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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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