arXiv subjects

François Golse

Publications and source records attributed to François Golse.

At least 55 records · Page 3Linked to original sources

arXiv2016

A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude.

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arXiv2016

The Schrödinger 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 $N$-body linear Schrödinger 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 Schrödinger 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.

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arXiv2016

Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case

In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.

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arXiv2016

The Steady Boltzmann and Navier-Stokes Equations

The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of the gas flow. While the steady Navier-Stokes equations with smooth divergence-free external force always have at least one smooth solutions, the Boltzmann equation with the same external force set in the torus, or in a bounded domain with specular reflection of gas molecules at the boundary may fail to have any solution, unless the force field is identically zero. Viscous heating seems to be of key importance in this situation. The nonexistence of any steady solution of the Boltzmann equation in this context seems related to the increase of temperature for the evolution problem, a phenomenon that we have established with the help of numerical simulations on the Boltzmann equation and the BGK model.

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arXiv2015

On the Mean Field and Classical Limits of Quantum Mechanics

The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.

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arXiv2015

Hölder regularity for hypoelliptic kinetic equations with rough diffusion coefficients

This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a Hölder regularity version of the classical averaging lemmas.

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arXiv2015

A gradient flow approach to quantization of measures

In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.

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arXiv2014

De Newton à Boltzmann et Einstein: validation des modèles cinétiques et de diffusion

The kinetic theory of Maxwell and Boltzmann has been the subject of major scientific controversies. The alleged incompatibility between the reversible nature of the equations of classical mechanics and the increase of entropy, which, in the kinetic theory of gases, is a mathematical property of the Boltzmann equation known as the H Theorem, was one of the most basic arguments against the validity of kinetic theory. About a century later, in 1974, O. Lanford proposed a strategy to prove that the Boltzmann equation describes a particular asymptotic limit of Newton's equations of classical mechanics for a system made of a large number N of spherical particles interacting by elastic collisions. A recent work by I. Gallagher, L. Saint-Raymond and B. Texier completes Lanford's proof and extends it to the case where the interaction between particles is given by a short range potential. In a later article, T. Bodineau, I. Gallagher and L. Saint-Raymond study the dynamics of a tagged particle among N other identical particles in the same asymptotic regime, and prove the validity of the linear Boltzmann equation on arbitrary large periods of time as N tends to infinity. Using classical results on the asymptotic theory of the linear Boltzmann equation, the same authors prove that the stochastic process known as the "Brownian motion" describes a particular limit of the deterministic dynamics of interacting particles.

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arXiv2014

On the Classical Limit of the Schrödinger Equation

This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index.

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arXiv2014

Global Solutions of the Boltzmann Equation over $\mathbb{R}^D$ near Global Maxwellians with Small Mass

We study the dynamics defined by the Boltzmann equation set in the Euclidean space $\mathbb{R}^D$ in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.

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arXiv2013

Empirical Measures and Vlasov Hierarchies

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, Math. Meth. Appl. Sci. 3 (1981), 445-455].

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arXiv2013

The Diffusion Approximation for the Linear Boltzmann Equation with Vanishing Scattering Coefficient

The present paper discusses the diffusion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our results apply for instance to the case of radiative transfer in a composite medium with optically thin inclusions in an optically thick background medium. The equation governing the evolution of the approximate particle density coincides with the limit of the diffusion equation with infinite diffusion coefficient in the optically thin inclusions.

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arXiv2013

Derivation of a homogenized two-temperature model from the heat equation

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat: Collège de France Seminar vol. 2. (Paris 1979-1980) Res. Notes in Math. vol. 60, pp. 98-138. Pitman, Boston, London, 1982.]

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arXiv2013

Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.

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arXiv2013

On the Dynamics of Large Particle Systems in the Mean Field Limit

This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics - can be rigorously derived from first principles, i.e. from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin's stability estimate in the Monge-Kantorovich distance for the empirical measures built on the solution of the N-particle motion equations in classical mechanics, or the theory of BBGKY hierarchies in the case of classical as well as quantum problems. We explain in detail how these different approaches are related; in particular we insist on the notion of chaotic sequences and on the propagation of chaos in the BBGKY hierarchy as the number of particles tends to infinity.

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arXiv2012

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.

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arXiv2011

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.

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arXiv2011

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.

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