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François Golse

Bio: François Golse is an academic researcher from École Polytechnique. The author has contributed to research in topics: Boltzmann equation & Boltzmann constant. The author has an hindex of 30, co-authored 112 publications receiving 3285 citations. Previous affiliations of François Golse include École Normale Supérieure & Ben-Gurion University of the Negev.


Papers
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Journal ArticleDOI
TL;DR: In this article, a Navier-Stokes limit for the Boltzmann equation considered over the infinite spatial domain R 3 is established, whose limit points (in the w-L 1 topology) are governed by Leray solutions of the limiting Navier Stokes equations.
Abstract: The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R 3. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L 1 topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.

310 citations

Book ChapterDOI
TL;DR: In this paper, the authors explain how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics can be rigorously derived from the fundamental microscopic equations that govern the evolution of large, interacting particle systems.
Abstract: 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 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.

200 citations

Journal ArticleDOI
TL;DR: In this paper, the one-dimensional cubic nonlinear Schrodinger equation was derived from a many-body quantum dynamics and the interaction potential was rescaled through a weak coupling limit together with a short-range one.
Abstract: We derive rigorously the one-dimensional cubic nonlinear Schrodinger equation from a many-body quantum dynamics. The interaction potential is rescaled through a weak-coupling limit together with a short-range one. We start from a factorized initial state, and prove propagation of chaos with the usual two-step procedure: in the former step, convergence of the solution of the BBGKY hierarchy associated to the many-body quantum system to a solution of the BBGKY hierarchy obtained from the cubic NLS by factorization is proven; in the latter, we show the uniqueness for the solution of the infinite BBGKY hierarchy.

194 citations

Journal ArticleDOI
TL;DR: In this paper, a nonlinear 1-particle equation from a linear Schrodinger equation in the time dependent case is derived, and convergence of solutions of the first to the second is established by using physically relevant estimates (L and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential.
Abstract: This work is devoted to the derivation of a nonlinear 1-particle equation from a linear AT-particle Schrodinger equation in the time dependent case. It emphazises the role of a so-called \"finite Schrodinger hierarchy\" and of a limiting (infinite) \"Schrodinger hierarchy\". Convergence of solutions of the first to solutions of the second is established by using \"physically relevant\" estimates (L and energy conservation) under very general assumptions on the interaction potential, including in particular the Coulomb potential. In the case of bounded potentials, a stability theorem for the infinite Schrodinger hierarchy is proved, based on Spohn's idea of using the trace norm and elementary techniques pertaining to the abstract Cauchy-Kowalewskaya theorem. The core of this program is to prove that if the limiting AT-particle distribution function is factorized at time t = 0, it remains factorized for all later times. We offer this contribution to Cathleen Morawetz as an expression of our admiration and friendship and in recognition of the influence that her work on the interaction of mathematics and physics has exerted on us.

173 citations

Journal ArticleDOI
TL;DR: In this article, the authors constructed a trace on the algebra of classical elements in Boutet de Monvel's calculus on a compact manifold with boundary of dimensionn>2, which coincides with Wodzicki's noncommutative residue if the boundary is reduced to the empty set.

160 citations


Cited by
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Book ChapterDOI
01 Jan 2011
TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

3,554 citations

Journal ArticleDOI
TL;DR: In this paper, the authors describe the rules of the ring, the ring population, and the need to get off the ring in order to measure the movement of a cyclic clock.
Abstract: 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

3,424 citations

Book ChapterDOI
01 Jan 1997
TL;DR: The boundary layer equations for plane, incompressible, and steady flow are described in this paper, where the boundary layer equation for plane incompressibility is defined in terms of boundary layers.
Abstract: The boundary layer equations for plane, incompressible, and steady flow are $$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

2,598 citations

Journal ArticleDOI
TL;DR: In this article, a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory is derived, and the first member of the hierarchy is the Euler system based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions.
Abstract: This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a “14-moment” closure, which also recovers the behavior of the Grad “13-moment” system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.

903 citations