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This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.

These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum $N$-particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul., Comment: 46 pages. Lecture notes of a course given at the CIRM (Centre International de Recherche Math\'ematique), Luminy (France), during the Research School "Scaling Limits from Microscopic to Macroscopic Physics", January 18th-22nd 2021, organized by the Jean-Morlet Chair, with Shi Jin (chair) and Mihai Bostan (local project leader)

In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48--55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit., Comment: 18 pages

Using theoretical and numerical arguments we discuss some of the commonly accepted approximations for the radiative transfer equations in climatology., 22 pages, 5 figures, one computer listing

New mathematical results are given for the Radiative Transfer equations alone and coupled with the temperature equation of a fluid: existence, uniqueness, a maximum principle and a convergent monotone iterative scheme. Numerical tests for Earth's atmosphere and the heating of a pool by the Sun are included.

New mathematical and numerical results are given for the coupling of the temperature equation of a fluid with Radiative Transfer: existence and uniqueness and a convergent monotone numerical scheme. The technique is shown to be feasible for studying the temperature of lake Leman heated by the sun and for the earth atmosphere to study the effects of greenhouse gases.; Un nouveau résultat d'existence et d'unicité est donné pour le système formé par les équations du transfert radiatif couplées à l'équation de la température d'un fluide. Une méthode numérique convergente et monotone en découle. La technique est appliquée au calcul de la température du lac Leman ainsi qu'à la température de l'atmosphère terrestre pour étudier l'influence des gaz à effet de serre.

We establish an observation inequality for the Schr\"odinger equation on $\bR^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94].This inequality involves only effective constants which are computed explicitly in their dependence in $\hbar$ and all parameters involved.

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003])., Comment: 48 pages, no figure. This new version includes a more detailed discussion of the case of finite rank densities

International audience; By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent 2 between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant $\hbar$. We obtain explicit uniform in $\hbar$ error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.

The present paper establishes the mean-field limit of the Abraham system for a system of identical charged particles subject to the electromagnetic interaction. This mean-field limit leads to a regularized variant of the relativistic Vlasov-Maxwell system. The result obtained here is similar to the one obtained in [F. Golse, Commun. Math. Phys. 310 (2012), 789-816], but is obtained by a purely Eulerian description of the particle dynamics.

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.

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., Comment: 55 pages, 4 multiple figures

30 pages; We study the dynamics defined by the Boltzmann equation set in the Euclidean space RD 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.

17 pages, 3 figures; proceedings of a a conference on "Hyperbolic Conservation Laws and Fluid Dynamics" held in Parma in February 2010; International audience; This paper reviews recent results obtained in collaboration with E. Bernard and E. Caglioti on the homogenization problem for the linear Boltzmann equation for a monokinetic population of particles set in a periodically perforated domain, assuming that particles are absorbed by the holes. We distinguish a critical scale for the hole radius in terms of the distance between neighboring holes, derive the homogenized equation under this scaling assumption, and study the asymptotic mass loss rate in the long time limit. The homogenized equation so obtained is set on an extended phase space as it involves an extra time variable, which is the time since the last jump in the stochastic process driving the linear Boltzmann equation. The present paper proposes a new proof of exponential decay for the mass which is based on a priori estimates on the homogenized equation instead of the renewal theorem used in [Bernard-Caglioti-Golse, SIAM J. Math. Anal. 42 (2010), 2082--2113].

23 pages; International audience; The present paper discusses an asymptotic theory for the Boltzmann equation leading to either the Prandtl incompressible boundary layer equations, or the incompressible hydrostatic equations. These results are formal, and based on the same moment method used in [C. Bardos, F. Golse, D. Levermore, J. Stat. Phys 63 (1991), pp. 323--344] to derive the incompressible Euler and Navier-Stokes equations from the Boltzmann equation.

International audience; The Boltzmann equation of the kinetic theory of gases involves two competing processes. Dissipation(or entropy production) due to the collisions between gas molecules drives the gas towards local thermodynamic (Maxwellian) equilibrium. If the spatial domain is the Euclidean space $\mathbf{R}^D$, the ballistic transport of gas molecules between collisions results in a dispersion effect which enhances the rarefaction of the gas, and offsets the effect of dissipation. The competition between these two effects leads to a scattering regime for the Boltzmann equation over $\mathbf{R}^D$ with molecular interaction satisfying Grad's angular cutoff assumption. The present paper reports on results in this direction obtained in collaboration with Bardos, Gamba and Levermore [Comm. Math. Phys. 346 (2016), 435–467] and discusses a few open questions related to this work.

29 pages; volume dedicated to Prof. Yoshio Sone; International audience; The present paper completes an earlier result by S. Ukai, T. Yang, and S.-H. Yu [Commun. Math. Phys. 236 (2003), 373--393] on weakly nonlinear half-space problems for the steady Boltzmann equation with hard-sphere potential.

National audience

This paper establishes the Rosseland approximation of the radiative transfer equations in a gray atmosphere, i.e. assuming that the opacity is independent of the radiation frequency.

International audience

This book is focused on the recent developments on the mathematical theory of partial differential equations of kinetic type. During the last few years, this domain has received a lot of attention and has given rise to several developments. The use of general advanced mathematical tools (regularity theory, compactness averaging lemmas, dispersion lemmas) is described together with more introductory topics. They are used to analyze the Boltzmann equation and its various hydrodynamical limits: convergence towards the Euler equations of incompressible fluids, models or scallings which allow to recover parabolic or hyperbolic limits. The last part in this book concerns the derivation of kinetic equations in the limit of large systems of interacting particles. Here, the purpose is to justify rigorously the so-called Boltzmann-Grad limit which allows to recover kinetic equations from the BBGKY hierarchy.

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 pages

7 pages, invited conference at the International Congress of Mathematical Physics, Prague 2009; International audience; The Tartar-DiPerna compensated compactness method, used initially to construct global weak solutions of hyperbolic systems of conservation laws for large data, can be adapted in order to provide some regularity estimates on these solutions. This note treats two examples: (a) the case of scalar conservation laws with convex flux, and (b) the Euler system for a polytropic, compressible fluid, in space dimension one.

20 pages; International audience; Compactness of families of solutions --- or of approximate solutions --- is a feature that distinguishes certain classes of nonlinear hyperbolic equations from the case of linear hyperbolic equations, in space dimension one. This paper shows that some classical compactness results in the context of hyperbolic conservation laws, such as the Lax compactness theorem for the entropy solution semigroup associated with a nonlinear scalar conservation laws with convex flux, or the Tartar-DiPerna compensated compactness method, can be turned into quantitative compactness estimates --- in terms of epsilon-entropy, for instance --- or even nonlinear regularization estimates. This regularizing effect caused by the nonlinearity is discussed in detail on two examples: a) the case of a scalar conservation law with convex flux, and b) the case of isentropic gas dynamics, in space dimension one.