Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on <!-- MATH $L^p_X(\mathbf{R}^2)$ --> with <!-- MATH $p\in (1,\infty).$ --> Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals with <!-- MATH $p^*:= \max \{p, (p/(p-1))\}$ --> , where the novelty is the lower bound.

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so-called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space $\dot B^{-1,\infty }_\infty$ . We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam.