• 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.  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.  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).  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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.  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.  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.  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.  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.  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.  doi: 10.1063/1.523491.  Google Scholar

[11]

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

[18]

E. Lieb, Analysis,, 2nd ed., (2001).  doi: 10.1090/gsm/014.  Google Scholar

[19]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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).  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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.  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.  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.  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.  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.  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.  doi: 10.1063/1.523491.  Google Scholar

[11]

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

[18]

E. Lieb, Analysis,, 2nd ed., (2001).  doi: 10.1090/gsm/014.  Google Scholar

[19]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[1]

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

[2]

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

[3]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[4]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[5]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[6]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[8]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[9]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[10]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[11]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[14]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[15]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[16]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[17]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332

[18]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[19]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293

[20]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (23)

Other articles
by authors

[Back to Top]