Remarks on the semirelativistic Hartree equations
Yonggeun Cho Tohru Ozawa Hironobu Sasaki Yongsun Shim
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.
keywords: radial solutions semirelativistic Hartree type equations global well-posedness scattering
On the orbital stability of fractional Schrödinger equations
Yonggeun Cho Hichem Hajaiej Gyeongha Hwang Tohru Ozawa
We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
keywords: Fractional Schrödinger equation finite time blowup. Hartree type nonlinearity Strichartz estimates
On radial solutions of semi-relativistic Hartree equations
Yonggeun Cho Tohru Ozawa
We consider the semi-relativistic Hartree type equation with nonlocal nonlinearity $F(u) = \lambda (|x|^{-\gamma} * |u|^2)u, 0 < \gamma < n, n \ge 1$. In [2, 3], the global well-posedness (GWP) was shown for the value of $\gamma \in (0, \frac{2n}{n+1}), n \ge 2$ with large data and $\gamma \in (2, n), n \ge 3$ with small data. In this paper, we extend the previous GWP result to the case for $\gamma \in (1, \frac{2n-1}n), n \ge 2$ with radially symmetric large data. Solutions in a weighted Sobolev space are also studied.
keywords: radially symmetric solution. global well-posedness semi-relativistic Hartree type equation
On small amplitude solutions to the generalized Boussinesq equations
Yonggeun Cho Tohru Ozawa
We study the existence and scattering of global small amplitude solutions to generalized Boussinesq (Bq) and improved modified Boussinesq (imBq) equations with nonlinear term $f(u)$ behaving as a power $u^p$ as $u \to 0$ in $\mathbb{R}^n, n \ge 1$.
keywords: global existence generalized Bq and imBq equations small amplitude solution scattering.
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations
Yonggeun Cho Gyeongha Hwang Soonsik Kwon Sanghyuk Lee
We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.
keywords: ill-posedness. Fractional Schrödinger equation cubic nonlinearity well-posedness
On small data scattering of Hartree equations with short-range interaction
Yonggeun Cho Gyeongha Hwang Tohru Ozawa
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
Remarks on some dispersive estimates
Yonggeun Cho Tohru Ozawa Suxia Xia
In this paper we consider the initial value problem for $i\partial_t u + \omega(|\nabla|) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].
keywords: dispersive estimate oscillatory integral time decay Dispersive equations Bessel functions. Strichartz estimate
Global well-posedness of critical nonlinear Schrödinger equations below $L^2$
Yonggeun Cho Gyeongha Hwang Tohru Ozawa
The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below $L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.
keywords: critical nonlinearity below $L^2$ global well-posedness weighted Strichartz estimate Hartree equations angular regularity.
Corrigendum to "On small data scattering of Hartree equations with short-range interaction" [Comm. Pure. Appl. Anal., 15 (2016), 1809-1823]
Yonggeun Cho Gyeongha Hwang Tohru Ozawa
keywords: Hartree equations short range potential small data scattering $U^p$ and $V^p$ space

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