American Institute of Mathematical Sciences

Advanced
September  2016, 15(5): 1809-1823. doi: 10.3934/cpaa.2016016

On small data scattering of Hartree equations with short-range interaction

Yonggeun Cho 1, , Gyeongha Hwang 2, and  Tohru Ozawa 3,

1. 

Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2. 

National Center for Theoretical Sciences, No. 1 Sec. 4 Roosevelt Rd., National Taiwan University, Taipei, 106, Taiwan
3. 

Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received  October 2015 Revised  May 2016 Published  July 2016

In this note we study Hartree type equations with $|\nabla|^\alpha (1 < \alpha \le 2)$ and potential whose Fourier transform behaves like $|\xi|^{-(d-\gamma_1)}$ at the origin and $|\xi|^{-(d-\gamma_2)}$ at infinity. We show non-existence of scattering when $0 < \gamma_1 \le 1$ and small data scattering in $H^s$ for $s > \frac{2-\alpha}2$ when $2 < \gamma_1 \le d$ and $0 < \gamma_2 \le 2$. For this we use $U^p-V^p$ space argument and Strichartz estimates.
Keywords: Hartree equations, short range potential, $U^p$ and $V^p$ spaces., small data scattering.
Mathematics Subject Classification: Primary: M35Q55; Secondary: 35Q4.
Citation: Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016
References:
[1]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkacialaj Ekvacioj}, 56 (2013), 193.  doi: 10.1619/fesi.56.193.  Google Scholar

[2]

Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060.  doi: 10.1137/060653688.  Google Scholar

[3]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[4]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 917.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Remarks on Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential,, \emph{SUT J. Math.}, 34 (1998), 13.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Scattering theory and asymptotics for large time of solutions to the Hartree type equations with a long range potential,, \emph{Hokkaido Math. J.}, 30 (2001), 137.  doi: 10.14492/hokmj/1350911928.  Google Scholar

[7]

N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential,, \emph{Diff. Int. Equations}, 28 (2015), 1085.   Google Scholar

[8]

N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, \emph{SIAM J. Math. Anal.}, 29 (1998), 1256.  doi: 10.1137/S0036141096312222.  Google Scholar

[9]

N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations,, \emph{Ann. Inst. H. Poincare Phys. Theor.}, 46 (1987), 187.   Google Scholar

[10]

S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity,, \emph{J. Differential Equations}, 259 (2015), 5510.  doi: 10.1016/j.jde.2015.06.037.  Google Scholar

[11]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 2265.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[12]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves,, Oberwolfach Seminars, 45 (2014).   Google Scholar

[13]

K. Nakanishi and T. Ozawa, Scattering Problem for Nonlinear Schrodinger and Hartree Equations,, Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, (2001).   Google Scholar

[14]

F. Pusateri, Modified scattering for the Boson star equation,, \emph{Commun. Math. Phys.}, 332 (2014), 1203.  doi: 10.1007/s00220-014-2094-x.  Google Scholar

show all references

References:
[1]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity,, \emph{Funkacialaj Ekvacioj}, 56 (2013), 193.  doi: 10.1619/fesi.56.193.  Google Scholar

[2]

Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation,, \emph{SIAM J. Math. Anal.}, 38 (2006), 1060.  doi: 10.1137/060653688.  Google Scholar

[3]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1121.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[4]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 917.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[5]

N. Hayashi and P. I. Naumkin, Remarks on Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential,, \emph{SUT J. Math.}, 34 (1998), 13.   Google Scholar

[6]

N. Hayashi and P. I. Naumkin, Scattering theory and asymptotics for large time of solutions to the Hartree type equations with a long range potential,, \emph{Hokkaido Math. J.}, 30 (2001), 137.  doi: 10.14492/hokmj/1350911928.  Google Scholar

[7]

N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential,, \emph{Diff. Int. Equations}, 28 (2015), 1085.   Google Scholar

[8]

N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation,, \emph{SIAM J. Math. Anal.}, 29 (1998), 1256.  doi: 10.1137/S0036141096312222.  Google Scholar

[9]

N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations,, \emph{Ann. Inst. H. Poincare Phys. Theor.}, 46 (1987), 187.   Google Scholar

[10]

S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity,, \emph{J. Differential Equations}, 259 (2015), 5510.  doi: 10.1016/j.jde.2015.06.037.  Google Scholar

[11]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces,, \emph{Commun. Pure Appl. Anal.}, 14 (2015), 2265.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[12]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves,, Oberwolfach Seminars, 45 (2014).   Google Scholar

[13]

K. Nakanishi and T. Ozawa, Scattering Problem for Nonlinear Schrodinger and Hartree Equations,, Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, (2001).   Google Scholar

[14]

F. Pusateri, Modified scattering for the Boson star equation,, \emph{Commun. Math. Phys.}, 332 (2014), 1203.  doi: 10.1007/s00220-014-2094-x.  Google Scholar

[1]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Corrigendum to "On small data scattering of Hartree equations with short-range interaction" [Comm. Pure. Appl. Anal., 15 (2016), 1809-1823]. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1939-1940. doi: 10.3934/cpaa.2017094
[2]

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
[3]

Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
[4]

Vladimir Ejov, Anatoli Torokhti. How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293
[5]

Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87
[6]

Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084
[7]

Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure & Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751
[8]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667
[9]

Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905
[10]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715
[11]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[12]

Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020029
[13]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[14]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058
[15]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[16]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[17]

Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037
[18]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[19]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675
[20]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences