Weighted estimates for stationary Navier-Stokes equations

We show Morrey-type estimates for the weak solution of the periodic Navier-Stokes equations in dimensionN, 5 <N < 10. ForN < 8, we prove the existence of a maximum solution.     

Even dimensional higher class groups of orders

Let F be a number field, and let denote the ring of integers in F. Let A be a finite-dimensional central simple F-algebra, and let Λ be an -order in A. In this paper it is shown that the p-torsion of the even dimensional higher class group Cl _2n (Λ) can only occur for primes p, which lie under prime ideals , at which is not maximal, or which divide the dimension of A. Supported by NSFC 10401014, and partially funded by Irish Research Council for Science, Engineering and Technology Basic Research Grant SC/02/265.     

Hyers-Ulam stability for linear equations of higher orders

We show that, in the classes of functions with values in a real or complex Banach space, the problem of Hyers-Ulam stability of a linear functional equation of higher order (with constant coefficients) can be reduced to the problem of stability of a first order linear functional equation. As a consequence we prove that (under some weak additional assumptions) the linear equation of higher order, with constant coefficients, is stable in the case where its characteristic equation has no complex roots of module one. We also derive some results concerning solutions of the equation.     

Non-reversibility and self-joinings of higher orders for ergodic flows

By studying the weak closure of multidimensional off-diagonal self-joinings, we provide a sufficient condition for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on $\mathbb{T}^2$ , so-called von Neumann flows and some special flows with piecewise polynomial roof functions. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without non-trivial topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.     

Positivity of solutions to inhomogeneous mixed-type equations of higher orders

We study the solution to inhomogeneous mixed-type elliptic-hyperbolic equations of higher orders. We prove that the solution sign is constant and depends on the sign of the right-hand side of the equation.     

On stochastic setting of stationary phase method

A probability model _∫Rexp(ι[nP(x)])dΦ_n(x) with Φ_n(x) the distribution function of random variable (ξ_k is i.i.d. sequence of r.v.’s with zero expectation and unit variance), being in a framework of stationary phase method is analyzed. The asymptotic expansion in CLT and Hörmander’s theorem play crucial role in asymptotic analysis of the model.     

Weighted estimates for commutators of higher dimensional Hardy operators

得到了由加权Lipschitz函数(或加权CMO函数)和n维Hardy算子生成的交换子在一些函数空间的有界性,例如加权Lebesgue空间,加权Herz型空间.     

The Riemann problem for functions with polar lines of higher orders

We consider solutions of the jump problem, homogeneous and inhomogeneous Riemann problems for functions with polar lines of orders p _k +1 (k = 1, 2, … ), p _k ? 0. We study the cases of continuous and discontinuous coefficients. In the special case p _k = 0 the obtained results turn into the earlier known ones.     

Weighted extended B-spline method for the approximation of the stationary Stokes problem

A new stabilized, mesh-free method for the approximation of the Stokes problem, using weighted extended B-splines (WEB-splines) as shape functions has been proposed. The web-spline based bilinear velocity–constant pressure element satisfies the so called inf–sup condition or Ladyshenskaya–Babus˘ka–Brezzi (LBB) condition. The main advantage of this method over standard finite element methods is that it uses regular grids instead of irregular partitions of domain, thus eliminating the difficult and time consuming pre-processing step. Convergence and Condition number estimates are derived. Numerical experiments in two space dimensions confirm the theoretical predictions.     

局部平稳扩散模型中时变参数的加权最小二乘估计

基于离散观测样本,利用局部线性拟合,得到了局部平稳扩散模型中时变漂移参数的加权最小二乘估计,并讨论了估计量的相合性,渐近正态性和一致收敛速度.同时,通过模拟研究说明了估计量的有效性.