On the mathematical side, the problem concerns the behavior of solutions of the hydrodynamical equations describing the time evolution of the velocities and, possibly, other quantities like density, temperature, magnetic field, etc. The simplest example is the almost two century old Navier-Stokes equations. Despite a long study, the open questions abound here. Let us mention only the best known problem, one of the seven Millennium Prize Problems of the Clay Mathematics Institute, about the global existence of solutions for smooth initial data, a genuine high Reynolds number problem. Different variations of hydrodynamical equations describe variety of physical phenomena (turbulent transport or diffusion, environmental dispersion, combustion, multiphase flows, geophysical and astrophysical flows etc.) and their mathematical study, besides common features, poses specific problems.
On the physical side, one usually deals with questions about the statistical properties of the flows, in particular, about the decay of a turbulent statistical state or about existence and properties of steady states maintained by continuously stirring the fluid. Such states are observed experimentally (e.g. in wind tunnels or water jets) and they seem to be characterized by non-vanishing fluxes of energy and other conserved quantities, a typical feature of non-equilibrium situations. The high Reynolds number turbulent states are strongly coupled and defy analytic approaches which proved successful in the equilibrium statistical mechanics, like the perturbative expansions and/or the renormalization group analysis. The statistics of such quantities as the velocity or temperature differences is highly non-Gaussian at short distances, as signaled by the spatial and the temporal intermittency of signals measured in natural flows, in experiments and in numerical simulations. The detailed mechanism of intermittency remains to be understood, despite many quasi-phenomenological models. An analytic control of the turbulent states remains still an elusive goal.
An important progress has been achieved in the last decade in understanding some simpler systems exhibiting behaviors similar to developed turbulence. These include the so-called weak or wave turbulence , the advection of passive scalar and vector fields by random velocities that mimic the turbulent ones, and, to certain extent, the so-called burgulence , the phenomena described by the Burgers equation.
Weak turbulence concerns the behavior of a spatially homogeneous ensemble of weakly interacting dispersive waves and occurs in several Hamiltonian PDE's, like the nonlinear Schrödinger equation. These Hamiltonian systems, once driven by external sources, exhibit non-equilibrium steady states with nonzero fluxes of conserved quantities (energy and particle number in the case of the Schrödinger equation). The difference with the "strong turbulence" of the Navier-Stokes equation is that the weakly turbulent state is accessible via perturbation theory in the strength of the adjustable nonlinear coupling constant which is absent in the case of strong turbulence.
Examples of passively advected quantities are the temperature or the impurity concentration in a fluid. Ideally one would be interested in the statistical properties of the advected field in the case where the underlying flow is turbulent. Significant progress has been achieved when the velocity field is taken random, with Gaussian statistics but decorrelated (white) in time. One mimics the important feature of turbulent flows by taking the velocities rough, i.e. only Hölder continuous, in space. For such an ensemble of velocities (called the Kraichnan model), it was possible to study the ensuing steady state of the advected fields both analytically and numerically. It appears to be a nonequilibrium state with nonzero flux of a conserved quantity, again in analogy to hydrodynamical turbulence. Moreover it exhibits intermittency in the form of anomalous scaling of moments of scalar differences in nearby points, the first (and so far only) nontrivial model where the anomalous scaling has been established analytically.
The Burgers equation is a pressureless version of the Navier-Stokes equation. Its 1-dimensional version, in particular, has been extensively studied. When randomly stirred, it exhibits the energy cascade with the short-distance dissipation dominated by the shock-like structures and strong intermittency.
Although good quantitative experimental measurements and numerical
simulations involving developed turbulence are notoriously difficult,
there has been a considerable progress on that side as well and new sets
and types of data become available. As for experiments, optical non-intrusive
methods start to give access to the multi-point structure of turbulent flows.
The importance of multi-point objects is one of the key lessons that emerged
from theoretical studies in the past few years. Going beyond structure functions
will also permit to highlight the presence and the role of coherent structures.
Another fast developing experimental tool is the particle-tracking technique,
whose aim is to follow fluid particles and hence give information on the
Lagrangian velocity statistics. Both types of experiments are strongly coupled
with the theoretical issues of current debate in the turbulence community.
Numerical simulations are natural candidates to permit the resolution of
the whole velocity field and of the Lagrangian and multi-point statistics,
although at more limited Reynolds numbers.
back to index
MONDAY, MAY 27, MORNING
10:00-10:40 KHANIN
Burgers turbulence in unbounded domains.
10:40-11:20 BEC
Hyperbolicity and statistics in forced Burgers turbulence
11:20-12:00 GIRAUD
Burgers turbulence (homogeneous/ space-periodic / non-homogeneous)
and its evolution in one dimension
12:00-14:00 LUNCH BREAK
MONDAY, MAY 27, AFTERNOON
14:00-14:40 WEHR
Front speed in the Burgers equation with a random flux
14:40-15:40 Burgulence: discussion
15:20-16:00 COFFEE BREAK
16:00-18:00 BRENIER
The Monge-Ampere equation (lecture 1)
TUESDAY, MAY 28, MORNING
10:00-12:00 BRENIER
The Monge-Ampere equation (lecture 2)
12:00-14:00 LUNCH BREAK
TUESDAY, MAY 28, AFTERNOON
14:00-14:40 FRISCH
Reconstruction of the primordial Universe: cosmological background, from
the Burgers/adhesion model to the Monge-Ampere
equation; presentation of the results.
14:40-15:20 SOBOLEVSKI
Reconstruction of the primordial Universe: implementation of the reconstruction
hypothesis, cyclic monotonicity, mass
transportation and the assignment problem.
15:20-16:00 COFFEE BREAK
16:00-18:00 Discussion on Monge-Ampere and Burgers
WEDNESDAY, MAY 29, MORNING
10:00-10:40 AURELL
Burgers equation and the dynamics of stratified self-gravitating particles.
10:40-11:20 BLANK
Dynamics of traffic jams
11:20-12:00 YAKHOT
Similarities of Burgers and real turbulence
12:00-14:00 LUNCH BREAK
WEDNESDAY, MAY 29, AFTERNOON
14:00-17:00 Discussion on Burgers and applications (with a coffee break
around 15:30)
THURSDAY, MAY 30, MORNING
10:00-10:40 EYINK
An Historical Account of Onsager's Dissipation Anomaly
10:40-11:40 DUCHON
Dissipation in weak Euler and Burgers solutions
11:40-12:20 VANDEN-EINJDEN
Topic: dissipation and anomaly
12:20-14:00 LUNCH BREAK
THURSDAY, MAY 30, AFTERNOON
14:00-14:40 SINAI
Quasi-linear approximations of the 3d Navier-Stokes system.
14:40-17:00 discussion on dissipation and Navier-Stokes (with a coffee
break around 15:30)
FRIDAY, MAY 31, MORNING
10:00-10:40 JENSEN
Pulses in the Parisi continuum shell equations
10:40-11:20 VASSILICOS
Lagrangian properties of Kinematic Simulations and their relation to
Eulerian statistics
11:20-12:00 BERNARD
Influence of friction on 2D enstrophy cascade
12:00-14:00 LUNCH BREAK
FRIDAY, MAY 31, AFTERNOON
14:00-14:40 GAWEDZKI
Variations on Lagrangian flow
15:30-16:00 COFFE BREAK
16:00-17:30 Final discussion
Uriel Frisch (with Yasha Sinai)
May June July
15 19 26 2 9 16 23 30 7 14
| ! ! ! ! ! ! !
Burg. worksh. Celani
Bec -------- ---------------------------
------------------------ Bernard Falkovich
------------------- Chertkov -----------
Duchon ----------------- Dombre
------------------- ---------------
Bricmont Bauer Constantin Arponen
---------- -------- ------------ Papanicolaou ---------
Frisch --------
---------------------- Dubrulle Nazarenko
Khanin --------- ------------------
--------------------- Moffatt Cencini
----------------- ------------
Biferale
Sobolevski -------------------------- Antonov
----------------------- Horvai --------------------------
Giraud ----------------------------------- Lukkarinen
----- Gawedzki ---------
-----------------------------------------------------
Sinai Le Jan
--------- ---------------
Blank Fannjiang
--------- --------------
Kup- -iai- -nen Villermauxback to index
--------------- -------- ------ -----------
Lohse
Eyink ---------
----------------- Newell Olla
Chabanol --------- ----------
-------- Komorowski Raimond
Aurell ----------------- -----------------
------ Hakulinen
Kurien Ciliberto --------
------------ --------
Vanden Eijnden Leveque Vulpiani
-------- ----------- -----------------
Brenier Steinberg
----------------- -------------------
Masmoudi
------------------ --------
Shnirelman
Pumir -------------
Vassilicos -------- Muratore-G
-------- ---------
Jensen Connaughton Mazzino
--------- -------- ---------------
Wehr Vincenzi
---------------------------------- ------------
Weinan E Zakharov
----------- Vergassola -----------------
--------------------- --------------------
| ! ! ! ! ! ! !
15 19 26 2 9 16 23 30 7 14
May June July
Tuesday June 4 at 10:15
Gregory EYINK (Tucson)
"On the joint cascade of energy and helicity"
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Thursday June 6 at 14:00
Uriel FRISCH (Nice)
"Extention to more than one dimension of the polar
decomposition for viscous Burgers equation"
Tuesday June 11 at 14:00
Peter CONSTANTIN (Chicago)
"Remarks on rotating fluid turbulence"
Abstract: I will discuss Lagrangian transport, energy dissipation and
energy spectra of rotating flow.
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Wednesday June 12 at 10:15 ESI lecture hall
Emmanuel LEVEQUE (Lyon)
"A model of rapidly depleted energy cascade in three-dimensional
Navier-Stokes turbulence"
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Wednesday June 12 at 14:00 ESI lecture hall
Mikhael CHERTKOV (Los Alamos)
"Decay of scalar turbulence revisited"
Abstract: The most efficient mixing of a scalar, passively advected by
a random flow, occurs if the flow is spatially smooth. However, any
realistic turbulent, or simply chaotic, flow cannot be smooth at all
scales. We demonstrate that at long times the rate of scalar
decay is dominated by regions (in real space or in inverse space)
where mixing is not as efficient as in an ideal smooth flow. We
examine two situations. The first is a spatially homogeneous stationary
turbulent flow with both viscous and inertial scales
present. It is shown that at large times scalar fluctuations decay
algebraically in time at all spatial scales (particularly in a
the viscous range, where the velocity is smooth). The second example
explains chaotic stationary flow in a disk/pipe. The boundary
region of the flow controls the long-time decay, which is algebraic
at some transient (asymptotically large, if diffusion is weak)
times, but becomes exponential, with the decay rate dependent on the
scalar diffusion coefficient, at longer times. This is joint work
with V. Lebedev (Landau I.)
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Thursday June 13 at 11:15 ESI lecture hall
Nader MASMOUDI (Courant)
"Existence and uniqueness of invariant measures
fort the Navier-Stokes equations"
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Thursday June 13 at 14:30 ESI lecture hall
Tomasz KOMOROWSKI (Lublin)
"On the superdiffusive behavior of passive tracer
with a Gaussian drift"
Abstract: We say that the bahavior of a passive tracer particle is
superdiffusive when the mean square of its displacement grows faster
than linearly. In this talk we characterize the family of Gaussian
drifts with power-law spectra, for which the motion of the passive
tracer is superdiffusive. Using variational principles we provide
upper and lower estimates of the respective Hurst exponents. We
shall consider both steady and time dependent drifts. The results
generalize those obtained by Avellaneda-Majda for the shear layer
flow.
back to index
Monday June 17 at 11:00 ESI lecture hall
Victor STEINBERG (Weizmann)
"Mixing by polymers"
Abstract: As we showed recently a flow of a visco-elastic polymer
solution can become quite irregular even at low velocity, high viscosity,
and in a small tank. So while the Reynolds number may be arbitrary low,
the observed flow has all main features of developed turbulence. This
elastic turbulence is accompanied by significant stretching of the polymer
molecules.We study mixing of viscous fluids at very low Reynolds
numbers in a curved channel in this flow regime. Polymer concentration
of only 0.001% suffices for efficient mixing. We showed that this regime
of mixing is particular well suited to study so called Batchelor regime
of mixing. It is one of the two simple cases, where the problem of
mixing can be solved analytically. Several important theoretical
predictions were verified for the first time.
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Monday June 17 at 14:30 ESI lecture hall
Mikhael CHERTKOV (Los Alamos)
"Acceleration of diffusion-limited chemical reaction by chaotic mixing"
Abstract: A comprehensive theory of the binary chemical reaction A+B->0
in chaotic flows at large Schmidt and Damkohler number, and with initially
injected equal (or close to equal) amount of both chemicals, is developed.
This is a joint work with V. Lebedev (Landau Inst., Moscow).
--------------------------------------------------------------------------------
Tuesday June 18 ESI lecture hall
Convection day (organized by Detlef Lohse)
10:15 - 10:50: Sergio CILIBERTO:
"Open problems in RB convection"
10:50 - 11:25: Detlef LOHSE:
"A unifying theory of thermal convection"
11:25 - 12:10: Berangere DUBRULLE: "Scaling in
convection from a quasilinear model of
turbulence"
14:00 - 15:45
discussion
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Tuesday June 18 16:00 ESI lecture hall
Albert Fannjiang (UC Davis)
"Noise Induced Dissipation in Discrete Time Dynamical Systems"
Abstract: We consider conservative dynamical systems on the torus
of d dimensions. We study the dissipation time scale and its physical
implications as the noise level psilon vanishes. We show that
non-mixing maps give rise to an O(1/psilon) dissipation time
whereas ergodic toral automorphisms, including cat maps and their
d-dimensional generalizations, have an O(n{(1/psilon)}) dissipation
time with a constant related to the entropy per dimension of the
automorphisms. Our approach reduces the calculation of the dissipation
time to a nonlinear, integer programming problem which is solved
asymptotically by means of certain fundamental theorems in Diophantine
approximations.
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Wednesday June 19 at 11:00 ESI lecture hall
George PAPANICOLAOU (Stanford)
"Scaling limits for the random Schroedinger equation"
--------------------------------------------------------------------------------
Wednesday June 19 at 14:30 ESI lecture hall
Alan NEWELL (Tucson)
"First part of an introductory mini-course on "Wave Turbulence"
--------------------------------------------------------------------------------
Thursday June 20 at 11:00 ESI lecture hall
Sergio CILIBERTO (ENS Lyon)
"Analogies and differences of crack formation with critical phenomena "
Abstract: The failure time of samples of heterogeneous materials (wood,
fiberglass) are studied as a function of the load features and geometry.
It is
shown that in these materials the failure time is predicted with a good
accuracy
by a model of microcrack nucleation proposed by Pomeau. We find that
the
time interval between events (precursors) and the energy are power
law
distributed and that the exponents of these power laws depend on the
load
history and on the material. In contrast, the cumulated acoustic energy
E
presents a critical divergence near the breaking time. A simple model
is also
discsussed.
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Thursday June 20 at 14:30 ESI lecture hall
Alain PUMIR (INLN, Nice)
"Statistical geometry in turbulence"
Abstract: I shall discuss the deformation of a set of Lagrangian particles
advected by a turbulent flow.
--------------------------------------------------------------------------------
Thursday June 20 at 15:15 ESI lecture hall
Detlef LOHSE (University of Twente)
"The effect of micro-bubles on developed turbulence"
--------------------------------------------------------------------------------
Friday June 21 at 14:00-16:00 Institute of Mathematics, Room 4
(next to the ESI lecture hall)
Alan NEWELL (Tucson)
Second part of a mini-course on "Wave Turbulence"
Monday June 24 at 11:15 ESI lecture hall
Luca BIFERALE (University Rome II)
"Anisotropic turbulence"
Abstract: We present a numerical study of anisotropic statistical
fluctuations in homogeneous turbulent flows. We predict the dimensional
scaling exponents zeta^j_d(p)=(p+j)/3 for the projections of the p-th
order
structure function in the j-th angular momentum sector. We show that
the measured exponents are anomalous exhibiting a clear deviation from
the dimensional prediction. Dimensional scaling is subleading and connected
to the dynamical fluctuations without phase correlations. Universality
of the observed anomalous scaling is discussed both theoretically and
by
means of numerical simulations at different Reynolds numbers and with
different forcing.
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Tuesday June 25 at 11:15 ESI lecture hall
Keith MOFFATT (DAMTP, University of Cambridge)
"Magneto-hydrodynamic turbulence: a brief review"
Abstract: MHD turbulence is characterised by transfer of energy between
the magnetic field B and the velocity field u. If this transfer is
systematically from u to B, then we have a turbulent dynamo, for which
the helicity of the velocity field plays a crucial role; if from
B to u,
then we have a "magnetic relaxation" process, for which the magnetic
helicity, conserved in the perfect conductivity limit, plays an equally
crucial role. These two aspects of MHD turbulence will be reviewed, and
some open problems will be identified.
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Wednesday June 26 at 11:15 ESI lecture hall
Keith MOFFATT (DAMTP, University of Cambridge)
"Magnetostrophic turbulence in the Earth's liquid core"
Abstract: Turbulence generated by a random buoyancy field in a rapidly
rotating medium permeated by a strong magnetic field is considered.
The velocity vield can be represented as a linear functional of the buyancy
field, and the problem takes the form of "an active scalar problem".
The helicity and the corresponding alpha-effect are calculated, and also
the Reynolds stress distribution. These are the essential ingredients
for maintenance of the magnetic field (i.e.dynamo action).
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Thursday June 27 at 11:15 ESI lecture hall
Yves LE JAN and Olivier RAIMOND (University Paris XI)
"Flows, coalescence and noise"
Abstract: We are interested in stationary ``fluid'' random evolutions
with independent increments. Under some mild assumptions, we show they
are
solutions of a stochastic differential equation (SDE). There are situations
where these evolutions are not described by flows of diffeomorphisms,
but
by coalescing flows or by flows of probability kernels. In an intermediate
phase, for which there exists a coalescing flow and a flow of kernels
solution of the SDE, a classification is given: all solutions of the
SDE
can be obtained by filtering the coalescing motion with respect to a
sub-noise containing the Gaussian part of its noise. Thus, the coalescing
motion cannot be described by a white noise.
-------------------------------------------------------------------------------
Thursday June 27 at 14:30 ESI lecture hall
Peter MARKOWICH (University of Vienna)
"Entropy dissipation methods for diffusive systems"
Wolfgang Pauli Seminar
-------------------------------------------------------------------------------
Friday June 28 at 14:30 ESI lecture hall
Alexander SHNIRELMAN (Tel Aviv University)
"Inverse cascade solutions of 2D incompressible Euler equations"
Abstract: The most prominent feature of 2-d turbulence is the inverse
energy cascade. This means that if the fluid is stirred on small scale,
then
the energy is irreversibly transferred to bigger scales. If, for example,
we
push the fluid at t=0, and the initial velocity field u(x,0) is small-scale
(i.e. its Fourier transform is concentrated in the area of high frequencies),
then we observe that the scale of the flow field u(x,t) grows, and the
flow
ends up as one of few vortices having the scale of the whole flow domain.
This phenomenon has been observed in numerical simulations and physical
experiments, and there exist speculative physical theories of it. But
there
has been no mathematical justification of the concept of inverse cascade.
In fact we are talking about some property of generic solutions of 2-d
Navier-Stokes of Euler equations. But no single solution has been
constructed which has the above inverse cascade property.
Monday July 1 at 14:30 ESI lecture hall
Nikolai ANTONOV (St. Petersburg University)
"Renormalization group and anomalous scaling in a model of passive
scalar advection"
Abstract: Field-theoretic renormalization group and operator-product
expansion are applied to Kraichnan's model of a passively advected scalar
field. Inertial-range anomalous scaling emerges as a consequence of the
existence in the model of "dangerous" composite operators, whose negative
dimensions are identified with the anomalous exponents. This allows one
to construct for them a systematic perturbation expansion, similar to
the well-known epsilon-expansion of the critical indicies. The practical
calculations are performed up to the third order (three-loop approximation).
---------------------------------------------------------------------------------
Tuesday July 2 at 11:15 ESI lecture hall
Antonio CELANI (INLN Nice)
"Active versus passive scalar turbulence"
Abstract: Active and passive scalar transported by an incompressible
two-dimensional conductive fluid are investigated. It is shown that
passive scalar displays a direct cascade towards small scales while
active magnetic potential builds up large scale structures in an inverse
cascade process. Correlations between scalar input and particle trajectories
are found to be responsible for those dramatic differences as well as
for
the behavior of dissipativen anomalies.
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Wednesday July 3 at 11:15 ESI lecture hall
Angelo VULPIANI (University Rome I)
"Front propagation in stirring media"
Abstract: The problem of front propagation in a stirring medium
is addressed in the case of cellular flows in three different regimes:
slow reaction, fast reaction and geometrical optics limits
--------------------------------------------------------------------------------
Wednesday July 3 at 14:30 ESI lecture hall
Vladimir ZAKHAROV (Landau Institute and University of Arizona)
"Bose condensation in wind driven sea"
Abstract: The theory of weak turbulence can explain the basic experimental
observations on wind-driven sea, accumulated during the last three decades.
The dominating process of frequency downshift can be compared with Bose
condensation in quantum systems.
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Thursday July 4 at 11:15 ESI lecture hall
Segey NAZARENKO (University of Warwick)
"Intermittency ans scale separation in turbulence"
Abstract: I will present numerical results of a study of the role
of distant wavenumber triads in shaping the structure functions
characteristic of Navier-Stokes intermittency.
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Thursday July 4 at 14:30 ESI lecture hall
Ville HAKULINEN (Helsinki University)
"Degenerate elliptic operators in passive advection"
Abstract: I present some applications of the Harnack inequality
for degenerate elliptic parabolic equations to the zero molecular
diffusivity limit of the Kraichnan model.
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Friday July 5 at 11:15 ESI lecture hall
Thierry DOMBRE (Grenoble University)
"Instantons and intermittency in 1D-cascade models of developed
turbulence"
Abstract: We shall present a semi-classical picture of intermittency
in the Gledzer-Okkitani-Yamada (GOY) shell model of turbulence, relying
on the interaction of soliton-like structures with a featureless random
background. The genet
al scheme developed for constructing instantons
in spatially discrete dynamical systems will be shown to be useful in
other contexts.
Monday July 8 at 14:30 ESI lecture hall
Gregory FALKOVICH (Weizmann Institute)
"Acceleration of rain start by cloud turbulence"
Abstract: I shall describe the statistics of concentration and collisions
of heavy inertial particles in a turbulent flow. The possible role of
turbulence in accelerating collisional droplet growth in clouds will
be
discussed.
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Tuesday July 9 at 14:30 ESI lecture hall
Vladimir ZAKHAROV (Landau Institute and University of Arizona)
"Quasi twodimensional hydrodynamics"
Abstract: The technique of quasi two dimensional hydrodynamics makes
it possible
to solvethe problem of vortex tubes.
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Wednesday July 10 at 14:30 ESI lecture hall
Emmanuel VILLERMAUX (University Aix-Marseille I)
"Mixing as an aggregation process"
Abstract: Basing on several experiments, it is shown how a stirred
scalar mixture relaxes towards uniformity through an aggregation process.
The elementary bricks are stretched sheets whose rates of diffusive
smoothing and coalescence build up the overall mixture concentration
distribution. The cases studied in particular include the ever
dispersing
mixture, and the confined mixture in two and three dimensions.
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Thursday July 11 at 14:30 ESI lecture hall
Piero OLLA (University of Lecce)
"Fokker-Planck equation formalism for random velocity fields"
Abstract: I will present a generalization of the Fokker-Planck
equation from random signals to random field and describe an
application to te transport of solid particles in turbulent
flows.
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Friday July 12 at 14:30 ESI lecture hall
Paolo MURATORE-GINANNESCHI (University of Helsinki)
"Gutzwiller's trace formula"
Abstract: A review of the formula is presented together with a new
derivation based on field theory techniques.
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Latest update on 5/07/2002.