DCDS
Schrödinger equations with nonlinearity of integral type
Nakao Hayashi Tohru Ozawa
Discrete & Continuous Dynamical Systems - A 1995, 1(4): 475-484 doi: 10.3934/dcds.1995.1.475
We consider the Cauchy problem for the nonlinear Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. Concerning the problem with finite initial time, we prove the global well-posedness in the largest space $L^2(\mathbb R^2)$. Concerning the problem with infinite initial time, we prove the existence of modified wave operators on a dense set of small and sufficiently regular asymptotic states.
keywords: propagation of laser beams. Cauchy problem Schrödinger equation
PROC
Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model
Jishan Fan Tohru Ozawa
Conference Publications 2011, 2011(Special): 400-409 doi: 10.3934/proc.2011.2011.400
The global Cauchy problem for an approximation model for the ideal density-dependent MHD- $\alpha$ model is studied. The vanishing limit on is also discussed.
keywords: ideal density-dependent MHD-$\alpha$
PROC
An approximation model for the density-dependent magnetohydrodynamic equations
Jishan Fan Tohru Ozawa
Conference Publications 2013, 2013(special): 207-216 doi: 10.3934/proc.2013.2013.207
The global Cauchy problem for an approximation model for the density-dependent MHD system is studied. The vanishing limit on $\alpha$ is also discussed.
keywords: MHD-alpha. MHD Density-dependent
PROC
A regularity criterion for 3D density-dependent MHD system with zero viscosity
Jishan Fan Tohru Ozawa
Conference Publications 2015, 2015(special): 395-399 doi: 10.3934/proc.2015.0395
This paper proves a regularity criterion $\nabla u,\nabla b\in L^\infty(0,T;L^\infty)$ for 3D density-dependent MHD system with zero viscosity and positive initial density.
keywords: MHD ideal fluid. regularity criterion
CPAA
Preface
T. Ogawa Tohru Ozawa
Communications on Pure & Applied Analysis 2015, 14(4): i-iii doi: 10.3934/cpaa.2015.14.4i
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Professor Gustavo Ponce on the occasion of his sixtieth birthday.

Gustavo Ponce was born on April 20, 1952, in Venezuela. He received his B.A. in 1976 from Universidad Central de Venezuela and his Ph. D. in 1982 with the dissertation entitled ``Long time stability of solutions of nonlinear evolution equations" under the supervision of Sergiu Klainerman and Louis Nirenberg at Courant Institute, New York University. After professional experiences at University of California at Berkely (1982-1984), Universidad Central de Venezuela (1984-1986), University of Chicago (1986-1989), and Pennsylvania State University (1989-1991), he was appointed to a full professorship at Department of Mathematics, University of California at Santa Barbara in 1991, where he has remained up until now.
keywords:
DCDS
Remarks on the semirelativistic Hartree equations
Yonggeun Cho Tohru Ozawa Hironobu Sasaki Yongsun Shim
Discrete & Continuous Dynamical Systems - A 2009, 23(4): 1277-1294 doi: 10.3934/dcds.2009.23.1277
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
CPAA
On the orbital stability of fractional Schrödinger equations
Yonggeun Cho Hichem Hajaiej Gyeongha Hwang Tohru Ozawa
Communications on Pure & Applied Analysis 2014, 13(3): 1267-1282 doi: 10.3934/cpaa.2014.13.1267
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
KRM
Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model
Jishan Fan Tohru Ozawa
Kinetic & Related Models 2009, 2(2): 293-305 doi: 10.3934/krm.2009.2.293
We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-$\alpha$-MHD model. Since the solutions to the Leray-$\alpha$-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field $B$ only.
keywords: MHD equations regularity criterion interpolation inequality in Besov spaces. Leray-$\alpha$-MHD model
DCDS
Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals
Jishan Fan Tohru Ozawa
Discrete & Continuous Dynamical Systems - A 2009, 25(3): 859-867 doi: 10.3934/dcds.2009.25.859
We consider the hydrodynamic theory of liquid crystals. We prove some regularity criteria for a simplified Ericksen-Leslie system. The existence and uniqueness of global smooth solutions is also proved for a regularization model of this simplified system.
keywords: liquid crystals Ericksen-Leslie system regularity condition Navier-Stokes equations.
DCDS-S
On radial solutions of semi-relativistic Hartree equations
Yonggeun Cho Tohru Ozawa
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 71-82 doi: 10.3934/dcdss.2008.1.71
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

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