174 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
Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak
The penalization method is used to take account of obstacles in a tokamak,
such as the limiter. We study a non linear hyperbolic system modelling the
plasma transport in the area close to the wall. A penalization which cuts the
transport term of the momentum is studied. We show numerically that this
penalization creates a Dirac measure at the plasma-limiter interface which
prevents us from defining the transport term in the usual sense. Hence, a new
penalty method is proposed for this hyperbolic system and numerical tests
reveal an optimal convergence rate without any spurious boundary layer.Comment: 8 pages; International Symposium FVCA6, Prague : Czech Republic
(2011
On Nonlinear Stochastic Balance Laws
We are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in of the approximations, uniformly in the
viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish an
error estimate for the stochastic viscosity method, as well as an explicit
estimate for the continuous dependence of stochastic entropy solutions on the
flux and random source functions. Various further generalizations of the
results are discussed
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
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
Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior
We consider the spatially homogeneous Boltzmann equation for inelastic hard
spheres, in the framework of so-called constant normal restitution
coefficients. We prove the existence of self-similar solutions, and we give
pointwise estimates on their tail. We also give general estimates on the tail
and the regularity of generic solutions. In particular we prove Haff 's law on
the rate of decay of temperature, as well as the algebraic decay of
singularities. The proofs are based on the regularity study of a rescaled
problem, with the help of the regularity properties of the gain part of the
Boltzmann collision integral, well-known in the elastic case, and which are
extended here in the context of granular gases.Comment: 41 page
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
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
A mathematical model for unsteady mixed flows in closed water pipes
We present the formal derivation of a new unidirectional model for unsteady
mixed flows in non uniform closed water pipe. In the case of free surface
incompressible flows, the \FS-model is formally obtained, using formal
asymptotic analysis, which is an extension to more classical shallow water
models. In the same way, when the pipe is full, we propose the \Pres-model,
which describes the evolution of a compressible inviscid flow, close to gas
dynamics equations in a nozzle. In order to cope the transition between a free
surface state and a pressured (i.e. compressible) state, we propose a mixed
model, the \PFS-model, taking into account changes of section and slope
variation
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
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