微极流体方程组在Besov空间中弱解的正则性准则（英文）

In this paper,we investigate the regularity criterion via the pressure of weak solutions to the micropolar fluid equations in three dimensions.We obtain that for 0 α1 if p∈L~(2/α)(0,T;■),then the weak solution(u,w) is regular on(0,T].     

局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性

In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^αΩ on a bounded open set Ω ■R(N ≥ 2) with C^(1,1) boundary ■Ω. We prove that when f ∈ L^p(Ω), and g ∈ C(Ω), the following problem (-△)^αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.     

BOUNDEDNESS OF VECTOR-VALUED CALDERN-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calder6n-Zygmund operator, simply CZO on Herz space and weak Herz space. In particular.we obtain vector-valued inequalities for CZO on Lq(Rd, |x|αdμ) space, with 1< q <∞, -n <α< n(q -1), and on L1,∞(Rd,|x|αdμ) space, with -n <α< 0.     

三维不可压霍尔磁流体方程组弱解的正则性准则(英文)

In this paper,we study the regularity criterion of weak solutions to the3 D incompressible Hall-magnetohydrodynamics,which is ifu and Bsatisfy the condition∫_0~T‖■_(x3)u(t)‖~q_(LP)+‖▽B‖~γ_(Lβ)dt∞ with 3/p+2/q≤1,3/β+2/γ≤1,p3,β3,then the weak solution(u,B) is a smooth one on(0,T].     

HARNACK ESTIMATES FOR WEAK SOLUTIONS OF EQUATIONS OF NON-NEWTONIAN POLYTROPIC FILTRATION

51. IntroductionIn this paper we investigate the local behavior of nonnegative weak solutions for theequations of non-Newtonian polytropic filtrationac -- div(IDumlp--'Dam) = 0,m(p -- 1) < 1,m > 0,p > l, in D'(flT),u E CI..(0, T; L?o.(n)), urn E Lro.(0, T; WI:)l(fl)), (1.1)where n is an open set of R", N 2 1, 0 < T < ool fiT ~ fix (0, T), and D ~ (k,', e).Equation (1.1) appears in a number of applications to describe the evolution of diffusion.The evolution HLaplacian equationut ~ …     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article,we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R~3.We obtain the classical blow-up criteria for smooth solutions(u,ω,b),i.e.,u ∈ L q(0,T;L p(R 3)) for 2 q + 3 p ≤ 1 with 3p≤∞,u ∈ C([0,T);L 3(R 3)) or u ∈L q(0,T;L p) for 3 2p≤∞ satisfying 2 q + 3p≤2.Moreover,our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid.In the end-point case p = ∞,the blow-up criteria can be extended to more general spaces u∈ L~1(0,T;B_(∞,∞)~0(R~3)).     

Remarks on the Regularity Criteria of Three-Dimensional Navier-Stokes Equations in Margin Case

In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity u satisfies u∈L2（0,T;BMO（R3）） or u∈L^2/1＋r（0,T;B∞,∞（R3）） for 0 〈 r 〈 1 are considered.     

Limiting case for the regularity criterion of the Navier-Stokes equations and the magnetohydrodynamic equations

Any weak solution u to the Navier-Stokes equations is showed to be regular under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small.     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix

In this paper, the authors cosider the derivation of the exact distributions of the ratios of the extreme roots to the trace of the Wishart matrix. Also, exact percentage points of these distributions are given and their applications are discussed.     

On the comparison between the APV and the NPV computed via the WACC

The valuation of levered investment in the practice is made with the WACC approach, even if the superior technique of the APV is available. The paper shows that the APV can be interpreted as the arbitrage free value of the portfolio made by an investment and a supporting loan. Therefore the WACC evaluation, generally different from the APV, allows for arbitrage. We provide various conditions which completely characterize the sign of the error as a functions of all the variables entering the model.     

用常微分方程模型分析预防和隔离措施对SARS发病率的影响

通过建立常微分方程模型 ,分析了预防和隔离措施对 SARS发病率的影响 ,并把计算结果与实际统计数据进行了比较 ,结果表明 ,及时高效的预防和隔离措施能够有效地控制 SARS的传播 .     

From the neutrino to the edge of the universe

Two recent findings necessitate a closer look at the existing standard models of particle physics and cosmology. These are the discovery of neutrino oscillation, and hence a non-zero mass on the one hand and, on the other, observations of distant supernovae which indicate that contrary to popular belief, the universe would continue to expand for ever, possibly accelerating in the process. In this paper it is pointed out that relatively recent studies which indicate a stochastic, quantum vacuum underpinning and a fractal structure for space time, reconcile both of the recent observations, harmoniously.     

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m₁ there is no gap in the spectrum which is larger than 2m₁. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.     

From the subalgebras of the square to the discriminator

Presented by A. F. Pixley.     

On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.