Analysis of optimal bivariate symmetric refinable Hermite interpolants
Bin Han Qun Mo
Multivariate refinable Hermite interpolants with high smoothness and small support are of interest in CAGD and numerical algorithms. In this article, we are particularly interested in analyzing some univariate and bivariate symmetric refinable Hermite interpolants, which have some desirable properties such as short support, optimal smoothness and spline property. We shall study the projection method for multivariate refinable function vectors and discuss some properties of multivariate spline refinable function vectors. Here a compactly supported multivariate spline function on $\mathbb R^s$ just means a function of piecewise polynomials supporting on a finite number of polygonal partition subdomains of $\mathbb R^s$. We shall discuss spline refinable function vectors by investigating the structure of the eigenvalues and eigenvectors of the transition operator. To illustrate the results in this paper, we shall analyze the optimal smoothness and spline properties of some univariate and bivariate refinable Hermite interpolants. For the regular triangular mesh, we obtain a bivariate $C^2$ symmetric dyadic refinable Hermite interpolant of order $2$ whose mask is supported inside $[-1,1]^2$.
keywords: Refinable Hermite interpolants interpolatory Hermite masks interpolatory Hermite subdivision schemes smoothness symmetry projection method subdivision triplets. multivariate splines
Local and global existence results for the Navier-Stokes equations in the rotational framework
Daoyuan Fang Bin Han Matthias Hieber
Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
keywords: global solution Rotational flows Fourier-Besov space Chemin-Lerner space. Littlewood-Paley decomposition
Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation
Bin Han Changhua Wei
This paper is devoted to studying the global well-posedness for 3D inhomogeneous logarithmical hyper-dissipative Navier-Stokes equations with dissipative terms $D^2u$. Here we consider the supercritical case, namely, the symbol of the Fourier multiplier $D$ takes the form $h(\xi)=|\xi|^{\frac{5}{4}}/g(\xi)$, where $g(\xi)=\log^{\frac{1}{4}}(2+|\xi|^2)$. This generalizes the work of Tao [17] to the inhomogeneous system, and can also be viewed as a generalization of Fang and Zi [12], in which they considered the critical case $h(\xi)=|\xi|^{\frac{5}{4}}$.
keywords: Critical spaces energy estimates. global solution
Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle
Shuguang Shao Shu Wang Wen-Qing Xu Bin Han
We consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle. Bogovski$\check{i}$ operator on a subset of $\mathbb{R}^2$ is used in this paper. One important thing is to show that the solution of the equations does not blow up in finite time in the sense of some $L^2$ norm. We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.
keywords: Global existence Navier-Stokes flow Bogovski$\check{i}$ operator.

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