The Eigenfunctions of the Stokes Operator in Special Domains. III

Abstract

Palladium‐catalyzed reactions in general are carried out under an inert atmosphere because the palladium intermediates involved in the catalytic cycles are often known to be sensitive to oxygen. In this paper, we report that various palladium‐catalyzed reductive couplings proceed smoothly under an air atmosphere and in aqueous medium. Under the air atmosphere reaction conditions, palladium‐triphenylphosphine complex was found to be inactive. By using zinc as the reducing reagent, aryl halides were homo‐coupled to give symmetrical biaryls either in aqueous acetone or in water in the presence of a catalytic amount of 18‐crown‐6 at room temperature. Both aryl iodides and aryl bromides reacted efficiently under the current reaction conditions. The reaction of arylhalosilanes with aryl halides under refluxing conditions in air and water catalyzed by palladium generated unsymmetrical biaryls efficiently in the presence of either KOH or NaF. Such air‐stable couplings are also suitable for a Suzuki‐type coupling and a Stille‐type coupling.

Constraint-induced restriction and extension operators with applications

“…In general, due to the divergence-free constraint-induced noncommutativity of the projection operator and the Laplacian, it is not an easy task to find an explicit expression of the eigenvalues [6][7][8] . Only when Ω = R n or T n (n-torus) (i.e., u and f show some spatial periodicity), we have the Leray explicit formula, and can apply the Fourier analysis to u.…”

“…Various aspects of recent progress on the theory of the operator were reported in Refs. [4][5][6][7][8]. In this paper, starting from the Helmholtz minimum dissipation principle and using the Hodge decomposition technique, we adopt a variational point of view to carry out an analysis of the Stokes operator.…”

On the vanishing viscosity limit in a disk

“…In [10], a complete set of eigenfunctions for the annulus is derived in terms of Bessel functions of the first and second kind, J n and Y n . By ignoring the terms involving Y n and modifying somewhat the calculation of the eigenvalues, one can easily obtain the eigenfunctions for a disk.…”

“…We will, however, derive the vorticity of the eigenfunctions directly, as this is quite easy. In determining the eigenvalues and the velocity of the eigenfunctions, which is more difficult, we will rely on the results in [10].…”

“…Thus, the eigenfunctions have vorticity of the form J n (λ 1/2 r)e inθ and it remains to determine the eigenvalues that satisfy u = 0 on the boundary. The easiest way to do this is to use the expressions in [10]. For n = 0, we drop the term involving Y 1 in Equation (30) p. 406 of [10], giving…”