KRM
Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations
Dongfen Bian Boling Guo
Kinetic & Related Models 2013, 6(3): 481-503 doi: 10.3934/krm.2013.6.481
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 $\gamma>3/2$.
keywords: large-time behavior. global existence isentropic fluids Electric-magnetohydrodynamic equations
DCDS
Stochastic Korteweg-de Vries equation driven by fractional Brownian motion
Guolian Wang Boling Guo
Discrete & Continuous Dynamical Systems - A 2015, 35(11): 5255-5272 doi: 10.3934/dcds.2015.35.5255
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.
keywords: bilinear estimate. stochastic convolution Hurst parameter fractional Brownian motion Korteweg-de Vries equation
DCDS
Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities
Boling Guo Zhengde Dai
Discrete & Continuous Dynamical Systems - A 1998, 4(4): 783-793 doi: 10.3934/dcds.1998.4.783
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 $E_1$.
keywords: global attractor dissipation. Hamiltonian amplitude equation
DCDS
The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$
Boling Guo Zhaohui Huo
Discrete & Continuous Dynamical Systems - A 2006, 16(1): 121-136 doi: 10.3934/dcds.2006.16.121
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 $H^3( \mathbb{R})$.
keywords: damping system the Fourier restriction norm asymptotic smoothing. Global attractor
DCDS
Global existence of weak solutions for Landau-Lifshitz-Maxwell equations
Shijin Ding Boling Guo Junyu Lin Ming Zeng
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 867-890 doi: 10.3934/dcds.2007.17.867
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.
keywords: Landau-Lifshitz-Maxwell equations global weak solution existence.
DCDS
Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions
Boling Guo Bixiang Wang
Discrete & Continuous Dynamical Systems - A 1996, 2(4): 455-466 doi: 10.3934/dcds.1996.2.455
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.
keywords: global attractor approximate inertial manifold Ginzburg-Landau equation. Gevrey class regularity
DCDS
Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements
Jie Jiang Boling Guo
Discrete & Continuous Dynamical Systems - A 2012, 32(1): 167-190 doi: 10.3934/dcds.2012.32.167
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.
keywords: global attractor. longtime behaviors microscopic movements First order phase transitions existence and uniqueness
DCDS
Existence of the solution for the viscous bipolar quantum hydrodynamic model
Boling Guo Guangwu Wang
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3183-3210 doi: 10.3934/dcds.2017136

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.

keywords: Viscous bipolar quantum hydrodynamic ir-rotational fluid local classical solution global existence
DCDS
Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation
Xinglong Wu Boling Guo
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 3211-3223 doi: 10.3934/dcds.2013.33.3211
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.
keywords: exponential decay infinite propagation speed. compact support The modified 2-component Camassa--Holm equation persistence properties
DCDS
Smooth solution of the generalized system of ferro-magnetic chain
Boling Guo Haiyang Huang
Discrete & Continuous Dynamical Systems - A 1999, 5(4): 729-740 doi: 10.3934/dcds.1999.5.729
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.

keywords: general Landau-Lifshitz equation. Smooth solution ferromagnetic chain

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