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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 |
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.
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. Duyckaerts, J. 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. Fröhlich, T.-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. Killip, D. Li, M. 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. Killip, T. 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. Killip, M. 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. Krieger, E. 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. Li, C. 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. Miao, G. 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. Miao, G. Xu and L. Zhao,
The Cauchy problem of the Hartree equation, J. Partial Differential Equations, 21 (2008), 22-44.
|
[28] |
C. Miao, G. 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. Miao, G. 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. Miao, G. 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. Duyckaerts, J. 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. Fröhlich, T.-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. Killip, D. Li, M. 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. Killip, T. 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. Killip, M. 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. Krieger, E. 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. Li, C. 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. Miao, G. 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. Miao, G. Xu and L. Zhao,
The Cauchy problem of the Hartree equation, J. Partial Differential Equations, 21 (2008), 22-44.
|
[28] |
C. Miao, G. 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. Miao, G. 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. Miao, G. 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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