Temperature spectra in shear flow and thermal convection

We show that the P_u(\om) \propto \om^{-7/3} shear velocity power spectrum gives rise to a P_\Theta (\om ) \propto \om^{-4/3} power spectrum for a passively advected scalar, as measured in experiment [K. Sreenivasan, Proc. R. Soc. London A {\bf 434}, 165 (1991)]. Applying our argument to high Rayleigh number Rayleigh Benard flow, we can account for the measured scaling exponents equally well as the Bolgiano Obukhov theory (BO59). Yet, of the two explanations, only the shear approach might be able to explain why no classical scaling range is seen in between the shear (or BO59) range and the viscous subrange of the experimental temperature spectrum [I. Procaccia {\it et al.}, Phys. Rev. A {\bf 44}, 8091 (1991)].Comment: 9 pages, 1 figure, 1 tabl

Problems of Harmonic Analysis related to finite type hypersurfaces in R^3, and Newton polyhedra

This article, which grew out of my lecture at the conference "Analysis and Applications: A Conference in Honor of Elias M. Stein" in May 2011, is intended to give an overview on a collection of results which have been obtained jointly with I.I. Ikromov, and in parts also with M. Kempe, and at the same time to give a kind of guided tour through the rather comprehensive proofs of the major results that I shall address. All of our work is highly influenced by the pioneering ideas developed by E.M. Stein.Comment: 44 pages, a picture