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Small data scattering of Hartree type fractional Schr\"odinger equations in dimension 2 and 3
조용근,Tohru Ozawa 대한수학회 2018 대한수학회지 Vol.55 No.2
In this paper we study the small-data scattering of the $d$ dimensional fractional Schr\"odinger equations with $d = 2, 3$, L\'evy index $1 < \al < 2$ and Hartree type nonlinearity $F(u) = \mu(|x|^{-\gamma}*|u|^2)u$ with $ \max(\al, \frac{2d}{2d-1}) < \gamma \le 2$, $\gamma < d$. This equation is scaling-critical in $\dot H^{s_c}$, $s_c = \frac{\gamma-\al}2$. We show that the solution scatters in $H^{s,1}$ for any $s > s_c$, where $H^{s, 1}$ is a space of Sobolev type taking in angular regularity with norm defined by $\|\varphi\|_{H^{s, 1}} = \|\varphi\|_{H^s} + \|\nabla_{\mathbb S} \varphi\|_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere $\mathbb S^{d-1}$ and utilize $U^p$-$V^p$ space argument.
Global solutions of semirelativistic Hartree type equations
조용근,Tohru Ozawa 대한수학회 2007 대한수학회지 Vol.44 No.5
We consider initial value problems for the semirelativisticHartree type equations with cubic convolution nonlinearity F(u) = (V juj2)u. HereV is a sum of two Coulomb type potentials. Under a specieddecay condition and a symmetric condition for the potential V we showthe global existence and scattering of solutions.
Finite time blowup for the fourth-order NLS
Yonggeun Cho,Tohru Ozawa,Chengbo Wang 대한수학회 2016 대한수학회보 Vol.53 No.2
We consider the fourth-order Schr\"odinger equation with focusing inhomogeneous nonlinearity $(-|x|^{-2}|u|^\frac4n u)$ in high space dimensions. Extending Glassey's virial argument, we show the finite time blow-up of solutions when the energy is negative.
SMALL DATA SCATTERING OF HARTREE TYPE FRACTIONAL SCHRÖDINGER EQUATIONS IN DIMENSION 2 AND 3
Cho, Yonggeun,Ozawa, Tohru Korean Mathematical Society 2018 대한수학회지 Vol.55 No.2
In this paper we study the small-data scattering of the d dimensional fractional $Schr{\ddot{o}}dinger$ equations with d = 2, 3, $L{\acute{e}}vy$ index 1 < ${\alpha}$ < 2 and Hartree type nonlinearity $F(u)={\mu}({\mid}x{\mid}^{-{\gamma}}{\ast}{\mid}u{\mid}^2)u$ with max(${\alpha}$, ${\frac{2d}{2d-1}}$) < ${\gamma}{\leq}2$, ${\gamma}$ < d. This equation is scaling-critical in ${\dot{H}}^{s_c}$, $s_c={\frac{{\gamma}-{\alpha}}{2}}$. We show that the solution scatters in $H^{s,1}$ for any s > $s_c$, where $H^{s,1}$ is a space of Sobolev type taking in angular regularity with norm defined by ${\parallel}{\varphi}{\parallel}_{H^{s,1}}={\parallel}{\varphi}{\parallel}_{H^s}+{\parallel}{\nabla}_{{\mathbb{S}}{\varphi}}{\parallel}_{H^s}$. For this purpose we use the recently developed Strichartz estimate which is $L^2$-averaged on the unit sphere ${\mathbb{S}}^{d-1}$ and utilize $U^p-V^p$ space argument.
FINITE TIME BLOWUP FOR THE FOURTH-ORDER NLS
Cho, Yonggeun,Ozawa, Tohru,Wang, Chengbo Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
We consider the fourth-order $Schr{\ddot{o}}dinger$ equation with focusing inhomogeneous nonlinearity ($-{\mid}x{\mid}^{-2}{\mid}u{\mid}^{\frac{4}{n}}u$) in high space dimensions. Extending Glassey's virial argument, we show the finite time blowup of solutions when the energy is negative.
Short-range scattering of Hartree type fractional NLS II
Cho, Yonggeun,Ozawa, Tohru Elsevier 2017 Nonlinear analysis Vol.157 No.-
<P>We prove the small data scattering for Hartree type fractional Schrodinger equation with inverse square potential. This is the border line problem between Strichartz range and weighted space range in view of the method of approach. To show this we carry out a subtle trilinear estimate via fractional Leibniz rule and Balakrishnan's formula. This paper is a sequel of the previous result (Cho, 2017). (C) 2017 Elsevier Ltd. All rights reserved.</P>
GLOBAL SOLUTIONS OF SEMIRELATIVISTIC HARTREE TYPE EQUATIONS
Cho, Yong-Geun,Ozawa, Tohru Korean Mathematical Society 2007 대한수학회지 Vol.44 No.5
We consider initial value problems for the semirelativistic Hartree type equations with cubic convolution nonlinearity $F(u)=(V*{\mid}u{\mid}^2)u$. Here V is a sum of two Coulomb type potentials. Under a specified decay condition and a symmetric condition for the potential V we show the global existence and scattering of solutions.
On the orbital stability of fractional Schrödinger equations
Cho, Yonggeun,Hajaiej, Hichem,Hwang, Gyeongha,Ozawa, Tohru American Institute of Mathematical Sciences 2014 COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Vol.13 No.3
We show the existence of ground state and orbital stability of standing waves of fractional Schrodinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.