# American Institute of Mathematical Sciences

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 - A, 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.   Google Scholar [2] L. Bergé, Soliton stability versus collapse,, Phys. Rev. E., 62 (2000), 3071.   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.   Google Scholar [4] N. L. Carothers, "A Short Course on Banach Space Theory,", London Mathematical Society Student Texts No. 64, (2005).   Google Scholar [5] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics 10, (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.   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.   Google Scholar [9] Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations,, RIMS Kokyuroku Bessatsu, B22 (2010), 145.   Google Scholar [10] Y. Cho and T. Ozawa, Sobolev inequalities with symmetry,, Commun. Contem. Math., 11 (2009), 355.   Google Scholar [11] Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations,, DCDS-A, 23 (2009), 1273.   Google Scholar [12] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Func. Anal., 179 (2001), 409.   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), (2007).   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.   Google Scholar [15] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181.   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.  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.  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.   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.   Google Scholar [21] K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity,, Funkcial. Ekvac., 51 (2008), 135.   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.  doi: 10.1002/cpa.20133.  Google Scholar [23] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  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.  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.   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.  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.   Google Scholar [28] _______, Global well-posedness and scattering for the mass-critical Hartree equation with radial data,, J. Math. Pure Appl., 91 (2009), 49.  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.   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.   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.   Google Scholar [32] M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case,, Math. Ann., 335 (2006), 645.  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.   Google Scholar [34] _______, "Nonlinear Dispersive Equations,", Local and global analysis, (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.   Google Scholar [36] Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups,, Funkcial. Ekvac., 30 (1987), 115.   Google Scholar [37] K. Yosida, "Functional Analysis,", Springer, (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.   Google Scholar [2] L. Bergé, Soliton stability versus collapse,, Phys. Rev. E., 62 (2000), 3071.   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.   Google Scholar [4] N. L. Carothers, "A Short Course on Banach Space Theory,", London Mathematical Society Student Texts No. 64, (2005).   Google Scholar [5] T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics 10, (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.   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.   Google Scholar [9] Y. Cho and K. Nakanishi, On the global existence of semirelativistic Hartree equations,, RIMS Kokyuroku Bessatsu, B22 (2010), 145.   Google Scholar [10] Y. Cho and T. Ozawa, Sobolev inequalities with symmetry,, Commun. Contem. Math., 11 (2009), 355.   Google Scholar [11] Y. Cho, T. Ozawa, H. Sasaki and Y. Shim, Remarks on the semirelativistic Hartree equations,, DCDS-A, 23 (2009), 1273.   Google Scholar [12] M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Func. Anal., 179 (2001), 409.   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), (2007).   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.   Google Scholar [15] D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications,, Forum Math., 23 (2011), 181.   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.  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.  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.   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.   Google Scholar [21] K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity,, Funkcial. Ekvac., 51 (2008), 135.   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.  doi: 10.1002/cpa.20133.  Google Scholar [23] M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  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.  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.   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.  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.   Google Scholar [28] _______, Global well-posedness and scattering for the mass-critical Hartree equation with radial data,, J. Math. Pure Appl., 91 (2009), 49.  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.   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.   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.   Google Scholar [32] M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case,, Math. Ann., 335 (2006), 645.  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.   Google Scholar [34] _______, "Nonlinear Dispersive Equations,", Local and global analysis, (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.   Google Scholar [36] Y. Tsutsumi, $L^2$-solutions for noninear Schrödinger equations and noninear groups,, Funkcial. Ekvac., 30 (1987), 115.   Google Scholar [37] K. Yosida, "Functional Analysis,", Springer, (1965).   Google Scholar
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