Claude Bardos

Bio: Claude Bardos is an academic researcher from University of Paris. The author has contributed to research in topics: Euler equations & Boltzmann equation. The author has an hindex of 43, co-authored 204 publications receiving 8572 citations. Previous affiliations of Claude Bardos include École Normale Supérieure & Département de Mathématiques.

Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary

Abstract: For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. ...

First order quasilinear equations with boundary conditions

Abstract: We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which chara...

Fluid dynamic limits of kinetic equations. I. Formal derivations

Abstract: The connection between kinetic theory and the macroscopic equations of fluid dynamics is described. In particular, our results concerning the incompressible Navier-Stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog. The moment formalism shows that the limit leading to the incompressible Navier-Stokes equations, like that leading to the compressible Euler equations, is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible Navier-Stokes equations. Some indications of the validity of these limits are given. More specifically, the connection between the DiPerna-Lions renormalized solution of the classical Boltzmann equation and the Leray solution of the Navier-Stokes equations is discussed.

Fluid dynamic limits of kinetic equations II convergence proofs for the boltzmann equation

Abstract: Using relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna-Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier-Stokes limit with an additional mild weak compactness assumption. The continuous time Navier-Stokes limit is also discussed. 01993 John Wiley & Sons., Inc.

Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data

Abstract: In this paper we consider the Vlasov Poisson equation in three space variables in the whole space. We show the existence of dispersion property. With this dispersion property we are able to prove the existence of a smooth solution for all times under the following assumption: the initial data are localised and small enough.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Boundary Layer Theory

Abstract: The boundary layer equations for plane, incompressible, and steady flow are $$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Finite Volume Methods

Abstract: Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

Monte Carlo and quasi-Monte Carlo methods

Abstract: Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.