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

Yonggeun Cho; Gyeongha Hwang; Tohru Ozawa

doi:10.3934/dcds.2013.33.1389

Previous Article
Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds
DCDS Home
This Issue
Next Article
Observable optimal state points of subadditive potentials

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
Related Papers

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 & 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. Google Scholar
[2]	L. Bergé, Soliton stability versus collapse, Phys. Rev. E., 62 (2000), 3071-3074. Google Scholar
[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. Google Scholar
[4]	N. L. Carothers, "A Short Course on Banach Space Theory," London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005. Google Scholar
[5]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003. Google Scholar
[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. Google Scholar
[7]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., (). Google Scholar
[8]	Y. Cho, S. Lee and T. Ozawa, On Hartree equations with derivatives, Nonlinear Analysis TMA, 74 (2011), 2098-2108. Google Scholar
[9]	Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations, RIMS Kokyuroku Bessatsu, B22 (2010), 145-166. Google Scholar
[10]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contem. Math., 11 (2009), 355-365. Google Scholar
[11]	Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations, DCDS-A, 23 (2009), 1273-1290. Google Scholar
[12]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Func. Anal., 179 (2001), 409-425. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. Google Scholar
[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. Google Scholar
[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. 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]	N. Hayashi and T. Ozawa, Smoothing effect for Schrödinger equations, J. Fuctional. Anal., 85 (1989), 307-348. Google Scholar
[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. Google Scholar
[21]	K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. Google Scholar
[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. Google Scholar
[23]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	_______, The Cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Commun. Partial Differential Equations, 25 (2000), 1471-1485. Google Scholar
[34]	_______, "Nonlinear Dispersive Equations," Local and global analysis, CBMS 106, eds: AMS, 2006. Google Scholar
[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. Google Scholar
[36]	Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups, Funkcial. Ekvac., 30 (1987), 115-125. Google Scholar
[37]	K. Yosida, "Functional Analysis," Springer, New York, 1965. Google Scholar

show all references

References:

[1]	C. Ahn and Y. Cho, Lorentz space extension of Strichartz estimate, Proc. Amer. Math. Soc., 133 (2005), 3497-3503. Google Scholar
[2]	L. Bergé, Soliton stability versus collapse, Phys. Rev. E., 62 (2000), 3071-3074. Google Scholar
[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. Google Scholar
[4]	N. L. Carothers, "A Short Course on Banach Space Theory," London Mathematical Society Student Texts No. 64, Cambridge University Press, 2005. Google Scholar
[5]	T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics 10, American Mathematical Society, 2003. Google Scholar
[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. Google Scholar
[7]	Y. Cho and S. Lee, Strichartz estimates in spherical coordinates,, Indiana Univ. Math. J., (). Google Scholar
[8]	Y. Cho, S. Lee and T. Ozawa, On Hartree equations with derivatives, Nonlinear Analysis TMA, 74 (2011), 2098-2108. Google Scholar
[9]	Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations, RIMS Kokyuroku Bessatsu, B22 (2010), 145-166. Google Scholar
[10]	Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contem. Math., 11 (2009), 355-365. Google Scholar
[11]	Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations, DCDS-A, 23 (2009), 1273-1290. Google Scholar
[12]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Func. Anal., 179 (2001), 409-425. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 23 (2011), 181-205. Google Scholar
[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. Google Scholar
[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. 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]	N. Hayashi and T. Ozawa, Smoothing effect for Schrödinger equations, J. Fuctional. Anal., 85 (1989), 307-348. Google Scholar
[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. Google Scholar
[21]	K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147. Google Scholar
[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. Google Scholar
[23]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	_______, The Cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Commun. Partial Differential Equations, 25 (2000), 1471-1485. Google Scholar
[34]	_______, "Nonlinear Dispersive Equations," Local and global analysis, CBMS 106, eds: AMS, 2006. Google Scholar
[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. Google Scholar
[36]	Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups, Funkcial. Ekvac., 30 (1987), 115-125. Google Scholar
[37]	K. Yosida, "Functional Analysis," Springer, New York, 1965. Google Scholar

[1]	Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725
[2]	P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[3]	Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3683-3702. doi: 10.3934/cpaa.2021126
[4]	Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023
[5]	Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[6]	Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
[7]	Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[8]	Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273
[9]	M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573
[10]	Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[11]	Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865
[12]	Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[13]	Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[14]	Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
[15]	Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903
[16]	Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L²-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084
[17]	Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[18]	Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[19]	Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021147
[20]	Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657

2020 Impact Factor: 1.392

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors