CPAA
Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain
Luigi C. Berselli Jishan Fan
Communications on Pure & Applied Analysis 2015, 14(2): 637-655 doi: 10.3934/cpaa.2015.14.637
In this paper, we prove some logarithmically improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain.
keywords: Regularity criteria MHD Boussinesq equations liquid crystals
CPAA
Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations
Yong Zhou Jishan Fan
Communications on Pure & Applied Analysis 2010, 9(3): 813-818 doi: 10.3934/cpaa.2010.9.813
We prove local in time existence and uniqueness of solutions of the ideal inhomogeneous magnetohydrodynamic equations.
keywords: local existence inhomogeneous Euler equations. Ideal inhomogeneous MHD
CPAA
Regularity criteria of strong solutions to a problem of magneto-elastic interactions
Yong Zhou Jishan Fan
Communications on Pure & Applied Analysis 2010, 9(6): 1697-1704 doi: 10.3934/cpaa.2010.9.1697
In this paper, various regularity criteria for the strong solutions to a problem arising in the study of magneto-elastic interactions are established. In particular, these regularity criteria are also true for the Landau-Lifshitz equation and give extensions of previous results.
keywords: Magneto-elastic materials regularity.
PROC
Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model
Jishan Fan Tohru Ozawa
Conference Publications 2011, 2011(Special): 400-409 doi: 10.3934/proc.2011.2011.400
The global Cauchy problem for an approximation model for the ideal density-dependent MHD- $\alpha$ model is studied. The vanishing limit on is also discussed.
keywords: ideal density-dependent MHD-$\alpha$
PROC
An approximation model for the density-dependent magnetohydrodynamic equations
Jishan Fan Tohru Ozawa
Conference Publications 2013, 2013(special): 207-216 doi: 10.3934/proc.2013.2013.207
The global Cauchy problem for an approximation model for the density-dependent MHD system is studied. The vanishing limit on $\alpha$ is also discussed.
keywords: MHD-alpha. MHD Density-dependent
PROC
A regularity criterion for 3D density-dependent MHD system with zero viscosity
Jishan Fan Tohru Ozawa
Conference Publications 2015, 2015(special): 395-399 doi: 10.3934/proc.2015.0395
This paper proves a regularity criterion $\nabla u,\nabla b\in L^\infty(0,T;L^\infty)$ for 3D density-dependent MHD system with zero viscosity and positive initial density.
keywords: MHD ideal fluid. regularity criterion
CPAA
Regularity criteria for a magnetohydrodynamic-$\alpha$ model
Yong Zhou Jishan Fan
Communications on Pure & Applied Analysis 2011, 10(1): 309-326 doi: 10.3934/cpaa.2011.10.309
We study the $n$-dimensional magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model in the whole space. Various regularity criteria are established. When $n=4$, uniqueness of weak solutions is also proved. As a corollary, the strong solution to this model exists globally, as $n \leq 4$.
keywords: 76D03. Primary: 35B40
CPAA
Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions
Jishan Fan Fei Jiang
Communications on Pure & Applied Analysis 2016, 15(1): 73-90 doi: 10.3934/cpaa.2016.15.73
In this paper, we study the large-time behavior of weak solutions to the initial-boundary problem arising in a simplified Ericksen-Leslie system for nonhomogeneous incompressible flows of nematic liquid crystals with a transformation condition of trigonometric functions (called by trigonometric condition for simplicity) posed on the initial direction field in a bounded domain $\Omega\subset \mathbb{R}^2$. We show that the kinetic energy and direction field converge to zero and an equilibrium state, respectively, as time goes to infinity. Further, if the initial density is away from vacuum and bounded, then the density, and velocity and direction fields exponential decay to an equilibrium state. In addition, we also show that the weak solutions of the corresponding compressible flows converge {an equilibrium} state.
keywords: Liquid crystals weak solutions exponential decay Navier-Stokes equations. large-time behavior
KRM
Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model
Jishan Fan Tohru Ozawa
Kinetic & Related Models 2009, 2(2): 293-305 doi: 10.3934/krm.2009.2.293
We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-$\alpha$-MHD model. Since the solutions to the Leray-$\alpha$-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field $B$ only.
keywords: MHD equations regularity criterion interpolation inequality in Besov spaces. Leray-$\alpha$-MHD model
DCDS
Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals
Jishan Fan Tohru Ozawa
Discrete & Continuous Dynamical Systems - A 2009, 25(3): 859-867 doi: 10.3934/dcds.2009.25.859
We consider the hydrodynamic theory of liquid crystals. We prove some regularity criteria for a simplified Ericksen-Leslie system. The existence and uniqueness of global smooth solutions is also proved for a regularization model of this simplified system.
keywords: liquid crystals Ericksen-Leslie system regularity condition Navier-Stokes equations.

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