• Previous Article
    Mathematics of nonlinear acoustics
  • EECT Home
  • This Issue
  • Next Article
    On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions
December  2015, 4(4): 431-445. doi: 10.3934/eect.2015.4.431

On the Cauchy problem for the Schrödinger-Hartree equation

1. 

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China, China

Received  August 2015 Revised  October 2015 Published  November 2015

In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation \begin{equation*} iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0, \end{equation*} where $I_\alpha$ is the Riesz potential. Firstly, we address questions related to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.
Citation: Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
References:
[1]

P. d'Avenia and M. Squassina, Soliton dynamics for the Schrödinger-Newton system, Math. Models Methods Appl. Sci., 24 (2014), 553-572. doi: 10.1142/S0218202513500590.  Google Scholar

[2]

C. Bonanno, P. d'Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl., 417 (2014), 180-199. doi: 10.1016/j.jmaa.2014.02.063.  Google Scholar

[3]

D. Cao and Y. Su, Minimal blow-up solutions of mass-critical inhomogeneous Hartree equation, Journal of Mathematical Physics, 54 (2013), 121511, 25pp. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[5]

J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D: Nonlinear Phenomena, 227 (2007), 142-148. doi: 10.1016/j.physd.2007.01.004.  Google Scholar

[6]

J. Frölich, 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.  Google Scholar

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768.  Google Scholar

[8]

H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923. doi: 10.3934/dcdss.2012.5.903.  Google Scholar

[9]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. doi: 10.1051/cocv:1998107.  Google Scholar

[10]

R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Theor., 46 (1987), 113-129. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, II.$H^s$-solutions and unconditional wellposedness, J. d'Analyse. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.  Google Scholar

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[14]

M. Lewin and N. Rougerie, Derivation of Pekar's polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal., 45 (2013), 1267-1301. doi: 10.1137/110846312.  Google Scholar

[15]

X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Diff. Eqns., 250 (2011), 2197-2226. doi: 10.1016/j.jde.2010.10.022.  Google Scholar

[16]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[17]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[18]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

C. Miao, G. Xu and L. Zhao, On the blow-up phenomenon for the mass-critical focusing Hartree equation in $\mathbbR^4$, Colloq. Math., 119 (2010), 23-50. doi: 10.4064/cm119-1-2.  Google Scholar

[24]

V. Moroz and J. V. Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[25]

R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.  Google Scholar

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

show all references

References:
[1]

P. d'Avenia and M. Squassina, Soliton dynamics for the Schrödinger-Newton system, Math. Models Methods Appl. Sci., 24 (2014), 553-572. doi: 10.1142/S0218202513500590.  Google Scholar

[2]

C. Bonanno, P. d'Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl., 417 (2014), 180-199. doi: 10.1016/j.jmaa.2014.02.063.  Google Scholar

[3]

D. Cao and Y. Su, Minimal blow-up solutions of mass-critical inhomogeneous Hartree equation, Journal of Mathematical Physics, 54 (2013), 121511, 25pp. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.  Google Scholar

[5]

J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D: Nonlinear Phenomena, 227 (2007), 142-148. doi: 10.1016/j.physd.2007.01.004.  Google Scholar

[6]

J. Frölich, 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.  Google Scholar

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136. doi: 10.1007/BF01214768.  Google Scholar

[8]

H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923. doi: 10.3934/dcdss.2012.5.903.  Google Scholar

[9]

P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. doi: 10.1051/cocv:1998107.  Google Scholar

[10]

R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys., 18 (1977), 1794-1797. doi: 10.1063/1.523491.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Theor., 46 (1987), 113-129. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[12]

T. Kato, On nonlinear Schrödinger equations, II.$H^s$-solutions and unconditional wellposedness, J. d'Analyse. Math., 67 (1995), 281-306. doi: 10.1007/BF02787794.  Google Scholar

[13]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[14]

M. Lewin and N. Rougerie, Derivation of Pekar's polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal., 45 (2013), 1267-1301. doi: 10.1137/110846312.  Google Scholar

[15]

X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Diff. Eqns., 250 (2011), 2197-2226. doi: 10.1016/j.jde.2010.10.022.  Google Scholar

[16]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[17]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[18]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[19]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[20]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[21]

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

[22]

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.  Google Scholar

[23]

C. Miao, G. Xu and L. Zhao, On the blow-up phenomenon for the mass-critical focusing Hartree equation in $\mathbbR^4$, Colloq. Math., 119 (2010), 23-50. doi: 10.4064/cm119-1-2.  Google Scholar

[24]

V. Moroz and J. V. Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[25]

R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.  Google Scholar

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[1]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[2]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[3]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[4]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[5]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[6]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[7]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[8]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[9]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052

[10]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[11]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[12]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[13]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[14]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[15]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[16]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[17]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[18]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203

[19]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[20]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (192)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]