January  2019, 39(1): 553-583. doi: 10.3934/dcds.2019023

The radial mass-subcritical NLS in negative order Sobolev spaces

1. 

Department of Mathematics, University of California Los Angeles, Los Angeles, CA, USA

2. 

Department of Systems Innovation, Graduate School of Engineering Sciences, Toyonaka, Osaka, Japan

3. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

* Corresponding author: Jason Murphy

Received  April 2018 Published  October 2018

We consider the mass-subcritical NLS in dimensions $d≥ 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

Citation: Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan. The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 553-583. doi: 10.3934/dcds.2019023
References:
[1]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, arXiv: math/0311048.

[4]

J. CollianderM. KeelG. StaffilaniT. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${\mathbb{R}}^3$, Ann. of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1, Amer. J. of Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), 3435-3516.  doi: 10.1215/00127094-3673888.

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.  doi: 10.1090/S0894-0347-2011-00727-3.

[8]

B. Dodson, Global well - posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, Preprint, arXiv: 1409.1950.

[9]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[10]

B. DodsonC. MiaoJ. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 759-787.  doi: 10.1016/j.anihpc.2016.05.004.

[11]

C. Gao, C. Miao and J. Yang, The intercritical defocusing nonlinear Schrödinger equation with radial initial data in dimensions four and higher, Preprint, arXiv: 1707.04686.

[12]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.  doi: 10.1007/BF02099195.

[13]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844.  doi: 10.1080/03605300008821569.

[14]

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, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.

[15]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.  doi: 10.1619/fesi.51.135.

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[17]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[18]

C. E. Kenig and F. Merle, Scattering for ${\dot H^{1/2}}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962.  doi: 10.1090/S0002-9947-09-04722-9.

[19]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (201), 617-633.  doi: 10.1215/S0012-7094-01-10638-8.

[20]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33pp. doi: 10.1007/s00030-017-0463-9.

[21]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258.  doi: 10.4171/JEMS/180.

[22]

R. Killip and M. Visan, Energy-supercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, Comm, Comm. Partial Differential Equations, 35 (2010), 945-987.  doi: 10.1080/03605301003717084.

[23]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[24]

R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.  doi: 10.2140/apde.2012.5.855.

[25]

R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Clay Math. Proc., 17 (2013), 325-437. 

[26]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266.  doi: 10.2140/apde.2008.1.229.

[27]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. xii+312 pp

[28]

J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263.  doi: 10.1016/0022-1236(78)90073-3.

[29]

C. Lu and J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differential Equations, 262 (2017), 4390-4414.  doi: 10.1016/j.jde.2017.01.005.

[30]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531.  doi: 10.3934/cpaa.2015.14.1481.

[31]

S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653.  doi: 10.1080/03605302.2017.1286672.

[32]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcrticial nonlinear Schrödinger equation, Preprin, t arXiv: 1605.09234.

[33]

S. Masaki, On the Scattering Problem of Mass-Subcritical Hartree Equation, Adv. Stud. Pure Math., in press.

[34]

S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the masssubcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré C, Anal. Non Lineairé, 35 (2018), 283-326.  doi: 10.1016/j.anihpc.2017.04.003.

[35]

S. Masaki and J. Segata, Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application, SIAM J. Math. Anal., 50 (2018), 2839-2866.  doi: 10.1137/17M1153893.

[36]

C. MiaoJ. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., 267 (2014), 1662-1724.  doi: 10.1016/j.jfa.2014.06.016.

[37]

J. Murphy, Intercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, SIAM J. Math. Anal., 46 (2014), 939-997.  doi: 10.1137/120898280.

[38]

J. Murphy, The defocusing ${\dot H^{1/2}}$-critical NLS in high dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 733-748.  doi: 10.3934/dcds.2014.34.733.

[39]

J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations, 40 (2015), 265-308.  doi: 10.1080/03605302.2014.949379.

[40]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in ${\mathbb{R}}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.

[41]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993.

[42]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[43]

T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80. 

[44]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202.  doi: 10.1215/S0012-7094-07-14015-8.

[45]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D Thesis, UCLA, 2006.

[46]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.

[47]

M. Visan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not., IMRN 2012, 1037–1067. doi: 10.1093/imrn/rnr051.

[48]

J. Xie and D. Fang, Global well-posedness and scattering for the defocusing ${\dot H^{s}}$-critical NLS, Chin. Ann. Math. Ser. B, 34 (2013), 801-842.  doi: 10.1007/s11401-013-0808-6.

[49]

T. Zhao, The defocusing energy-supercritical NLS in higher dimensions, Acta. Math Sin. (Eng. Ser.), 33 (2017), 911-925.  doi: 10.1007/s10114-017-6499-2.

show all references

References:
[1]

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  doi: 10.1090/S0894-0347-99-00283-0.

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, arXiv: math/0311048.

[4]

J. CollianderM. KeelG. StaffilaniT. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${\mathbb{R}}^3$, Ann. of Math., 167 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.

[5]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1, Amer. J. of Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2, Duke Math. J., 165 (2016), 3435-3516.  doi: 10.1215/00127094-3673888.

[7]

B. Dodson, Global well-posedness and scattering for the defocusing, L2-critical nonlinear Schrödinger equation when d ≥ 3, J. Amer. Math. Soc., 25 (2012), 429-463.  doi: 10.1090/S0894-0347-2011-00727-3.

[8]

B. Dodson, Global well - posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, Preprint, arXiv: 1409.1950.

[9]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030.

[10]

B. DodsonC. MiaoJ. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 759-787.  doi: 10.1016/j.anihpc.2016.05.004.

[11]

C. Gao, C. Miao and J. Yang, The intercritical defocusing nonlinear Schrödinger equation with radial initial data in dimensions four and higher, Preprint, arXiv: 1707.04686.

[12]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.  doi: 10.1007/BF02099195.

[13]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844.  doi: 10.1080/03605300008821569.

[14]

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, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.

[15]

K. Hidano, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac., 51 (2008), 135-147.  doi: 10.1619/fesi.51.135.

[16]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[17]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[18]

C. E. Kenig and F. Merle, Scattering for ${\dot H^{1/2}}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962.  doi: 10.1090/S0002-9947-09-04722-9.

[19]

C. E. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (201), 617-633.  doi: 10.1215/S0012-7094-01-10638-8.

[20]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33pp. doi: 10.1007/s00030-017-0463-9.

[21]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258.  doi: 10.4171/JEMS/180.

[22]

R. Killip and M. Visan, Energy-supercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, Comm, Comm. Partial Differential Equations, 35 (2010), 945-987.  doi: 10.1080/03605301003717084.

[23]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[24]

R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.  doi: 10.2140/apde.2012.5.855.

[25]

R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Clay Math. Proc., 17 (2013), 325-437. 

[26]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266.  doi: 10.2140/apde.2008.1.229.

[27]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves. Generalized Korteweg-de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. xii+312 pp

[28]

J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263.  doi: 10.1016/0022-1236(78)90073-3.

[29]

C. Lu and J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differential Equations, 262 (2017), 4390-4414.  doi: 10.1016/j.jde.2017.01.005.

[30]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531.  doi: 10.3934/cpaa.2015.14.1481.

[31]

S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653.  doi: 10.1080/03605302.2017.1286672.

[32]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcrticial nonlinear Schrödinger equation, Preprin, t arXiv: 1605.09234.

[33]

S. Masaki, On the Scattering Problem of Mass-Subcritical Hartree Equation, Adv. Stud. Pure Math., in press.

[34]

S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the masssubcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré C, Anal. Non Lineairé, 35 (2018), 283-326.  doi: 10.1016/j.anihpc.2017.04.003.

[35]

S. Masaki and J. Segata, Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application, SIAM J. Math. Anal., 50 (2018), 2839-2866.  doi: 10.1137/17M1153893.

[36]

C. MiaoJ. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., 267 (2014), 1662-1724.  doi: 10.1016/j.jfa.2014.06.016.

[37]

J. Murphy, Intercritical NLS: Critical ${\dot H^{s}}$-bounds imply scattering, SIAM J. Math. Anal., 46 (2014), 939-997.  doi: 10.1137/120898280.

[38]

J. Murphy, The defocusing ${\dot H^{1/2}}$-critical NLS in high dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 733-748.  doi: 10.3934/dcds.2014.34.733.

[39]

J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations, 40 (2015), 265-308.  doi: 10.1080/03605302.2014.949379.

[40]

E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in ${\mathbb{R}}^{1+4}$, Amer. J. Math., 129 (2007), 1-60.  doi: 10.1353/ajm.2007.0004.

[41]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993.

[42]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[43]

T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math., 11 (2005), 57-80. 

[44]

T. TaoM. Visan and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202.  doi: 10.1215/S0012-7094-07-14015-8.

[45]

M. Visan, The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D Thesis, UCLA, 2006.

[46]

M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.

[47]

M. Visan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not., IMRN 2012, 1037–1067. doi: 10.1093/imrn/rnr051.

[48]

J. Xie and D. Fang, Global well-posedness and scattering for the defocusing ${\dot H^{s}}$-critical NLS, Chin. Ann. Math. Ser. B, 34 (2013), 801-842.  doi: 10.1007/s11401-013-0808-6.

[49]

T. Zhao, The defocusing energy-supercritical NLS in higher dimensions, Acta. Math Sin. (Eng. Ser.), 33 (2017), 911-925.  doi: 10.1007/s10114-017-6499-2.

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