LPS＇S CRITERION FOR INCOMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper we derive LPS's criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in R3. We show that if0 T +∞ is the maximal time interval for the unique smooth solution u ∈ C∞([0, T), R3),then |u| + |▽d| /∈ Lq([0, T ], Lp(R3)), where p and q safisfy the Ladyzhenskaya-Prodi-Serrin's condition:3p+2q= 1 and p ∈(3, +∞].     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

GLOBAL EXISTENCE AND CONVERGENCE RATES OF SMOOTH SOLUTIONS FOR THE 3-D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITHOUT HEAT CONDUCTIVITY

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

(2,q)阶分数差分方程的解

首次提出了一种分数阶差分,分数阶和分以及分数阶差分方程的定义,并给出(2,q)阶常系数分数阶差分方程的具体解法.