American Institute of Mathematical Sciences

Advanced
September  2013, 2(3): 531-542. doi: 10.3934/eect.2013.2.531

Energy methods for Hartree type equations with inverse-square potentials

Toshiyuki Suzuki 1,

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  July 2012 Revised  February 2013 Published  July 2013

Nonlinear Schrödinger equations with nonlocal nonlinearities described by integral operators are considered. This generalizes usual Hartree type equations (HE)$_{0}$. We construct weak solutions to (HE)$_{a}$, $a\neq 0$, even if the kernel is of non-convolution type. The advantage of our methods is the applicability to the problem with strongly singular potential $a|x|^{-2}$ as a term in the linear part and with critical nonlinearity.
Keywords: integral operator, Hartree equations, nonlinear Schrödinger equations, inverse-square potentials., Energy methods.
Mathematics Subject Classification: Primary: 35Q55, 35Q40; Secondary: 81Q1.
Citation: Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

[3]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.  Google Scholar

[4]

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

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: 10.1007/BF01258601.  Google Scholar

[6]

J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys., 16 (1975), 1122-1130. doi: 10.1063/1.522642.  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]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. doi: 10.1007/BF01609845.  Google Scholar

[9]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. doi: 10.1080/00036811.2011.631914.  Google Scholar

[10]

N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evolution Equations and Control Theory, 1 (2012), 337-354. doi: 10.3934/eect.2012.1.337.  Google Scholar

[11]

V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392. doi: 10.1007/s00209-007-0279-0.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, "Fourier Analysis and Nonlinear Partial Differential Equations," Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203 (2003), 519-549. doi: 10.1016/S0022-1236(03)00238-6.  Google Scholar

[3]

N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680. doi: 10.1512/iumj.2004.53.2541.  Google Scholar

[4]

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

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477-494. doi: 10.1007/BF01258601.  Google Scholar

[6]

J. M. Chadam and R. T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys., 16 (1975), 1122-1130. doi: 10.1063/1.522642.  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]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194. doi: 10.1007/BF01609845.  Google Scholar

[9]

N. Okazawa, T. Suzuki and T. Yokota, Cauchy problem for nonlinear Schrödinger equations with inverse-square potentials, Appl. Anal., 91 (2012), 1605-1629. doi: 10.1080/00036811.2011.631914.  Google Scholar

[10]

N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evolution Equations and Control Theory, 1 (2012), 337-354. doi: 10.3934/eect.2012.1.337.  Google Scholar

[11]

V. Pierfelice, Weighted Strichartz estimates for the Schrödinger and wave equations on Damek-Ricci spaces, Math. Z., 260 (2008), 377-392. doi: 10.1007/s00209-007-0279-0.  Google Scholar

[1]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022
[2]

Toshiyuki Suzuki. Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space. Conference Publications, 2015, 2015 (special) : 1019-1024. doi: 10.3934/proc.2015.1019
[3]

Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65
[4]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[5]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623
[6]

Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221
[7]

Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
[8]

Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162
[9]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
[10]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723
[11]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[12]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[13]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541
[14]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215
[15]

Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[16]

Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286
[17]

Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237
[18]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
[19]

Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
[20]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences