125.
[37]	K. Yosida, "Functional Analysis," Springer, New York, 1965.

show all references

References:

[1]	C. Ahn and Y. Cho, Lorentz space extension of Strichartz estimate, Proc. Amer. Math. Soc., 133 (2005), 3497-3503.
[2]	L. Bergé, Soliton stability versus collapse, Phys. Rev. E., 62 (2000), 3071-3074.
[3]	A. Bouard and R. Fukuizumi, Stability of standing waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities, Annales de l'IHP., 6 (2005), 1-21.
[4]	N. L. Carothers, "A Short Course on Banach Space Theory," London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005.
[5]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003.
[6]	M. Chae, Y. Cho and S. Lee, Mixed norm estimates of Schrodinger waves and their applications, Commun. Partial Differential Equations, 35 (2010), 906-943.
[7]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., ().
[8]	Y. Cho, S. Lee and T. Ozawa, On Hartree equations with derivatives, Nonlinear Analysis TMA, 74 (2011), 2098-2108.
[9]	Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations, RIMS Kokyuroku Bessatsu, B22 (2010), 145-166.
[10]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contem. Math., 11 (2009), 355-365.
[11]	Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations, DCDS-A, 23 (2009), 1273-1290.
[12]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Func. Anal., 179 (2001), 409-425.
[13]	J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation in 2d, Int. Math. Res. Not. 23 (2007), Art. ID rnm090, 30 pp.
[14]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
[15]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205.
[16]	J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension $n \ge 2$, Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1080/15332969.1993.9985061.
[17]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[18]	Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations,, in preprint, ().
[19]	N. Hayashi and T. Ozawa, Smoothing effect for Schrödinger equations, J. Fuctional. Anal., 85 (1989), 307-348.
[20]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2} - Ze^2/r$, Commun. Math. Physics, 53 (1977), 285-294.
[21]	K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.
[22]	J. Kato, M. Nakamura and T. Ozawa, A generalization of the weighted Strichartz estimates for wave equations and an application to self-similar solutions, Comm. Pure Appl. Math., 60 (2007), 164-186. doi: 10.1002/cpa.20133.
[23]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[24]	S. Machihara, M. Nakamura, K. Nakanashi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Func. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.
[25]	F. Merle, Nonexistence of minimal blow-up solutions of equations $iu_t=-\Delta u - k(x)\|u\|^{4/N}u \text{ in } \mathbbR^n$, Ann. Inst. H. Poincaré Phys. Théor, 64 (1996), 33-85.
[26]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Func. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.
[27]	_______, The Cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44.
[28]	_______, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pure Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003.
[29]	_______, Global well-posedness and scattering for the defocusing $H^{1/2}$-subcritical Hartree equation in $R^d$, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1831-1852.
[30]	_______, Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation in $\mathbbR^{1+n}$, Commun. Partial Differential Equations, 36 (2011), 729-776.
[31]	K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space. II, Ann. Henri Poincaré, 3 (2002), 503-535.
[32]	M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann., 335 (2006), 645-673. doi: 10.1007/s00208-006-0757-4.
[33]	T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Commun. Partial Differential Equations, 25 (2000), 1471-1485.
[34]	_______, "Nonlinear Dispersive Equations," Local and global analysis, CBMS 106, eds: AMS, 2006.
[35]	I. Towers and B. A. Malomed, Stable, $(2 + 1)$dimensional solutions in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Am. B, 19 (2002), 537-543.
[36]	Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups, Funkcial. Ekvac., 30 (1987), 115-125.
[37]	K. Yosida, "Functional Analysis," Springer, New York, 1965.

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