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

Previous Article
Liouville type theorems for singular integral equations and integral systems
CPAA Home
This Issue
Next Article
A direct method of moving planes for fractional Laplacian equations in the unit ball

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

Abstract
Related Papers

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 and 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, Funkacialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193.
[2]	Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.
[3]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.
[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), 917-941. doi: 10.1016/j.anihpc.2008.04.002.
[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, SUT J. Math., 34 (1998), 13-24.
[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, Hokkaido Math. J., 30 (2001), 137-161. doi: 10.14492/hokmj/1350911928.
[7]	N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential, Diff. Int. Equations, 28 (2015), 1085-1104.
[8]	N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222.
[9]	N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213.
[10]	S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037.
[11]	Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.
[12]	H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, 45 2014.
[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).
[14]	F. Pusateri, Modified scattering for the Boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x.

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, Funkacialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193.
[2]	Y. Cho and T. Ozawa, On the semi-relativisitc Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074. doi: 10.1137/060653688.
[3]	Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121.
[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), 917-941. doi: 10.1016/j.anihpc.2008.04.002.
[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, SUT J. Math., 34 (1998), 13-24.
[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, Hokkaido Math. J., 30 (2001), 137-161. doi: 10.14492/hokmj/1350911928.
[7]	N. Hayashi, P. I. Naumkin and T. Ogawa, Scattering operator for semirelativistic Hartree type equation with a short range potential, Diff. Int. Equations, 28 (2015), 1085-1104.
[8]	N. Hayashi, P.I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29 (1998), 1256-1267. doi: 10.1137/S0036141096312222.
[9]	N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincare Phys. Theor., 46 (1987), 187-213.
[10]	S. Herr and T. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity, J. Differential Equations, 259 (2015), 5510-5532. doi: 10.1016/j.jde.2015.06.037.
[11]	Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265.
[12]	H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, 45 2014.
[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).
[14]	F. Pusateri, Modified scattering for the Boson star equation, Commun. Math. Phys., 332 (2014), 1203-1234. doi: 10.1007/s00220-014-2094-x.

[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 and 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 and Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081
[3]	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 $?. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293
[4]	Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289
[5]	Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87
[6]	Yanfang Gao, Zhiyong Wang. Minimal mass non-scattering solutions of the focusing L²-critical Hartree equations with radial data. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1979-2007. doi: 10.3934/dcds.2017084
[7]	Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure and Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667
[8]	Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905
[9]	Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure and Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751
[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 and Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715
[11]	Lingyu Diao, Jian Gao, Jiyong Lu. Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Advances in Mathematics of Communications, 2020, 14 (4) : 555-572. doi: 10.3934/amc.2020029
[12]	Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[13]	Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $\|u\|^{2p}u$. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[14]	Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[15]	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
[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]	Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028
[18]	Abdelbaki Selmi, Abdellaziz Harrabi, Cherif Zaidi. Nonexistence results on the space or the half space of $ -\Delta u+\lambda u = \|u\|^{p-1}u $ via the Morse index. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2839-2852. doi: 10.3934/cpaa.2020124
[19]	Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2022, 16 (3) : 643-665. doi: 10.3934/amc.2020127
[20]	Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Yonggeun Cho Gyeongha Hwang Tohru Ozawa
on Google Scholar
Yonggeun Cho Gyeongha Hwang Tohru Ozawa

[Back to Top]