75 research outputs found
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off
This paper deals with the long time behavior of solutions to the spatially
homogeneous Boltzmann equation. The interactions considered are the so-called
(non cut-off and non mollified) hard potentials. We prove an exponential in
time convergence towards the equilibrium, improving results of Villani from
\cite{Vill1} where a polynomial decay to equilibrium is proven. The basis of
the proof is the study of the linearized equation for which we prove a new
spectral gap estimate in a space with a polynomial weight by taking
advantage of the theory of enlargement of the functional space for the
semigroup decay developed by Gualdani and al in \cite{GMM}. We then get our
final result by combining this new spectral gap estimate with bilinear
estimates on the collisional operator that we establish.Comment: 22 page
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
Quantitative uniform in time chaos propagation for Boltzmann collision processes
This paper is devoted to the study of mean-field limit for systems of
indistinguables particles undergoing collision processes. As formulated by Kac
\cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1)
prove and quantify this property for Boltzmann collision processes with
unbounded collision rates (hard spheres or long-range interactions), (2) prove
and quantify this property \emph{uniformly in time}. This yields the first
chaos propagation result for the spatially homogeneous Boltzmann equation for
true (without cut-off) Maxwell molecules whose "Master equation" shares
similarities with the one of a L\'evy process and the first {\em quantitative}
chaos propagation result for the spatially homogeneous Boltzmann equation for
hard spheres (improvement of the %non-contructive convergence result of
Sznitman \cite{S1}). Moreover our chaos propagation results are the first
uniform in time ones for Boltzmann collision processes (to our knowledge),
which partly answers the important question raised by Kac of relating the
long-time behavior of a particle system with the one of its mean-field limit,
and we provide as a surprising application a new proof of the well-known result
of gaussian limit of rescaled marginals of uniform measure on the
-dimensional sphere as goes to infinity (more applications will be
provided in a forthcoming work). Our results are based on a new method which
reduces the question of chaos propagation to the one of proving a purely
functional estimate on some generator operators ({\em consistency estimate})
together with fine stability estimates on the flow of the limiting non-linear
equation ({\em stability estimates})
Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
In this paper we prove new constructive coercivity estimates for the
Boltzmann collision operator without cutoff, that is for long-range
interactions. In particular we give a generalized sufficient condition for the
existence of a spectral gap which involves both the growth behavior of the
collision kernel at large relative velocities and its singular behavior at
grazing and frontal collisions. It provides in particular existence of a
spectral gap and estimates on it for interactions deriving from the hard
potentials \phi(r) = r^{-(s−1)}, or the so-called moderately
soft potentials \phi(r) = r^{−(s−1)}, , (without
angular cutoff). In particular this paper recovers (by constructive means),
improves and extends previous results of Pao [46]. We also obtain constructive
coercivity estimates for the Landau collision operator for the optimal
coercivity norm pointed out in [34] and we formulate a conjecture about a
unified necessary and sufficient condition for the existence of a spectral gap
for Boltzmann and Landau linearized collision operators.Comment: 29 page
On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics
This thesis deals with some nonlinear and nonlocal effective equations arising in kinetic theory and nonlinear optics.
First, it is shown that the homogeneous non-cutoff Boltzmann equation for Maxwellian molecules enjoys strong smoothing properties:
In the case of power-law type particle interactions, we prove the Gevrey smoothing conjecture. For Debye-Yukawa type interactions, an analogous smoothing effect is shown.
In both cases, the smoothing is exactly what one would expect from an analogy to certain heat equations of the form , with a suitable function , which grows at infinity, depending on the interaction potential.
The results presented work in arbitrary dimensions, including also the one-dimensional Kac-Boltzmann equation.
In the second part we study the entropy decay of certain solutions of the Kac master equation, a probabilistic model of a gas of interacting particles. It is shown that for initial conditions corresponding to particles in a thermal equilibrium and particles out of equilibrium, the entropy relative to the thermal state decays exponentially to a fraction of the initial relative entropy, with a rate that is essentially independent of the number of particles.
Finally, we investigate the existence of dispersion management solitons. Using variational techniques, we prove that there is a threshold for the existence of minimisers of a nonlocal variational problem, even with saturating nonlinearities, related to the dispersion management equation
Regularity theory for the spatially homogeneous Boltzmann equation with cut-off
We develop the regularity theory of the spatially homogeneous Boltzmann
equation with cut-off and hard potentials (for instance, hard spheres), by (i)
revisiting the Lp-theory to obtain constructive bounds, (ii) establishing
propagation of smoothness and singularities, (iii) obtaining estimates about
the decay of the sin- gularities of the initial datum. Our proofs are based on
a detailed study of the "regularity of the gain operator". An application to
the long-time behavior is presented.Comment: 47 page
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