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 $\gamma>3/2$.

DCDS

We consider the Cauchy problem for the Korteweg-de Vries equation driven by a cylindrical fractional Brownian motion (fBm) in this paper. With Hurst parameter $H\geq\frac{7}{16}$ of the fBm, we obtain the local existence results with initial value in classical Sobolev spaces $H^s$ with $s\geq -\frac{9}{16}$. Furthermore, we give the relation between the Hurst parameter $H$ and the index $s$ to the Sobolev spaces $H^s$, which finds out the regularity between the driven term fBm and the initial value for the stochastic Korteweg-de Vries equation.

DCDS

In this paper, the long time behavior of
solution for the dissipative Hamiltonian amplitude equation
governing modulated wave instabilities is considered. First the
global weak attractor for this equation in $E_1$ is constructed.
And then by exact analysis of energy equations, it is showed that
the global weak attractor is actually the global strong attractor
in $E_1$.

DCDS

The existence of the global attractor of the damped, forced
generalized KdV-Benjamin-Ono equation in $L^2( \mathbb{R})$ is
proved for forces in $L^2( \mathbb{R})$. Moreover, the global
attractor in $L^2( \mathbb{R})$ is actually a compact set in $H^3(
\mathbb{R})$.

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 investigate the existence of classical solution of the viscous bipolar quantum hydrodynamic(QHD) models for ir-rotational fluid in a periodic domain. By applying the iteration method, we prove that the viscous bipolar QHD model has a local classical solution. Then we prove this solution is global with small initial data, based on a series of a priori estimates. Finally, we obtained the inviscid limit of this viscous quantum hydrodynamic model.

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.