24,156 research outputs found
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 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
. 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
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