On the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity

In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition

Mechanistic studies on reactivities of organometallic macrocyclic complexes of chromium and cobalt

Reaction pathways leading to the formation and cleavage of a transition metal-carbon bond at various oxidation states of the metal occupy a central position in understanding many enzymatic reactions and designing catalysts;Part I focuses on the homolysis vs. heterolysis of a C-Cr(III) bond. Rate constants of homolysis and heterolysis of the C-Cr(III) bonds along with corresponding [delta]H[superscript]\ddagger and [delta]S[superscript]\ddagger were measured by ESR and stopped flow techniques. Factors favoring one pathway over another to cleave a C-Cr(III) bond are discussed;Part II describes a unique chain reaction and a S[subscript] E2 reaction between I[subscript]2 and RCrL[superscript]2+. The molecular structure of RCrL[superscript]2+, determined by x-ray diffraction for the first time, provides clearly a structural basis for the reactivity discussions;Part III concerns the oxidation of organochromium(III) complexes by dihalide and pseudo-dihalide radical anions generated by pulse radiolysis. The reaction mechanism and reactivity trend are discussed;Part IV concentrates on the oxidation mechanism of RCrL(H[subscript]2O)[superscript]2+ and the fate of RCr(H[subscript]2O)[superscript]3+ as well as the corresponding reduction potentials. An outer-sphere electron transfer mechanism was established. A facile C-Cr(IV) bond hemolyses was discovered along with its negligible heterolysis. Reduction potential of E[subscript]1/2(RCr[superscript] IV/RCr[superscript] III) and the relevant self exchange rate constant k[subscript]11 were estimated by a kinetic method;In Part V the study is extended to organocobalt complexes with attention turned to reduction induced cleavages of a transition metal-carbon bond. Electrochemical methods in conjunction with ESR, NMR, and GC-MASS are used to reveal evidence of novel reactions--reduction induced hydrogen atom transfer and reduction induced alkyl group migration. Part V also concerns the kinetics of reductions of the organocobalt compounds by a nickel(I) complex;In Part VI, the crystallization of ((CH[subscript]3)[subscript]4N) (Co(dmgBF[subscript]2)[subscript]2py) and its molecular structure determined by X-ray diffraction are described. The Co[superscript] I-N[subscript] eq distances are unusually short (1.839 A), even shorter than the corresponding bond (1.878 A) in the cobalt(II) analogue. Reasons of this unusual Co-N bond shortening is discussed along with the electronic structure of the d[superscript]8 cobalt(I) anion