DCDS
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.
CPAA
The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films
Shijin Ding Qiang Du
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.
keywords: vortices thin films Superconductivity pinning.
DCDS
Compressible hydrodynamic flow of liquid crystals in 1-D
Shijin Ding Junyu Lin Changyou Wang Huanyao Wen
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.
keywords: Nematic liquid crystals compressible hydrodynamic flow classical and strong solutions.
DCDS
Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations
Yingshan Chen Shijin Ding Wenjun Wang
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.
keywords: the Cauchy problem global existence uniqueness Compressible Navier-Stokes-Smoluchowski equations optimal convergence rate.
DCDS-B
Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities
Yinghua Li Shijin Ding Mingxia Huang
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.
keywords: blow-up criterion. Allen-Cahn equation Diffuse interface model nonhomogeneous Navier-Stokes equations
DCDS-S
Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum
Bingyuan Huang Shijin Ding Huanyao Wen
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 $t>0$.
keywords: vacuum. Classical solution compressible Navier-Stokes-Smoluchowski equations
DCDS-B
Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one
Shijin Ding Changyou Wang Huanyao Wen
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 $n_0\in H^1$.
keywords: Compressible hydrodynamic flow weak solution. nematic liquid crystals

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