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Let$X$(-ϱ$B$^{$m$}×$C$^{$n$}be a compact set over the unit sphere ϱ$B$^{$m$}such that for each$z$∈ϱ$B$^{$m$}the fiber$X$_{$z$}={ω∈$C$^{$n$};($z, ω$)∈$X$} is the closure of a completely circled pseudoconvex domain in$C$^{$n$}. The polynomial hull $$\hat X$$ of$X$is described in terms of the Perron-Bremermann function for the homogeneous defining function of$X$. Moreover, for each point ($z$_{0},$w$_{0})∈Int $$\hat X$$ there exists a smooth up to the boundary analytic disc$F$:Δ→$B$^{$m$}×$C$^{$n$}with the boundary in$X$such that$F$(0)=($z$_{0},$w$_{0}).

ABSTRACT We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no-flux boundary condition for the distribution function, we establish the existence and uniqueness of a global weak solution to the two dimensional model involving the Stokes equation.

We study a fourth order finite difference method for the unsteady incompressible Navier-Stokes equations in vorticity formulation. The scheme is essentially compact and can be implemented very efficiently. Either Briley’s formula, or a new higher order formula, which will be derived in this paper, can be chosen as the vorticity boundary condition. By formal Taylor expansion, the new formula for the vorticity on the boundary gives 4th order accuracy; while Briley’s formula provides only 3rd order accuracy. However, the use of either formula results in a stable method and achieves full 4th order accuracy. The convergence analysis of the scheme with our new formula will be given in this paper, while that with Briley’s formula has been established in earlier literature. The consistency analysis is easier than that of Briley’s formula, no Strang type analysis is needed. In the stability analysis part, we adopt the technique of controlling some local terms by the diffusion term via discrete elliptic regularity. Physical no-slip boundary conditions are used throughout.