April  2017, 37(4): 1979-2007. doi: 10.3934/dcds.2017084

Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data

1. 

Department of Mathematics, Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications, Fujian Normal University, Fuzhou, 350007, China

* Corresponding author

Received  March 2016 Revised  November 2016 Published  December 2016

Fund Project: Both authors are supported by NFSC (Grant no.11501111), and the first author is partially supported by the program Nonlinear Analysis and Its Applications(IRTL1206) from Fujian Normal University.

We prove that for the Cauchy problem of focusing $L^2$-critical Hartree equations with spherically symmetric $H^1$ data in dimensions $3$ and $4$, the global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. The approach is a linearization analysis around the ground state combined with an in-out spherical wave decomposition technique.

Citation: Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084
References:
[1]

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

[2]

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.

[3]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[4]

J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in Séminaire: Equations aux Dérivées Partielles. 2003-2004, Sémin. Equ. Dériv. Partielles, École Polytech. , Palaiseau, 2004, Exp. No. XIX, 26pp.

[5]

J. FröhlichT.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.  doi: 10.1007/s002200100579.

[6]

Y. Gao and H. Wu, Scattering for the focusing $\dot H^{1/2}$-critical Hartree equation in energy space, Nonlinear Anal., 73 (2010), 1043-1056.  doi: 10.1016/j.na.2010.04.033.

[7]

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.

[8]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005 (2005), 2815-2828.  doi: 10.1155/IMRN.2005.2815.

[9]

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

[10]

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

[11]

S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.

[12]

R. KillipD. LiM. Visan and X. Zhang, Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS, SIAM J. Math. Anal., 41 (2009), 219-236.  doi: 10.1137/080720358.

[13]

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.

[14]

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.

[15]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer, Math. Soc., Providence, RI, (2013), 325-437. 

[16]

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.

[17]

J. KriegerE. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.  doi: 10.1007/s00023-009-0010-2.

[18]

M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\textbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[19]

D. LiC. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013.

[20]

D. Li and X. Zhang, On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation, Adv. Math., 220 (2009), 1171-1192.  doi: 10.1016/j.aim.2008.10.013.

[21]

D. Li and X. Zhang, On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions $d≥q 2$, Sci. China Math., 55 (2012), 385-434.  doi: 10.1007/s11425-012-4359-1.

[22]

E. H. Lieb and M. Loss, Analysis vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[23]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123.  doi: 10.1007/PL00001673.

[24]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), 427-454.  doi: 10.1215/S0012-7094-93-06919-0.

[25]

F. Merle and P. Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2), 161 (2005), 157-222.  doi: 10.4007/annals.2005.161.157.

[26]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627.  doi: 10.1016/j.jfa.2007.09.008.

[27]

C. MiaoG. Xu and L. Zhao, The Cauchy problem of the Hartree equation, J. Partial Differential Equations, 21 (2008), 22-44. 

[28]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl. (9), 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003.

[29]

C. MiaoG. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5.

[30]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbb R^{1+n}$, Comm. Partial Differential Equations, 36 (2011), 729-776.  doi: 10.1080/03605302.2010.531073.

[31]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

show all references

References:
[1]

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

[2]

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.

[3]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[4]

J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in Séminaire: Equations aux Dérivées Partielles. 2003-2004, Sémin. Equ. Dériv. Partielles, École Polytech. , Palaiseau, 2004, Exp. No. XIX, 26pp.

[5]

J. FröhlichT.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.  doi: 10.1007/s002200100579.

[6]

Y. Gao and H. Wu, Scattering for the focusing $\dot H^{1/2}$-critical Hartree equation in energy space, Nonlinear Anal., 73 (2010), 1043-1056.  doi: 10.1016/j.na.2010.04.033.

[7]

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.

[8]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005 (2005), 2815-2828.  doi: 10.1155/IMRN.2005.2815.

[9]

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

[10]

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

[11]

S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192.  doi: 10.1016/j.jfa.2005.10.005.

[12]

R. KillipD. LiM. Visan and X. Zhang, Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS, SIAM J. Math. Anal., 41 (2009), 219-236.  doi: 10.1137/080720358.

[13]

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.

[14]

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.

[15]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in Evolution equations, vol. 17 of Clay Math. Proc., Amer, Math. Soc., Providence, RI, (2013), 325-437. 

[16]

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.

[17]

J. KriegerE. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205.  doi: 10.1007/s00023-009-0010-2.

[18]

M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\textbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[19]

D. LiC. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163.  doi: 10.1016/j.jde.2008.05.013.

[20]

D. Li and X. Zhang, On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation, Adv. Math., 220 (2009), 1171-1192.  doi: 10.1016/j.aim.2008.10.013.

[21]

D. Li and X. Zhang, On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions $d≥q 2$, Sci. China Math., 55 (2012), 385-434.  doi: 10.1007/s11425-012-4359-1.

[22]

E. H. Lieb and M. Loss, Analysis vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[23]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123.  doi: 10.1007/PL00001673.

[24]

F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69 (1993), 427-454.  doi: 10.1215/S0012-7094-93-06919-0.

[25]

F. Merle and P. Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2), 161 (2005), 157-222.  doi: 10.4007/annals.2005.161.157.

[26]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627.  doi: 10.1016/j.jfa.2007.09.008.

[27]

C. MiaoG. Xu and L. Zhao, The Cauchy problem of the Hartree equation, J. Partial Differential Equations, 21 (2008), 22-44. 

[28]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl. (9), 91 (2009), 49-79.  doi: 10.1016/j.matpur.2008.09.003.

[29]

C. MiaoG. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5.

[30]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in $\mathbb R^{1+n}$, Comm. Partial Differential Equations, 36 (2011), 729-776.  doi: 10.1080/03605302.2010.531073.

[31]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

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