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### Open Access Journals

CPAA

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 $[-1,1]^2$.

CPAA

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 $r \in
[1,\infty)$.

DCDS

Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

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 $h(\xi)=|\xi|^{\frac{5}{4}}$.

KRM

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.

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