October  2009, 23(4): 1277-1294. doi: 10.3934/dcds.2009.23.1277

Remarks on the semirelativistic Hartree equations

1. 

Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea

2. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810

3. 

Department of Mathematics, Osaka University, Toyonaka 563-0043

4. 

Department of Mathematics, POSTECH, Pohang 790-784, South Korea

Received  January 2007 Revised  August 2007 Published  November 2008

We study the global well-posedness (GWP) and small data scattering of radial solutions of the semirelativistic Hartree type equations with nonlocal nonlinearity $F(u) = \lambda (|\cdot|^{-\gamma}$ * $|u|^2)u$, $\lambda \in \mathbb{R} \setminus \{0\}$, $0 < \gamma < n$, $n \ge 3$. We establish a weighted $L^2$ Strichartz estimate applicable to non-radial functions and some fractional integral estimates for radial functions.
Citation: Yonggeun Cho, Tohru Ozawa, Hironobu Sasaki, Yongsun Shim. Remarks on the semirelativistic Hartree equations. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1277-1294. doi: 10.3934/dcds.2009.23.1277
[1]

Kiyeon Lee. Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3683-3702. doi: 10.3934/cpaa.2021126

[2]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[3]

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923

[4]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1

[5]

Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725

[6]

Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084

[7]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[8]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[9]

Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289

[10]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[11]

Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71

[12]

Jinkai Li, Edriss Titi. Global well-posedness of strong solutions to a tropical climate model. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4495-4516. doi: 10.3934/dcds.2016.36.4495

[13]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657

[14]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

[15]

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, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

[16]

Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081

[17]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053

[18]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905

[19]

Márcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934/cpaa.2019117

[20]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (12)

[Back to Top]