TITLES AND ABSTRACTS OF LECTURES
CLAUDE BARDOS
Kinetic Equations and Averaging Lemmas
Monday April 13 4 :00 pm ; Graduate Lecture :
Kinetic equations have a huge range of application in physics and engineering.
They describe the evolution of the density probability distribution function, f , of
interacting/colliding particles and molecules. Specifically, f describes the probability density of having a particle at time t at spatial location x with velocity v. These
equations involve the advection operator
f 7→ ∂t f + v · ∇x f
and in the absence of interaction/collision the equations take the simple advection
form (known as Liouville equation)
∂t f + v · ∇x f = 0,
with initial value f (x, v, 0) = f0 (x, v).
This is a very simple equation which has an explicit solution as travelling wave
f (x, v, t) = f0 (x − vt, v).
In spite of that (or may be because of that) it plays an important role in the analysis
of partial differential equations.
1. It exhibits dispersion properties which are shared by other important equations, such as, for instance, d’Alembert and Schrödinger equations.
2. The solution f presented above enjoys exactly the same regularity of the
initial value. However, these dispersion properties lead to more regularity results
for the moments of f (with respect to any test functions φ(v)), that is, regularity
of
Z
f (x, v, t)φ(v)dv
which carry the name of averaging lemmas.
Such lemmas turned out to be instrumental in the understanding of general
kinetic equations in the presence of nonlinear collision/interaction between the particles. In this talk I will discuss these averaging Lemmas.
My interest in this problem started from the project of validating the diffusion
approximation of the spectral analysis of the neutron transport equation, BardosSantos-Sentis (1984). This in turn has motivated the first version of the averaging lemma, Golse-Perthame-Sentis (1984), followed by a paper of Golse-LionsPerthame-Sentis (1988), and then many other contributions.
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C. BARDOS
Appearance of Turbulence in the Euler Limit with Boundary Effects
Wednesday April 15 4 :00 pm ; Colloquium :
In his famous 1941 paper, in the framework of statistical theory of turbulence,
Kolmogorov relates the spectra of turbulent flows to the anomalous energy dissipation :
lim νh|∇x uν |2 i = > 0,
ν→0
where ν is the viscosity of the turbulent flow and u is its velocity. Keeping that
in mind, I observe that the deterministic counterpart of the statistical theory of
turbulence would be the weak convergence of solutions of the Navier-Stokes equations, when the Reynolds number goes to infinity, and the emergence of the Wigner
measure which is related to this weak convergence. As a result (and this is due to a
theorem of Kato (1983)) it appears that, in the presence of boundary effects, there
seems to be a connection between the anomalous energy dissipation, asserted by
Kolmogorov, and the lack of convergence of smooth solutions of the Navier-Stokes
equations to smooth solutions of the Euler equation, as the viscosity ν → 0.
An extension of the Kato theorem shows that the relation
lim ν∂n (uν )τ |∂Ω = u 6= 0,
ν→0
is equivalent to the appearance of “turbulence".
This may be better understood in the context of ‘wild solutions’ of De Lellis and
Szekelyhidi. Eventually one remarks that the quantity u coincides with the term
used by Prandtl and Von-Karman in the construction of the turbulent layer.
This is a report on joint works with Golse , Paillard, Szekelyhidi, Titi and Wiedemann.
TITLES AND ABSTRACTS OF LECTURES
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Moments Method for the 6th Hilbert Problem
Thursday April 16 ; 4 pm ; Colloquium :
In the Paris ICM 1900 Hilbert proposed as a 6 problem to use the approach of
Boltzmann to derive the macroscopic law of physics from first principle.
A systematic use of the moments method (and averaging lemmas), first for the
diffusion approximation, and then for the incompressible limit of the Boltzmann
equations has generated constant progress concerning this problem.
One of the main issue is the appearance of irreversibility in such process. This
in turn has eventually necessitated the introduction of some ad-hoc randomness.
I intend to review several issues related to this program and concluding with
the most recent contribution of Bodineau, Gallagher and Saint Raymond, where
it is shown that the equations for the Brownian motion can be derived from the
Newtonian dynamics with no other measure than the standard Lebesgue measure.
I was involved this programme with the systematic introduction of the moment
method first for the diffusion limit of the transport equation and for the formal
“incompressible limit" of the Boltzmann equations, Bardos-Golse-Levermore (1991),
and later for the construction of examples of systems of particles with pathological
limit.
(C.B.) Laboratoire J.-L. Lions, 4 place Jussieu, 75252 Paris Cedex 05, France
E-mail address: claude.bardos@gmail.com