Spectral instability of some non-selfadjoint anharmonic oscillators

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $\Pi_n$ associated with their eigenvalues. More precisely, we are concerned with the complex Airy operator and even anharmonic oscillator. We get asymptotic expansions for the norm of the spectral projections associated with the large eigenvalues, extending the results of Davies and Davies-Kuijlaars

Fourier analysis methods for the compressible Navier-Stokes equations

In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics equations in the whole space or in the torus. We here give an overview of results that we can get by Fourier analysis and paradifferential calculus, for the compressible Navier-Stokes equations. We focus on the Initial Value Problem in the case where the fluid domain is the whole space or the torus in dimension at least two, and also establish some asymptotic properties of global small solutions. The time decay estimates in the critical regularity framework that are stated at the end of the survey are new, to the best of our knowledge

Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on $\{0,1\}^n$. This new bound is shown to be asymptotically optimal.Comment: Published at http://dx.doi.org/10.1214/009117906000000287 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org