980 research outputs found
The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials
The present paper proves that all limit points of sequences of renormalized
solutions of the Boltzmann equation in the limit of small, asymptotically
equivalent Mach and Knudsen numbers are governed by Leray solutions of the
Navier-Stokes equations. This convergence result holds for hard cutoff
potentials in the sense of H. Grad, and therefore completes earlier results by
the same authors [Invent. Math. 155, 81-161(2004)] for Maxwell molecules.Comment: 56 pages, LaTeX, a few typos have been corrected, a few remarks
added, one uncited reference remove
On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic
distribution of scatterers cannot be described with a linear Boltzmann
equation. This is at variance with the case of a Poisson distribution of
scatterers, for which the convergence to the linear Boltzmann equation has been
proved by Gallavotti [Phys. Rev. (2) 185 (1969), p. 308]
The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions
The periodic Lorentz gas is the dynamical system corresponding to the free
motion of a point particle in a periodic system of fixed spherical obstacles of
radius centered at the integer points, assuming all collisions of the
particle with the obstacles to be elastic. In this Note, we study this motion
on time intervals of order and in the limit as , in the case of
two space dimensions
Optimal Regularizing Effect for Scalar Conservation Laws
We investigate the regularity of bounded weak solutions of scalar
conservation laws with uniformly convex flux in space dimension one, satisfying
an entropy condition with entropy production term that is a signed Radon
measure. The proof is based on the kinetic formulation of scalar conservation
laws and on an interaction estimate in physical space.Comment: 24 pages, assumption (11) in Theorem 3.1 modified together with the
example on p. 7, one remark added after the proof of Lemma 4.3, some typos
correcte
The Schr\"odinger Equation in the Mean-Field and Semiclassical Regime
In this paper, we establish (1) the classical limit of the Hartree equation
leading to the Vlasov equation, (2) the classical limit of the -body linear
Schr\"{o}dinger equation uniformly in N leading to the N-body Liouville
equation of classical mechanics and (3) the simultaneous mean-field and
classical limit of the N-body linear Schr\"{o}dinger equation leading to the
Vlasov equation. In all these limits, we assume that the gradient of the
interaction potential is Lipschitz continuous. All our results are formulated
as estimates involving a quantum analogue of the Monge-Kantorovich distance of
exponent 2 adapted to the classical limit, reminiscent of, but different from
the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343
(2016), 165-205]. As a by-product, we also provide bounds on the quadratic
Monge-Kantorovich distances between the classical densities and the Husimi
functions of the quantum density matrices.Comment: 33 page
On the distribution of free-path lengths for the periodic Lorentz gas III
In a flat 2-torus with a disk of diameter removed, let be the
distribution of free-path lengths (the probability that a segment of length
larger than with uniformly distributed origin and direction does not meet
the disk).
We prove that behaves like for each
and in the limit as , in some appropriate sense.
We then discuss the implications of this result in the context of kinetic
theory.Comment: 26 pages, 5 figures, to be published in Commun. Math. Phy
The Incompressible Euler Limit of the Boltzmann Equation with Accommodation Boundary Condition
The convergence of solutions of the incompressible Navier-Stokes equations
set in a domain with boundary to solutions of the Euler equations in the large
Reynolds number limit is a challenging open problem both in 2 and 3 space
dimensions. In particular it is distinct from the question of existence in the
large of a smooth solution of the initial-boundary value problem for the Euler
equations. The present paper proposes three results in that direction. First,
if the solutions of the Navier-Stokes equations satisfy a slip boundary
condition with vanishing slip coefficient in the large Reynolds number limit,
we show by an energy method that they converge to the classical solution of the
Euler equations on its time interval of existence. Next we show that the
incompressible Navier-Stokes limit of the Boltzmann equation with Maxwell's
accommodation condition at the boundary is governed by the Navier-Stokes
equations with slip boundary condition, and we express the slip coefficient at
the fluid level in terms of the accommodation parameter at the kinetic level.
This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat.
Phys. 63 (1991), 323-344]. Finally, we establish the incompressible Euler limit
of the Boltzmann equation set in a domain with boundary with Maxwell's
accommodation condition assuming that the accommodation parameter is small
enough in terms of the Knudsen number. Our proof uses the relative entropy
method following closely the analysis in [L. Saint-Raymond, Arch. Ration. Mech.
Anal. 166 (2003), 47-80] in the case of the 3-torus, except for the boundary
terms, which require special treatment.Comment: 40 page
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