## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
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DCDS

In this paper we study the model that the usual Maxwell's
equations are supplemented with a constitution relation in which
the electric displacement equals a constant time the electric
field plus an internal polarization variable and the magnetic
displacement equals a constant time the magnetic field plus the
microscopic magnetization. Using the Galerkin method and viscosity
vanishing approach, we obtain the existence of the global weak
solution for the Landau-Lifshitz-Maxwell equations. The main
difficulties in this study are due to the loss of compactness in
the system.

CPAA

In this paper, we discuss the global minimizers of a
free energy for the superconducting thin films placed in a magnetic
field $h_{e x}$ below the lower critical field $H_{c1}$ or between
$H_{c1}$ and the upper critical field $H_{c2}$.
For $h_{e x}$ is near but smaller than $H_{c1}$, we prove
that the global minimizer having no vortex is unique.
For $H_{c1}$<<$h_{e x}$<<$H_{c2}$, we
prove that the density of the vortices of the global minimizer is
proportional
to the applied field.

DCDS

We consider a simplified version of Ericksen-Leslie equation
modeling the compressible hydrodynamic flow of nematic liquid
crystals in dimension one. If the initial data $(\rho_0, u_0,n_0)\in
C^{1,\alpha}(I)\times C^{2,\alpha}(I)\times C^{2,\alpha}(I, S^2)$
and $\rho_0\ge c_0>0$, then we obtain both existence and uniqueness
of global classical solutions. For $0\le\rho_0\in H^1(I)$ and $(u_0,
n_0)\in H^1(I)\times H^2(I,S^2)$, we obtain both existence and
uniqueness of global strong solutions.

DCDS

This paper is concerned with the Cauchy
problem of the compressible Navier-Stokes-Smoluchowski equations in
$\mathbb{R}^3$. Under the smallness assumption on both the external
potential and the initial perturbation of the stationary solution in
some Sobolev spaces, the existence theory of global solutions in
$H^3$ to the stationary profile is established. Moreover, when the
initial perturbation is bounded in $L^p$-norm with $1\leq p<
\frac{6}{5}$, we obtain the optimal convergence rates of the
solution in $L^q$-norm with $2\leq q\leq 6$ and its first order
derivative in $L^2$-norm.

DCDS-B

This paper is concerned with a coupled Navier-Stokes/Allen-Cahn system describing
a diffuse interface model for two-phase flow of viscous incompressible fluids
with different densities in a bounded domain $\Omega\subset\mathbb R^N$($N=2,3$).
We establish a criterion for possible break down of such solutions at finite
time in terms of the temporal integral of both the maximum norm of the deformation tensor of
velocity gradient and the square of maximum norm of gradient of phase field variable in 2D.
In 3D, the temporal integral of the square of maximum norm of velocity is also needed.
Here, we suppose the initial density function $\rho_0$ has a positive lower bound.

DCDS-S

This paper is concerned with the Cauchy problem for compressible Navier-Stokes-Smoluchowski equations with vacuum in $\mathbb{R}^3$. We prove both existence and uniqueness of the local strong solution, and then obtain a local classical solution by
deriving the smoothing effect of the strong solution for $t>0$.

DCDS-B

We consider the equation modeling the compressible hydrodynamic flow of liquid
crystals in one dimension. In this paper, we establish the existence of a weak
solution $(\rho, u,n)$ of such a system when the initial density function $0\le \rho_0
\in L^\gamma$ for $\gamma>1$, $u_0\in L^2$, and $n_0\in H^1$. This extends a
previous result by [12], where the existence of a weak
solution was obtained under the stronger assumption that
the initial density function $0$<$c\le \rho_0\in H^1$,
$u_0\in L^2$, and $n_0\in H^1$.

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