Turbulence is a state of flows which is characterized by a combination of chaotic and random behaviours affecting a very large range of scales. It is governed by Navier-Stokes equations and corresponds to their solutions in the limit where the fluid viscosity becomes negligible, the nonlinearity dominant and the turbulent dissipation constant. In this regime one observes that fluctuations tend to self-organize into coherent structures which seem to have their own dynamics.

A prominent tool for multiscale decomposition are wavelets. A wavelet is a well localized oscillating smooth function, e.g. a wave packet, which is translated and dilated. The wavelet transform decomposes a flow field into scale-space contributions from which it can be reconstructed.

We will show how the wavelet transform can decompose turbulent flows into coherent and incoherent contributions presenting different statistical and dynamical properties. We will then propose a new way to analyze and predict the evolution of turbulent flows.

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The presentation will use different results obtained in collaboration with:

Kai Schneider (Universite de Provence, Marseille, France),
Naoya Okamoto, Katsunori Yoshimatsu and Yukio Kaneda (Nagoya University, Japan)

Related publications can be downloaded from the web page
http://wavelets.ens.fr

Bedford and Taylor showed in 1982 that the complex
Monge-Ampere operator can be well defined (as a regular measure) for locally bounded plurisubharmonic (psh) functions,
and is continuous (in the weak topology) for decreasing
sequences. On the other hand, it is known that this operator
cannot be well defined for all psh functions. We will give a
precise characterization of its domain of definition. It turns
out that in dimension 2 it consists precisely of those psh
functions that belong to the Sobolev space W^{1,2}_{loc}.
We will also discuss a related question on compact K\"ahler
manifolds.

In general, there exist no analytical methods for quantitative analysis of the nonlinear propagation of ultrashort optical pulses in fiber, which underlies the operation of femtosecond-pulse fiber lasers. Such methods are needed now as the current generation of fiber lasers promise to greatly enhance the performance of practical instruments. In general, a pulse undergoes large changes in its temporal shape, spectral shape, and phase or frequency as it traverses a fiber laser, which in turn pose severe challenges to mathematical models. Self-similar pulse evolution is remarkable because monotonically-evolving, asymptotic solutions of the governing wave equation exist, despite the periodic boundary condition of a laser resonator. Highly-chirped pulse solutions can also exist in the presence of strong dissipation, and these so-called dissipative solitons represent a new class of laser pulses that offers remarkable behavior and performance. Quantitative models for lasers based on these pulses will be developed from first principles. These models will be studied in appropriate parameter regimes where simplified nonlinear dynamical systems theory can be utilized. In all cases, stability of the pulse solutions is the crucial issue. The theoretical efforts are highly interdisciplinary: combining asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis with models that are based on, and validated by, experimental observations.