Global existence of weak solutions for Landau-Lifshitz-Maxwell equations
Shijin Ding Boling Guo Junyu Lin Ming Zeng
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.
Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions
Boling Guo Bixiang Wang
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
Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements
Jie Jiang Boling Guo
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
Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation
Xinglong Wu Boling Guo
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
Smooth solution of the generalized system of ferro-magnetic chain
Boling Guo Haiyang Huang
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
The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation
Zhaohui Huo Boling Guo
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.
keywords: [k;Z]-multiplier. the Fourier restriction norm The generalized nonlinear dispersive equation
Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction
Xueke Pu Boling Guo
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.
keywords: global solutions Quantum hydrodynamics semiclassical limit.
Well-posedness for the three-dimensional compressible liquid crystal flows
Xiaoli Li Boling Guo
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.
keywords: existence and uniqueness. strong solution vacuum compressible Liquid crystals
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations
Jianqing Chen Boling Guo
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.
keywords: global existence strong instability. blowing up Interpolation inequality
Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations
Dongfen Bian Boling Guo
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

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