45 research outputs found
Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models
In this lectures given at the Morning side center of Mathematics in October
2016, we present in a very simple framework Hilbertian hypocoercive methods in
the case of 1d kinetic inhomogeneous equations, and some illustrations
concerning short time or long time behavior in a linear or non-linear
perturbative settin
On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential
We prove a global existence result with initial data of low regularity, and
prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system
with small non linear term but with a possibly large exterior confining
potential in dimension and . The proof relies on a fixed point
argument using sharp estimates (at short and long time scales) of the
semi-group associated to the Fokker-Planck operator, which were obtained by the
first author.Comment: 29 pages. To appear in Journal of Functional Analysi
Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off
In this paper, we investigate the problems of Cauchy theory and exponential
stability for the inhomogeneous Boltzmann equation without angular cut-off. We
only deal with the physical case of hard potentials type interactions (with a
moderate angular singularity). We prove a result of existence and uniqueness of
solutions in a close-to-equilibrium regime for this equation in weighted
Sobolev spaces with a polynomial weight, contrary to previous works on the
subject, all developed with a weight prescribed by the equilibrium. It is the
first result in this more physically relevant frameworkfor this equation.
Moreover, we prove an exponential stability for such a solution, with a rate as
close as we want to the optimal rate given by the semigroup decay of the
linearized equation. Let us highlight the fact that a key point of the
development of our Cauchy theory is the proof of new regularization estimates
in short time for the linearized operator thanks to pseudo-differential tools.Comment: arXiv admin note: text overlap with arXiv:1709.0994
Global hypoelliptic and symbolic estimates for the linearized Boltzmann operator without angular cutoff
In this article we provide global subelliptic estimates for the linearized
inhomogeneous Boltzmann equation without angular cutoff, and show that some
global gain in the spatial direction is available although the corresponding
operator is not elliptic in this direction. The proof is based on a multiplier
method and the so-called Wick quantization, together with a careful analysis of
the symbolic properties of the Weyl symbol of the Boltzmann collision operator
Magnetic WKB Constructions
This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB
expansions for the eigenfunctions were only established in presence of a
non-zero electric potential. Here we tackle the pure magnetic case. Thanks to
Feynman-Hellmann type formulas and coherent states decomposition, we develop
here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the
problem, we are led to solve an effective eikonal equation in pure magnetic
cases and to obtain WKB expansions. We also investigate explicit examples for
which we can improve our general theorem: global WKB expansions, quasi-optimal
estimates of Agmon and upper bound of the tunelling effect (in symmetric
cases). We also apply our strategy to get more accurate descriptions of the
eigenvalues and eigenfunctions in a wide range of situations analyzed in the
last two decades
Holomorphic extension of the de Gennes function
This note is devoted to prove that the de Gennes function has a holomorphic
extension on a strip containing the real axis
Global hypoelliptic estimates for Landau-type operators with external potential
In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential
Semiclassical tunneling and magnetic flux effects on the circle
International audienceThis paper is devoted to semiclassical tunneling estimates induced on the circle by a double well electric potential in the case when a magnetic field is added. When the two electric wells are connected by two geodesics for the Agmon distance, we highlight an oscillating factor (related to the circulation of the magnetic field) in the splitting estimate of the first two eigenvalues