LIOUVILLE TYPE THEOREM FOR THE STATIONARY EQUATIONS OF MAGNETO-HYDRODYNAMICS

We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L~6(R~3) and BMO~(-1)(R~3) must be identically zero.This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L~2(R~3), i.e., finite Dirichlet integral.     

Weak type estimates for the maximal operators associated to plane curves

For the plane curves Γ,the maximal operator associated to it is defined byMf(x)=sup|∫f(x-Γ(t))(r~(-1)t)r~(-1)dt|where is a Schwartz function.For a certain class of curves in R~2,M is shown to boundedon (H(R~2),Weak L~1(R~2).This extends the theorem of Stein & Wainger and the theo-rem of Weinberg.     

The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.     

POSITIVE THIN PLATE SPLINES

We consider the problem of interpolating positive data at scat-tered data points of the plane R~2. To solve the problem, weintroduce the Positive Thin Plate Spline, i. e. the solution to Minimize integral from n=R~2{x~2/~2u}~2 + 2{xy/~2u}~2 + {y~2/~2u}~2, u ∈ D~(-2)L~2(R~2); u(t_j) = z_j, j=1…, n; u(t)≥0, t∈ Kwhere (t_j, z_j) are the data, K is a convex compact subset of R~2.We give existence, uniqueness, characterisation and convergenceresults. We also present a dual algorithm to compute this splineand show numerical experience with the method.     

L2-harmonic 1-forms on Complete Manifolds

We study the global behavior of complete minimal δ-stable hypersurfaces in R~(n+1) by using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ (n-1)~2/n~2)hypersurface in R~(n+1) has only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author's theorems on hypersurfaces and quaternionic K¨ahler manifolds.     

Some remarks on Wente’s inequality and the Lorentz-Sobolev space

In this note we consider Wente's type inequality on the Lorentz-Sobolev space.If▽f∈L~p1,q1(R~n),G ∈ L~(p2,q2)(R~n) and div G≡0 in the sense of distribution where(1/p1)+(1/P2)=(1/q1)+(1/q2)=1,1P1,P2∞,it is known that G·▽f belongs to the Hardy space H~1 and furthermore‖G·▽f‖H~1≤C‖▽f‖L~(p1,q1)(R~2)‖G‖L~(p2,q2)(R~2).Reader can see[9]Section 4.Here we give a new proof of this result.Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest.Finally,we use this inequality to get a generalisation of Bethuel's inequality.     

Non blow-up criterion for the 3-D Magneto-hydrodynamics equations in the limiting case

In this paper, we prove that suitable weak solution(u, b) of the 3-D MHD equations can be extended beyond T if u∈L~∞(0,T; L~3(R~3)) and the horizontal components b_h of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.     

基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析

We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f~δ. Instead of the observation L~2 space, we perform the reconstruction of the derivative in a weighted L~2 space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.     

FRACTIONAL INTEGRALS OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES

In this paper it has been systematically studied the imbedding properties o f fractional integral operatorsof periodic functions of several variables,and isomorphic properties of fractional intregral operators in thespaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration,thespace of Lipschitz continuous functions and the Sobolev space are identical in L~2-norm.Results obtainedhereare not true for fractional integrals(or Riesz potentials)in R~n.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

三维非牛顿流体动力学方程衰减性的一个注记

利用Stokes算子的谱分解方法和线性Lp-Lq估计研究一类三维不可压缩非牛顿流体弱解的最优代数衰减速率,证明了当初速度满足u0∈L2(R3)和∫R3(1+|x|)|u0(x)|dx∞,其弱解在L2范数下的衰减率为t-5/4.     

一类二元半正交小波包的构造

本文给出伸缩矩阵行列式为2的一类二元半正交小波包的构造算法.该小波包是以频域给出的,随着用于小波包分裂的滤波器选取的不同会得到L2(R2)中形态各异的Riesz基,这样使得L2(R2)中小波基的选择更灵活.     

Global Smooth Solutions to the 2-D Inhomogeneous Navier-Stokes Equations with Variable Viscosity

Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0 -- 1∈ H^s＋1（R^2）, u0 ∈ H^s（R^2） ∩ H^-ε（R^2） for s 〉 2 and 0 〈 ε 〈 1, the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid. Furthermore, the L^2 decay rate of the velocity field is obtained.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE COMPRESSIBLE NEMATIC LIQUID CRYSTAL SYSTEM IN R~3

In this paper,we study a nematic liquid crystals system in three-dimensional whole space R~3 and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L~1(R~3) additionally.     

R~2上一类非牛顿流体动力学方程组解的最优衰减率

本文利用Fourier分解方法讨论一类二维不可压缩非牛顿流体动力学方程组弱解的衰减性,证明了弱解在L~2范数下衰减率为(1+t)~(-1/2),和线性热传导方程解的衰减率一致,在此意义下本文的结果是最优的。     

GLOBAL EXISTENCE AND OPTIMAL CONVERGENCE RATES OF SOLUTIONS FOR THREE-DIMENSIONAL ELECTROMAGNETIC FLUID SYSTEM

In this article, we study the electromagnetic fluid system in three-dimensional whole space R~3. Under assumption of small initial data, we establish the unique global solution by energy method. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the L~p-L~q estimates for the linearized equations and an elaborate energy method when the L~1-norm of the perturbation is bounded.     

关于两个最小二乘解流形的逼近性

L1={X∈Rn×m|f(X)=‖XA1-B1‖2+‖CT1X-DT1‖2=min},L2={Y∈Rn×m|g(Y)=‖YA2-B2‖2+‖CT2Y-DT2‖2=min},其中A1∈Rm×k1,B1∈Rn×k1,C1∈Rn×l1,D1∈Rm×l1,A2∈Rm×k2,B2∈Rn×k2,C2∈Rn×l2,D2∈Rm×l2均为已知矩阵,本文讨论了L1,L2两个线性流形之间的逼近性,给出了d(L1,L2)=minX∈L1,Y∈L2‖X-Y‖的具体表达式.     

带弱奇异核非线性积分微分方程的有限元分析

讨论了带弱奇异核的非线性抛物积分微分方程的Hermite型各向异性矩形元逼近.在各向异性网格下导出了关于Riesz投影的L~2和H~1模的误差估计.在半离散和向后欧拉全离散格式下,基于Riesz投影的性质并利用平均值技巧,分别得到了L~2模意义下的最优误差估计.     

混合边值条件下Laplace方程的等谱问题以及分形鼓

本文将首先在R^2和R^3中各自构造出一对混合边值条件的等谱非等距同构的基本构件。并且在R^2中利用自相似的方法构造出相应的等谱非等距同构分形鼓．在此基础上，本文讨论了这类分形鼓的波数目函数的渐近估计．得到其第二项系数的上界和下界估计。     

一类奇异非线性双调和方程的正整解

该文建立了一类R~2上奇异非线性双调和方程正的径向对称整体解的存在性定理,并给出了解的有关性质,丰富和发展了文[1]」,[2]的理论和应用.