610 research outputs found
From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge
We study the asymptotic regime of strong electric fields that leads from the
Vlasov-Poisson system to the Incompressible Euler equations. We also deal with
the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The
originality consists in considering a situation with a finite total charge
confined by a strong external field. In turn, the limiting equation is set in a
bounded domain, the shape of which is determined by the external confining
potential. The analysis extends to the situation where the limiting density is
non-homogeneous and where the Euler equation is replaced by the Lake Equation,
also called Anelastic Equation.Comment: 39 pages, 3 figure
Kinetic equations with Maxwell boundary conditions
We prove global stability results of {\sl DiPerna-Lions} renormalized
solutions for the initial boundary value problem associated to some kinetic
equations, from which existence results classically follow. The (possibly
nonlinear) boundary conditions are completely or partially diffuse, which
includes the so-called Maxwell boundary conditions, and we prove that it is
realized (it is not only a boundary inequality condition as it has been
established in previous works). We are able to deal with Boltzmann,
Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace
theorems of the kind previously introduced by the author for the Vlasov
equations, new results concerning weak-weak convergence (the renormalized
convergence and the biting -weak convergence), as well as the
Darroz\`es-Guiraud information in a crucial way
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
Ill-posedness of the hydrostatic Euler and singular Vlasov equations
In this paper, we develop an abstract framework to establish ill-posedness in
the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable
spectra. We apply it to prove the ill-posedness for the hydrostatic Euler
equations as well as for the kinetic incompressible Euler equations and the
Vlasov-Dirac-Benney system
Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation
This work is concerned with the quasineutral limit of the one-dimensional
Vlasov-Poisson equation, for initial data close to stationary homogeneous
profiles. Our objective is threefold: first, we provide a proof of the fact
that the formal limit does not hold for homogeneous profiles that satisfy the
Penrose instability criterion. Second, we prove on the other hand that the
limit is true for homogeneous profiles that satisfy some monotonicity
condition, together with a symmetry condition. We handle the case of
well-prepared as well as ill- prepared data. Last, we study a stationary
boundary-value problem for the formal limit, the so-called quasineutral Vlasov
equation. We show the existence of numerous stationary states, with a lot of
freedom in the construction (compared to that of BGK waves for Vlasov-Poisson):
this illustrates the degeneracy of the limit equation.Comment: 50 page
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