Analysis of optimal bivariate symmetric refinable Hermite interpolants
Bin Han Qun Mo
Communications on Pure & Applied Analysis 2007, 6(3): 689-718 doi: 10.3934/cpaa.2007.6.689
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
Communications on Pure & Applied Analysis 2015, 14(2): 609-622 doi: 10.3934/cpaa.2015.14.609
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

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