Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ?3

We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10301014)     

Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ℝ3

We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces.     

Regularity criterion of axisymmetric weak solutions to the 3D Navier-Stokes equations

We consider the regularity of axisymmetric weak solutions to the Navier-Stokes equations in R³. Let u be an axisymmetric weak solution in R³×(0,T), w=curlu, and wθ be the azimuthal component of w in the cylindrical coordinates. Chae-Lee [D. Chae, J. Lee, On the regularity of axisymmetric solutions of the Navier-Stokes equations, Math. Z. 239 (2002) 645-671] proved the regularity of weak solutions under the condition wθ∈Lq(0,T;Lr), with , . We deal with the marginal case r=∞ which they excluded. It is proved that u becomes a regular solution if .     

Blow up problem for Landau-Lifshitz equations in two dimensions

In 1986, Zhou and Guo in [1] proved the global existence of weak solution for generalized Landau-Lifshitz equations without Gilbert term in multi-dimensions. They consider thehomogeneous boundary problemwith the initial value conditionfor the system of ferromagnetic chain with several variableswhere j(x, t, Z) is a given 3-dimensional vector funC-non in x e R", t E R+, Z e R', W(x)is a given 3-dimensional initial value function on fi, fi is a bounded domain in n-dimensionalEuclidean space…     

Regularity via one vorticity component for the 3D axisymmetric MHD equations

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that $w_{\theta }{e}_{\theta }\in L^1(0,T;{\dot{B}}_{\infty ,\infty }^0)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations.     

Global weak solutions to Landau-Lifshitz equations into compact Lie algebras

We consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra g; which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold T or a bounded domain in {ℝ}^n into a unit sphere {{\displaystyle S}}_g(\mathrm{1}) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of {ℝ}^{\mathrm{3}} and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.     

Regularity criteria of axisymmetric weak solutions to the 3D magnetohydrodynamic equations

In this paper, we study the regularity criteria for axisymmetric weak solutions to the MHD equations in ?³. Let ω _θ , J _θ and u _θ be the azimuthal component of ω, J and u in the cylindrical coordinates, respectively. Then the axisymmetric weak solution (u, b) is regular on (0, T) if (ω _θ , J _θ ) ∈ L ^q (0, T; L ^p ) or (ω _θ , ▽(u _θ e _θ )) ∈ L ^q (0, T; L ^p ) with $\tfrac{3} {p} + \tfrac{2} {q} \leqslant 2$ , $\tfrac{3} {2} < p < \infty$ . In the endpoint case, one needs conditions $\left( {\omega _\theta ,J_\theta } \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ or $\left( {\omega _\theta ,\nabla \left( {u_\theta e_\theta } \right)} \right) \in L^1 \left( {0,T;\dot B_{\infty ,\infty }^0 } \right)$ .     

Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations

In this paper we study the magneto-micropolar fluid equations in ℝ³, prove the existence of the strong solution with initial data in H^s(ℝ³) for , and set up its blow-up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda-type blow-up criterion for smooth solution (u, ω, b) that relies on the vorticity of velocity ∇ × u only. Copyright © 2007 John Wiley & Sons, Ltd.     

The regularity criterion of the weak solution to the 3D viscous Boussinesq equations in Besov spaces

In this paper, by using the Fourier localization technique and Bony's paraproduct decomposition, we give a regularity criterion of the weak solution to 3D viscous Boussinesq equations in Besov spaces. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence of the periodic solution to the nematic liquid crystal equations in weighted Sobolev spaces

This paper is concerned with the time-periodic solution to the simplified incompressible nematic liquid crystal equation. We prove the existence of the time-periodic solution of this equation with small external forces g₁ and g₂, satisfying the T-periodic conditions g_j(t)=g_j(t+T) for j=1,2 in weighted Sobolev spaces.     

Regularity condition of solutions to the quasi-geostrophic equations in Besov spaces with negative indices

With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ（t, x） on （0, T ） of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ（t, x） ∈L 2γ/γ-2δ （（0, T ）; B^δ-γ/2 ∞∞（R^2）） for 0 〈 δ 〈 γ/2 .     

三维各向异性Navier-Stokes方程一类大解的整体正则性——献给陆善镇教授75华诞

利用解析性估计和方程非线性项的特殊结构,本文证明了三维各向异Navier-Stokes方程对一类在垂直方向慢变的大初值的整体适定性.     

A new blowup criterion of strong solutions to the two-dimensional equations of compressible nematic liquid crystal flows

In this paper, we concern the Cauchy problem of two-dimensional (2D) compressible nematic liquid crystal flows with vacuum as far-field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem.     

Regularity Criteria for the Generalized MHD Equations

For Sobolev spaces in Lipschitz domains with no imposed boundary conditions, the Aronszajn–Smith theorem algebraically characterizes coercive formally positive integro-differential quadratic forms. Recently, linear elliptic differential operators with formally positive forms have been constructed with the property that no formally positive forms for these operators can be coercive in any bounded domain. In the present article 4th order operators of this kind are shown by perturbation to have coercive forms that are (necessarily) algebraically indefinite. The perturbation here from noncoercive formally positive forms to coercive algebraically indefinite forms requires Agmon's characterization of coerciveness in smoother domains than Lipschitz.