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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.

DCDS

In the present paper , we show the Gevrey class regularity of solutions
for the generalized Ginzburg-Landau equation in two spatial dimensions.
We also introduce an approximate inertial manifold for this
system.

DCDS

This paper is devoted to the study of long-time behavior of the solutions to
a one-dimensional full model for the first order phase transitions. Our system features
a

*strongly nonlinear*internal energy balance equation, governing the evolution of the absolute temperature $\theta$, which is coupled with an evolution equation for the phase change parameter $f$ with a third-order nonlinearity $G_2'(f)$ in place of the customarily constant latent heat. The main novelty of this paper is that we perform an argument to establish Lemma 3.1 which enables us to obtain uniform estimates of the global solutions with respect to time. Asymptotic behavior of the solutions as time goes to infinity and the compactness of the orbit are obtained. Furthermore, we investigate the dynamics of the system and prove the existence of global attractors.
DCDS

In this paper, we mainly study persistence properties and infinite
propagation for the modified 2-component Camassa--Holm equation. We
first prove that persistence properties of the solution to the
equation provided the initial potential satisfies a certain sign
condition. Finally, we get the infinite propagation if the initial
datas satisfy certain compact conditions, while the solution to
system (1.1) instantly loses compactly supported, the solution has
exponential decay as $|x|$ goes to infinity.

DCDS

In this paper, we consider the initial value problem with periodic boundary condition for a class of general systems of the ferromagnetic chain

$z_t=-\alpha z\times (z\times z_{x x})+ z\times z_{x x}+z\times f(z), \qquad (\alpha \geq 0).$

The existence of unique smooth solutions is proved by using the technique of spatial difference and a priori estimates of higher-order derivatives in Sobolev spaces.

DCDS

The well-posedness of the Cauchy problem
for a generalized nonlinear dispersive equation is studied. Local
well-posedness for data in $H^s(\mathbb R)(s>-\frac{1}{8})$ and
the global result for data in $ L^{2}(\mathbb{R})$ are obtained if
$l=2$. Moreover, for $l=3$, the problem is locally well-posed for
data in $H^s(s>\frac{1}{4}).$ The main idea is to use the Fourier
restriction norm method.

KRM

The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.

DCDS-S

This paper is concerned with
the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various
kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$.
Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and
uniqueness is established for the local strong solution with
large initial data and also for the global strong solution with
initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.

DCDS-B

In this paper, we first give an important interpolation
inequality. Secondly, we use this inequality to prove the
existence of local and global solutions of an inhomogeneous
Schrödinger equation. Thirdly, we construct several invariant
sets and prove the existence of blowing up solutions. Finally, we
prove that for any $\omega>0$ the standing wave $e^{i \omega
t} \phi (x)$ related to the ground state solution $\phi$ is strongly
unstable.

KRM

We are concerned with global existence and large-time behavior of solutions to the isentropic
electric-magnetohydrodynamic equations in a bounded domain
$\Omega\subseteq\mathbb{R}^{N}$, $N=2,\ 3$. We establish the existence and large-time behavior of global weak solutions
through a three-level approximation, energy estimates on condition that the adiabatic constant satisfies $\gamma>3/2$.

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