2,674 research outputs found
A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media
In this work, we are interested in the spectrum of the diffusively excited
granular gases equation, in a space inhomogeneous setting, linearized around an
homogeneous equilibrium.
We perform a study which generalizes to a non-hilbertian setting and to the
inelastic case the seminal work of Ellis and Pinsky about the spectrum of the
linearized Boltzmann operator. We first give a precise localization of the
spectrum, which consists in an essential part lying on the left of the
imaginary axis and a discrete spectrum, which is also of nonnegative real part
for small values of the inelasticity parameter. We then give the so-called
inelastic "dispersion relations", and compute an expansion of the branches of
eigenvalues of the linear operator, for small Fourier (in space) frequencies
and small inelasticity.
One of the main novelty in this work, apart from the study of the inelastic
case, is that we consider an exponentially weighted Banach
setting instead of the classical Hilbertian
case, endorsed with Gaussian weights. We prove in particular that the results
of Ellis and Pinsky holds also in this space.Comment: 30 pages, 2 figure
Blow up Analysis for Anomalous Granular Gases
We investigate in this article the long-time behaviour of the solutions to
the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for
hard spheres. This model describes a diluted gas composed of hard spheres under
statistical description, that dissipates energy during collisions. We assume
that the gas is "anomalous", in the sense that energy dissipation increases
when temperature decreases. This allows the gas to cool down in finite time. We
study existence and uniqueness of blow up profiles for this model, together
with the trend to equilibrium and the cooling law associated, generalizing the
classical Haff's Law for granular gases. To this end, we investigate the
asymptotic behaviour of the inelastic Boltzmann equation with and without drift
term by introducing new strongly "nonlinear" self-similar variables.Comment: 20
Residual equilibrium schemes for time dependent partial differential equations
Many applications involve partial differential equations which admits
nontrivial steady state solutions. The design of schemes which are able to
describe correctly these equilibrium states may be challenging for numerical
methods, in particular for high order ones. In this paper, inspired by
micro-macro decomposition methods for kinetic equations, we present a class of
schemes which are capable to preserve the steady state solution and achieve
high order accuracy for a class of time dependent partial differential
equations including nonlinear diffusion equations and kinetic equations.
Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure
Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation
In this paper we study the large-time behavior of the solution to a general
Rosenau type approximation to the heat equation, by showing that the solution
to this approximation approaches the fundamental solution of the heat equation
at a sub-optimal rate. The result is valid in particular for the central
differences scheme approximation of the heat equation, a property which to our
knowledge has never been observed before.Comment: 20 page
A Hierarchy of Hybrid Numerical Methods for Multi-Scale Kinetic Equations
In this paper, we construct a hierarchy of hybrid numerical methods for
multi-scale kinetic equations based on moment realizability matrices, a concept
introduced by Levermore, Morokoff and Nadiga. Following such a criterion, one
can consider hybrid scheme where the hydrodynamic part is given either by the
compressible Euler or Navier-Stokes equations, or even with more general
models, such as the Burnett or super-Burnett systems.Comment: 27 pages, edit: typo and metadata chang
Asymptotic Behavior of Thermal Non-Equilibrium Steady States for a Driven Chain of Anharmonic Oscillators
We consider a model of heat conduction which consists of a finite nonlinear
chain coupled to two heat reservoirs at different temperatures. We study the
low temperature asymptotic behavior of the invariant measure. We show that, in
this limit, the invariant measure is characterized by a variational principle.
We relate the heat flow to the variational principle. The main technical
ingredient is an extension of Freidlin-Wentzell theory to a class of degenerate
diffusions.Comment: 40 page
On steady-state preserving spectral methods for homogeneous Boltzmann equations
In this note, we present a general way to construct spectral methods for the
collision operator of the Boltzmann equation which preserves exactly the
Maxwellian steady-state of the system. We show that the resulting method is
able to approximate with spectral accuracy the solution uniformly in time.Comment: 7 pages, 3 figure
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