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

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

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

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.
Citation: Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
References:
[1]

Proc. Amer. Math. Soc., 133 (2005), 3497-3503.  Google Scholar

[2]

Phys. Rev. E., 62 (2000), 3071-3074.  Google Scholar

[3]

Annales de l'IHP., 6 (2005), 1-21.  Google Scholar

[4]

London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005.  Google Scholar

[5]

Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003.  Google Scholar

[6]

Commun. Partial Differential Equations, 35 (2010), 906-943.  Google Scholar

[7]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., ().   Google Scholar

[8]

Nonlinear Analysis TMA, 74 (2011), 2098-2108.  Google Scholar

[9]

RIMS Kokyuroku Bessatsu, B22 (2010), 145-166.  Google Scholar

[10]

Commun. Contem. Math., 11 (2009), 355-365.  Google Scholar

[11]

DCDS-A, 23 (2009), 1273-1290.  Google Scholar

[12]

J. Func. Anal., 179 (2001), 409-425.  Google Scholar

[13]

Int. Math. Res. Not. 23 (2007), Art. ID rnm090, 30 pp.  Google Scholar

[14]

Comm. Pure Appl. Math., 57 (2004), 987-1014.  Google Scholar

[15]

Forum Math., 23 (2011), 181-205.  Google Scholar

[16]

Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1080/15332969.1993.9985061.  Google Scholar

[17]

J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[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, ().   Google Scholar

[19]

J. Fuctional. Anal., 85 (1989), 307-348.  Google Scholar

[20]

Commun. Math. Physics, 53 (1977), 285-294.  Google Scholar

[21]

Funkcial. Ekvac., 51 (2008), 135-147.  Google Scholar

[22]

Comm. Pure Appl. Math., 60 (2007), 164-186. doi: 10.1002/cpa.20133.  Google Scholar

[23]

Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[24]

J. Func. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.  Google Scholar

[25]

Ann. Inst. H. Poincaré Phys. Théor, 64 (1996), 33-85.  Google Scholar

[26]

J. Func. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[27]

J. Partial Diff. Eqs., 21 (2008), 22-44.  Google Scholar

[28]

J. Math. Pure Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003.  Google Scholar

[29]

Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1831-1852.  Google Scholar

[30]

Commun. Partial Differential Equations, 36 (2011), 729-776.  Google Scholar

[31]

Ann. Henri Poincaré, 3 (2002), 503-535.  Google Scholar

[32]

Math. Ann., 335 (2006), 645-673. doi: 10.1007/s00208-006-0757-4.  Google Scholar

[33]

Commun. Partial Differential Equations, 25 (2000), 1471-1485.  Google Scholar

[34]

Local and global analysis, CBMS 106, eds: AMS, 2006.  Google Scholar

[35]

J. Opt. Soc. Am. B, 19 (2002), 537-543.  Google Scholar

[36]

Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar

[37]

Springer, New York, 1965.  Google Scholar

show all references

References:
[1]

Proc. Amer. Math. Soc., 133 (2005), 3497-3503.  Google Scholar

[2]

Phys. Rev. E., 62 (2000), 3071-3074.  Google Scholar

[3]

Annales de l'IHP., 6 (2005), 1-21.  Google Scholar

[4]

London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005.  Google Scholar

[5]

Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003.  Google Scholar

[6]

Commun. Partial Differential Equations, 35 (2010), 906-943.  Google Scholar

[7]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., ().   Google Scholar

[8]

Nonlinear Analysis TMA, 74 (2011), 2098-2108.  Google Scholar

[9]

RIMS Kokyuroku Bessatsu, B22 (2010), 145-166.  Google Scholar

[10]

Commun. Contem. Math., 11 (2009), 355-365.  Google Scholar

[11]

DCDS-A, 23 (2009), 1273-1290.  Google Scholar

[12]

J. Func. Anal., 179 (2001), 409-425.  Google Scholar

[13]

Int. Math. Res. Not. 23 (2007), Art. ID rnm090, 30 pp.  Google Scholar

[14]

Comm. Pure Appl. Math., 57 (2004), 987-1014.  Google Scholar

[15]

Forum Math., 23 (2011), 181-205.  Google Scholar

[16]

Comm. Math. Phys., 151 (1993), 619-645. doi: 10.1080/15332969.1993.9985061.  Google Scholar

[17]

J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[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, ().   Google Scholar

[19]

J. Fuctional. Anal., 85 (1989), 307-348.  Google Scholar

[20]

Commun. Math. Physics, 53 (1977), 285-294.  Google Scholar

[21]

Funkcial. Ekvac., 51 (2008), 135-147.  Google Scholar

[22]

Comm. Pure Appl. Math., 60 (2007), 164-186. doi: 10.1002/cpa.20133.  Google Scholar

[23]

Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[24]

J. Func. Anal., 219 (2005), 1-20. doi: 10.1016/j.jfa.2004.07.005.  Google Scholar

[25]

Ann. Inst. H. Poincaré Phys. Théor, 64 (1996), 33-85.  Google Scholar

[26]

J. Func. Anal., 253 (2007), 605-627. doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[27]

J. Partial Diff. Eqs., 21 (2008), 22-44.  Google Scholar

[28]

J. Math. Pure Appl., 91 (2009), 49-79. doi: 10.1016/j.matpur.2008.09.003.  Google Scholar

[29]

Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1831-1852.  Google Scholar

[30]

Commun. Partial Differential Equations, 36 (2011), 729-776.  Google Scholar

[31]

Ann. Henri Poincaré, 3 (2002), 503-535.  Google Scholar

[32]

Math. Ann., 335 (2006), 645-673. doi: 10.1007/s00208-006-0757-4.  Google Scholar

[33]

Commun. Partial Differential Equations, 25 (2000), 1471-1485.  Google Scholar

[34]

Local and global analysis, CBMS 106, eds: AMS, 2006.  Google Scholar

[35]

J. Opt. Soc. Am. B, 19 (2002), 537-543.  Google Scholar

[36]

Funkcial. Ekvac., 30 (1987), 115-125.  Google Scholar

[37]

Springer, New York, 1965.  Google Scholar

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