212 research outputs found
Extended Rearrangement inequalities and applications to some quantitative stability results
In this paper, we prove a new functional inequality of Hardy-Littlewood type
for generalized rearrangements of functions. We then show how this inequality
provides {\em quantitative} stability results of steady states to evolution
systems that essentially preserve the rearrangements and some suitable energy
functional, under minimal regularity assumptions on the perturbations. In
particular, this inequality yields a {\em quantitative} stability result of a
large class of steady state solutions to the Vlasov-Poisson systems, and more
precisely we derive a quantitative control of the norm of the
perturbation by the relative Hamiltonian (the energy functional) and
rearrangements. A general non linear stability result has been obtained in
\cite{LMR} in the gravitational context, however the proof relied in a crucial
way on compactness arguments which by construction provides no quantitative
control of the perturbation. Our functional inequality is also applied to the
context of 2D-Euler system and also provides quantitative stability results of
a large class of steady-states to this system in a natural energy space
Stable ground states and self-similar blow-up solutions for the gravitational Vlasov-Manev system
In this work, we study the orbital stability of steady states and the
existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM)
system. This system is a kinetic model which has a similar Vlasov structure as
the classical Vlasov-Poisson system, but is coupled to a potential in (Manev potential) instead of the usual gravitational potential in
, and in particular the potential field does not satisfy a Poisson
equation but a fractional-Laplacian equation. We first prove the orbital
stability of the ground states type solutions which are constructed as
minimizers of the Hamiltonian, following the classical strategy: compactness of
the minimizing sequences and the rigidity of the flow. However, in driving this
analysis, there are two mathematical obstacles: the first one is related to the
possible blow-up of solutions to the VM system, which we overcome by imposing a
sub-critical condition on the constraints of the variational problem. The
second difficulty (and the most important) is related to the nature of the
Euler-Lagrange equations (fractional-Laplacian equations) to which classical
results for the Poisson equation do not extend. We overcome this difficulty by
proving the uniqueness of the minimizer under equimeasurabilty constraints,
using only the regularity of the potential and not the fractional-Laplacian
Euler-Lagrange equations itself. In the second part of this work, we prove the
existence of exact self-similar blow-up solutions to the Vlasov-Manev equation,
with initial data arbitrarily close to ground states. This construction is
based on a suitable variational problem with equimeasurability constraint
A particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling
In this work, we derive particle schemes, based on micro-macro decomposition,
for linear kinetic equations in the diffusion limit. Due to the particle
approximation of the micro part, a splitting between the transport and the
collision part has to be performed, and the stiffness of both these two parts
prevent from uniform stability. To overcome this difficulty, the micro-macro
system is reformulated into a continuous PDE whose coefficients are no longer
stiff, and depend on the time step in a consistent way. This
non-stiff reformulation of the micro-macro system allows the use of standard
particle approximations for the transport part, and extends the work in [5]
where a particle approximation has been applied using a micro-macro
decomposition on kinetic equations in the fluid scaling. Beyond the so-called
asymptotic-preserving property which is satisfied by our schemes, they
significantly reduce the inherent noise of traditional particle methods, and
they have a computational cost which decreases as the system approaches the
diffusion limit
Asymptotic preserving schemes for highly oscillatory kinetic equation
This work is devoted to the numerical simulation of a Vlasov-Poisson model
describing a charged particle beam under the action of a rapidly oscillating
external electric field. We construct an Asymptotic Preserving numerical scheme
for this kinetic equation in the highly oscillatory limit. This scheme enables
to simulate the problem without using any time step refinement technique.
Moreover, since our numerical method is not based on the derivation of the
simulation of asymptotic models, it works in the regime where the solution does
not oscillate rapidly, and in the highly oscillatory regime as well. Our method
is based on a "double-scale" reformulation of the initial equation, with the
introduction of an additional periodic variable
Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium
In this work, we propose some numerical schemes for linear kinetic equations
in the diffusion and anomalous diffusion limit. When the equilibrium
distribution function is a Maxwellian distribution, it is well known that for
an appropriate time scale, the small mean free path limit gives rise to a
diffusion type equation. However, when a heavy-tailed distribution is
considered, another time scale is required and the small mean free path limit
leads to a fractional anomalous diffusion equation. Our aim is to develop
numerical schemes for the original kinetic model which works for the different
regimes, without being restricted by stability conditions of standard explicit
time integrators. First, we propose some numerical schemes for the diffusion
asymptotics; then, their extension to the anomalous diffusion limit is studied.
In this case, it is crucial to capture the effect of the large velocities of
the heavy-tailed equilibrium, so that some important transformations of the
schemes derived for the diffusion asymptotics are needed. As a result, we
obtain numerical schemes which enjoy the Asymptotic Preserving property in the
anomalous diffusion limit, that is: they do not suffer from the restriction on
the time step and they degenerate towards the fractional diffusion limit when
the mean free path goes to zero. We also numerically investigate the uniform
accuracy and construct a class of numerical schemes satisfying this property.
Finally, the efficiency of the different numerical schemes is shown through
numerical experiments
Models of dark matter halos based on statistical mechanics: I. The classical King model
We consider the possibility that dark matter halos are described by the
Fermi-Dirac distribution at finite temperature. This is the case if dark matter
is a self-gravitating quantum gas made of massive neutrinos at statistical
equilibrium. This is also the case if dark matter can be treated as a
self-gravitating collisionless gas experiencing Lynden-Bell's type of violent
relaxation. In order to avoid the infinite mass problem and carry out a
rigorous stability analysis, we consider the fermionic King model. In this
paper, we study the non-degenerate limit leading to the classical King model.
This model was initially introduced to describe globular clusters. We propose
to apply it also to large dark matter halos where quantum effects are
negligible. We determine the caloric curve and study the thermodynamical
stability of the different configurations. Equilibrium states exist only above
a critical energy in the microcanonical ensemble and only above a
critical temperature in the canonical ensemble. For , the system
undergoes a gravothermal catastrophe and, for , it undergoes an
isothermal collapse. We compute the profiles of density, circular velocity, and
velocity dispersion. We compare the prediction of the classical King model to
the observations of large dark matter halos. Because of collisions and
evaporation, the central density increases while the slope of the halo density
profile decreases until an instability takes place. We show that large dark
matter halos are relatively well-described by the King model at, or close to,
the point of marginal microcanonical stability. At that point, the King model
generates a density profile that can be approximated by the modified Hubble
profile. This profile has a flat core and decreases as at large
distances, like the observational Burkert profile. Less steep halos are
unstable
Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations
We introduce a new numerical strategy to solve a class of oscillatory
transport PDE models which is able to captureaccurately the solutions without
numerically resolving the high frequency oscillations {\em in both space and
time}.Such PDE models arise in semiclassical modeling of quantum dynamics with
band-crossings, and otherhighly oscillatory waves. Our first main idea is to
use the nonlinear geometric optics ansatz, which builds theoscillatory phase
into an independent variable. We then choose suitable initial data, based on
the Chapman-Enskog expansion, for the new model. For a scalar model, we prove
that so constructed model will have certain smoothness, and consequently, for a
first order approximation scheme we prove uniform error estimates independent
of the (possibly small) wave length. The method is extended to systems arising
from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic
phenomenon. Numerous numerical examples demonstrate that the method has the
desired properties
Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime
We apply the two-scale formulation approach to propose uniformly accurate
(UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic
limit regime. The nonlinear Dirac equation involves two small scales
and with in the nonrelativistic
limit regime. The small parameter causes high oscillations in time which brings
severe numerical burden for classical numerical methods. We transform our
original problem as a two-scale formulation and present a general strategy to
tackle a class of highly oscillatory problems involving the two small scales
and . Suitable initial data for the two-scale
formulation is derived to bound the time derivatives of the augmented solution.
Numerical schemes with uniform (with respect to )
spectral accuracy in space and uniform first order or second order accuracy in
time are proposed. Numerical experiments are done to confirm the UA property.Comment: 22 pages, 6 figures. To appear on Communications in Mathematical
Science
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