Generalized surface quasi‐geostrophic equations with singular velocities

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$ , where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$ , which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.     

The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid

The velocity fields corresponding to an incompressible fluid of Oldroyd-B type subject to a linear flow within an infinite edge are determined for all values of the relaxation and retardation times. The well known solution for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appears as a limiting case of our solutions. To cite this article: C. Fetecau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 979–984.     

Partitions lineaires arguesiennes d'un espace vectoriel

In this work, we study linear and desarguesian partitions of a finite dimensional vector space over a skew-field K. When K is finite, we describe the set of all these partitions as a homogeneous space of the general linear group and we give an enumeration formula.     

Blends of a new thermoplastic in a thermoset epoxy matrix

Properties of thermoplastic modified epoxy network have been studied. The particularity of this work is the use of new thermoplastic epoxies whose structure is close to the final matrix. Blends of thermoset epoxy (Diglycidyl Ether of Bisphenol A/4-4′ methylenebis [3-chloro 2,6-diethylaniline]) with a thermoplastic content from 5 to 40%w have been realised. Initial miscibility in the thermoset precursors shows an UCST behaviour with a maximal value near 130°C for a thermoplastic content of 10%. Due to the presence of tertiary amine and pendant hydroxyl groups on the thermoplastic backbone, epoxy amine reactions are faster than for the neat system but the thermoplastic seems not to have reacted with the thermoset network. The final blends are transparent but toughening increase is rather low.     

ChemInform Abstract: Crystal Structures of New Alkali Metal-Rich Oxometallates: Rubidium Aluminate Tetrahydroxide,Rb9(AlO4)(OH)4, Rubidium Orthogallate,Rb5GaO4, Cesium bis-Chromate(IV) Oxide,Cs10(CrO4)2O,and Cesium Diindate,Cs8In2O7.

The crystal structures of several new alkali metal oxometallates are determined by single crystal XRD.