5,619 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
Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials
This paper deals with the long time behaviour of solutions to the spatially
homogeneous Landau equation with hard potentials . We prove an exponential in
time convergence towards the equilibrium with the optimal rate given by the
spectral gap of the associated linearized operator. This result improves the
polynomial in time convergence obtained by Desvillettes and Villani
\cite{DesVi2}. Our approach is based on new decay estimates for the semigroup
generated by the linearized Landau operator in weighted (polynomial or
stretched exponential) -spaces, using a method develloped by Gualdani,
Mischler and Mouhot \cite{GMM}.Comment: 20 pages. Minor corrections, improvement on the presentatio
A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization
A deterministic method is proposed for solving the Boltzmann equation. The
method employs a Galerkin discretization of the velocity space and adopts, as
trial and test functions, the collocation basis functions based on weights and
roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or
full-range Hermite polynomials depending whether or not the distribution
function presents a discontinuity in the velocity space. The resulting
semi-discrete Boltzmann equation is in the form of a system of hyperbolic
partial differential equations whose solution can be obtained by standard
numerical approaches. The spectral rate of convergence of the results in the
velocity space is shown by solving the spatially uniform homogeneous relaxation
to equilibrium of Maxwell molecules. As an application, the two-dimensional
cavity flow of a gas composed by hard-sphere molecules is studied for different
Knudsen and Mach numbers. Although computationally demanding, the proposed
method turns out to be an effective tool for studying low-speed slightly
rarefied gas flows
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