126 research outputs found
The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics
The present work establishes the mean-field limit of a N-particle system
towards a regularized variant of the relativistic Vlasov-Maxwell system,
following the work of Braun-Hepp [Comm. in Math. Phys. 56 (1977), 101-113] and
Dobrushin [Func. Anal. Appl. 13 (1979), 115-123] for the Vlasov-Poisson system.
The main ingredients in the analysis of this system are (a) a kinetic
formulation of the Maxwell equations in terms of a distribution of
electromagnetic potential in the momentum variable, (b) a regularization
procedure for which an analogue of the total energy - i.e. the kinetic energy
of the particles plus the energy of the electromagnetic field - is conserved
and (c) an analogue of Dobrushin's stability estimate for the
Monge-Kantorovich-Rubinstein distance between two solutions of the regularized
Vlasov-Poisson dynamics adapted to retarded potentials.Comment: 34 page
Regularizing effect and local existence for non-cutoff Boltzmann equation
The Boltzmann equation without Grad's angular cutoff assumption is believed
to have regularizing effect on the solution because of the non-integrable
angular singularity of the cross-section. However, even though so far this has
been justified satisfactorily for the spatially homogeneous Boltzmann equation,
it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates
for the collision operator, establishing the hypo-ellipticity of the Boltzmann
operator based on a generalized version of the uncertainty principle, and
analyzing the commutators between the collision operator and some weighted
pseudo differential operators, we prove the regularizing effect in all (time,
space and velocity) variables on solutions when some mild regularity is imposed
on these solutions. For completeness, we also show that when the initial data
has this mild regularity and Maxwellian type decay in velocity variable, there
exists a unique local solution with the same regularity, so that this solution
enjoys the regularity for positive time
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB)
model of plasma physics. This model consists of the pressureless gas dynamics
equations coupled with the Poisson equation and where the Boltzmann relation
relates the potential to the electron density. If the quasi-neutral assumption
is made, the Poisson equation is replaced by the constraint of zero local
charge and the model reduces to the Isothermal Compressible Euler (ICE) model.
We compare a numerical strategy based on the EPB model to a strategy using a
reformulation (called REPB formulation). The REPB scheme captures the
quasi-neutral limit more accurately
Global existence and full regularity of the Boltzmann equation without angular cutoff
We prove the global existence and uniqueness of classical solutions around an
equilibrium to the Boltzmann equation without angular cutoff in some Sobolev
spaces. In addition, the solutions thus obtained are shown to be non-negative
and in all variables for any positive time. In this paper, we study
the Maxwellian molecule type collision operator with mild singularity. One of
the key observations is the introduction of a new important norm related to the
singular behavior of the cross section in the collision operator. This norm
captures the essential properties of the singularity and yields precisely the
dissipation of the linearized collision operator through the celebrated
H-theorem
A Wasserstein approach to the one-dimensional sticky particle system
We present a simple approach to study the one-dimensional pressureless Euler
system via adhesion dynamics in the Wasserstein space of probability measures
with finite quadratic moments.
Starting from a discrete system of a finite number of "sticky" particles, we
obtain new explicit estimates of the solution in terms of the initial mass and
momentum and we are able to construct an evolution semigroup in a
measure-theoretic phase space, allowing mass distributions with finite
quadratic moment and corresponding L^2-velocity fields. We investigate various
interesting properties of this semigroup, in particular its link with the
gradient flow of the (opposite) squared Wasserstein distance.
Our arguments rely on an equivalent formulation of the evolution as a
gradient flow in the convex cone of nondecreasing functions in the Hilbert
space L^2(0,1), which corresponds to the Lagrangian system of coordinates given
by the canonical monotone rearrangement of the measures.Comment: Added reference
Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain
We study the emergent dynamics for the hydrodynamic Cucker--Smale system arising in the modeling of flocking dynamics in interacting many-body systems. Specifically, the initial value problem with a moving domain is considered to investigate the global existence and time-asymptotic behavior of classical solutions, provided that the initial mass density has bounded support and the initial data are in an appropriate Sobolev space. In order to show the emergent behavior of flocking, we make use of an appropriate Lyapunov functional that measures the total fluctuation in the velocity relative to the mean velocity. In our analysis, we present the local well-posedness of the smooth solutions via Lagrangian coordinates, and we extend to the global-in-time solutions by establishing the uniform flocking estimates.close
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
Kinetic Theory of a Dilute Gas System under Steady Heat Conduction
The velocity distribution function of the steady-state Boltzmann equation for
hard-core molecules in the presence of a temperature gradient has been obtained
explicitly to second order in density and the temperature gradient. Some
thermodynamical quantities are calculated from the velocity distribution
function for hard-core molecules and compared with those for Maxwell molecules
and the steady-state Bhatnagar-Gross-Krook(BGK) equation. We have found
qualitative differences between hard-core molecules and Maxwell molecules in
the thermodynamical quantities, and also confirmed that the steady-state BGK
equation belongs to the same universality class as Maxwell molecules.Comment: 36 pages, 4 figures, 5 table
Two-way multi-lane traffic model for pedestrians in corridors
We extend the Aw-Rascle macroscopic model of car traffic into a two-way
multi-lane model of pedestrian traffic. Within this model, we propose a
technique for the handling of the congestion constraint, i.e. the fact that the
pedestrian density cannot exceed a maximal density corresponding to contact
between pedestrians. In a first step, we propose a singularly perturbed
pressure relation which models the fact that the pedestrian velocity is
considerably reduced, if not blocked, at congestion. In a second step, we carry
over the singular limit into the model and show that abrupt transitions between
compressible flow (in the uncongested regions) to incompressible flow (in
congested regions) occur. We also investigate the hyperbolicity of the two-way
models and show that they can lose their hyperbolicity in some cases. We study
a diffusive correction of these models and discuss the characteristic time and
length scales of the instability
Stability of flows associated to gradient vector fields and convergence of iterated transport maps
In this paper we address the problem of stability of flows
associated to a sequence of vector fields under minimal regularity requirements
on the limit vector field, that is supposed to be a gradient. We apply this
stability result to show the convergence of iterated compositions of optimal
transport maps arising in the implicit time discretization (with respect to the
Wasserstein distance) of nonlinear evolution equations of a diffusion type.
Finally, we use these convergence results to study the gradient flow of a
particular class of polyconvex functionals recently considered by Gangbo, Evans
ans Savin. We solve some open problems raised in their paper and obtain
existence and uniqueness of solutions under weaker regularity requirements and
with no upper bound on the jacobian determinant of the initial datum
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