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### Open Access Journals

DCDS

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.

PROC

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.

PROC

The global Cauchy problem for an approximation model for the
density-dependent MHD system is studied. The vanishing limit on
$\alpha$ is also discussed.

PROC

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.

CPAA

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.

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

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.

CPAA

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.

KRM

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.

DCDS

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.

DCDS-S

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.

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