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, 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
 [1] Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227 [2] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393 [3] Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057 [4] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006 [5] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 [6] Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $n$-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018 [7] Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001 [8] Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005 [9] Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022 [10] Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $2$D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093 [11] Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237 [12] Jiangang Qi, Bing Xie. Extremum estimates of the $L^1$-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243 [13] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [14] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [15] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 [16] Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 [17] A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121. [18] Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270 [19] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [20] Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060

2019 Impact Factor: 1.338

Metrics

• PDF downloads (78)
• HTML views (0)
• Cited by (7)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]