Claude Bardos

Sorbonne Université | UPMC · Laboratoire Jacques-Louis Lions (LJLL)

We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the stan...

Using theoretical and numerical arguments we discuss some of the commonly accepted approximations for the radiative transfer equations in climatology.

In this paper we study the Hamiltonian dynamics of charged particles subject to a non-self-consistent stochastic electric field, when the plasma is in the so-called weak turbulent regime. We show that the asymptotic limit of the Vlasov equation is a diffusion equation in the velocity space, but homogeneous in the physical space. We obtain a diffusi...

In this paper, we study the Hamiltonian dynamics of charged particles subject to a non-self-consistent stochastic electric field when the plasma is in the so-called weak turbulent regime. We show that the asymptotic limit of the Vlasov equation is a diffusion equation in the velocity space but homogeneous in the physical space. We obtain a diffusio...

Radiative transfer is at the heart of the mechanism to explain the greenhouse effect based on the partial infrared opacity of carbon dioxide, methane and other greenhouse gases in the atmosphere. In absence of thermal diffusion, the mathematical model consists of a first order integro-differential equation coupled with an integral equation for the...

This contribution, built on the companion paper [1], is focused on the different mathematical approaches available for the analysis of the quasilinear approximation in plasma physics.

We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order one-third in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the bound...

We consider the incompressible Euler equations in a bounded domain in three space dimensions. Recently, the first two authors proved Onsager's conjecture for bounded domains, i.e., that the energy of a solution to these equations is conserved provided the solution is H\"older continuous with exponent greater than 1/3, uniformly up to the boundary....

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the “universality” of the Onsager exponent, α> 1 / 3 , concerning the regularity of the solutions, say in C⁰,α, that guarantees the conservation of the generalized entropy, re...

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In par...

We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order 1/3 in the interior of the domain, and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary. T...

This contribution is based on a theorem of Kato which relates for time dependent problems the appearance of turbulence with the anomalous energy dissipation, giving for the Cauchy problem an avatar of a basic idea of the statistical theory of turbulence. Some variant of this theorem are given and then it is shown how this point of view is in full a...

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the "universality" of the Onsager exponent, $\alpha > 1/3$, concerning the regularity of the solutions, say in $C^{0,\alpha}$, that guarantees the conservation of the generali...

The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t$, with the impermeability boundary condition: $u\cdot \vec n =0$ on $\p...

Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the bounda...

The aim of this paper is to provide a justification of the Maxwell-Boltzmann approximation of electron density from kinetic models. First, under reasonable regularity assumption, we rigorously derive a reduced kinetic model for the dynamics of ions, while electrons satisfy the Maxwell-Boltzmann relation. Second, we prove that equilibria of the elec...

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker-Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof re...

This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path $\varepsilon$ and the heterogeneity length scale $\eta$ vanish. No periodicity assumption is made on the absorption cross section of the background material. There is an assumption that the heterogeneity lengt...

We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(\epsilon,\delta)$-domains$\Omega\subset\mathbb{R}^n$ with a $d$-set boundary (for instance, aself-similar fractal), we find the first term of the small-timeasymptotic expansion of the...

This contribution is an element of a research program devoted to the analysis of a variant of the Vlasov–Poisson equation that we dubbed the Vlasov–Dirac–Benney equation or in short V–D–B equation. As such it contains both new results and efforts to synthesize previous observations. One of main links between the different issues is the use of the e...

We establish various criteria, which are known in the incompressible case, for the validity of the inviscid limit for the compressible Navier-Stokes flows considered in a general domain $\Omega$ in $\mathbb{R}^n$ with or without a boundary. In the presence of a boundary, a generalized Navier boundary condition for velocity is assumed, which in part...

This paper provides an elementary proof of the classical limit of the Schr\"{o}dinger 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 Schr\"{o}dinger type equations [A. Laptev, I. Sigal, Review of Math....

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 th...

This contribution concerns a one-dimensional version of the Vlasov equation dubbed the Vlasov-Dirac-Benney equation (in short V-D-B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with...

We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In t...

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 gove...

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such prob...

We consider rotational initial data for the two-dimensional incompressible Euler equations on an annulus. Using the convex integration framework, we show that there exist infinitely many admissible weak solutions (i.e. such with non-increasing energy) for such initial data. As a consequence, on bounded domains there exist admissible weak solutions...

This contribution covers the topics presented by the authors at the {\it ``Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961"} meeting that took place in Marseille in 2011. It focuses on some of the mathematical approaches to fluid dynamics and turbulence. This contribution does not pretend to cover or answer, as the...

Well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic differences bet...

We show that for a certain family of initial data, there exist non-unique weak solutions to the 3D incompressible Euler equations satisfying the weak energy inequality, whereas the weak limit of every sequence of Leray-Hopf weak solutions for the Navier-Stokes equations, with the same initial data, and as the viscosity tends to zero, is uniquely de...

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 tr...

The kinetic boundary layer for gas mixtures is described by a half space boundary value problem for the two-species steady Boltzmann equation with an incoming distribution. We consider it around a drifting normalized bi-Maxwellian and prove that the boundary layer problem is well-posed when the drifting velocity u exceeds the sound speed , but one...

The authors investigate the stability of a steady ideal plane flow in an arbitrary domain in terms of the L2 norm of the vorticity. Linear stability implies nonlinear instability provided the growth rate of the linearized system exceeds the Liapunov exponent of the flow. In contrast, a maximizer of the entropy subject to constant energy and mass is...

In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations preserve initial...

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report). It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the...

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-b...

The paper deals with numerical analysis of an inverse boundary transport problem. A class of iterative algorithms for solving this problem is considered, the algorithms convergence conditions are studied, and the convergence rate estimates are derived. Numerical examples are presented.

In this paper, we study, on a very simple kinetic model, the flow structure induced by a discontinuity of the boundary data. The model considered is a stationary one-speed transport equation posed in a half-plane; for simplicity, the boundary data consist of the number density of incoming particles. The propagation of singularities is studied with...

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 characterizes the obtained solution.

In this paper, we formulate and analyze the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations for molecular systems with pairwise interaction. This set of coupled nonlinear PDEs and ODEs is an approximation of the N-particle time-dependent Schrödinger equation based on (time-dependent) linear combinations of (time-dependent) Slater...

We present an α-regularization of the Birkhoff-Rott equation (BR-α equation), induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-α equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vo...

We present here a survey of recent results concerning the mathematical analysis of instabilities of the interface between two incompressible, non viscous, fluids of constant density and vorticity concentrated on the interface. This configuration includes the so-called Kelvin-Helmholtz (the two densities are equal), Rayleigh-Taylor (two different, n...

Over the recent years, mostly under the impetus of the late JL Lions, important progress have been made for the control of distributed systems. This has contributed to the understanding of the duality which exist between the modal analysis and the need of very localized actuators. This duality leads to the phenomena of overspilling (excitation of h...

This contribution is devoted to the mathematical analysis of more or less sophisticated approximations of the time evolution of systems of N quantum particles. New results for the Multiconfiguration Time Dependent Hartree Fock (MCTDHF) method are summarized and compared with the simpler Hartree and Hartree Fock equations.

We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e|k|/kd{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Us...

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear...

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex...

The multiconfiguration time-dependent Hartree–Fock (MCTDHF for short) system is an approximation of the linear many-particle Schrödinger equation with a binary interaction potential by nonlinear “one-particle” equations. MCTDHF methods are widely used for numerical calculations of the dynamics of few-electron systems in quantum physics and quantum...

This paper is devoted to the analysis of solutions of fluid equations with interfaces and the purpose is to show how recent results on analyticity are related to the instabilities of the interface.

In this paper we consider the problem of estimating the singular support of the Green's function of the wave equation in a bounded region by cross correlating noisy signals. A collection of sources with unknown spatial distribution emit stationary random signals into the medium, which are recorded at two observation points. We show that the cross c...

In the long-time scale, we consider the fluid dynamical lim- its for the kinetic equations when the fluctuation is decomposed into even and odd parts with respect to the microscopic veloc- ity with different scalings. It is shown that when the background state is an absolute Maxwellian, the limit fluid dynamical equa- tions are the incompressible N...

Instability in Models Connected with Fluid Flows II presents chapters from world renowned specialists. The stability of mathematical models simulating physical processes is discussed in topics on control theory, first order linear and nonlinear equations, water waves, free boundary problems, large time asymptotics of solutions, stochastic equations...

This contribution describes a part of the programm realised in collaboration with G. Lebeau and J. Rauch (cf. [BLR]) and [Le1, Le2]. It has been observed that in the framework of linear hyperbolic problems, the questions of uniqueness of solutions, “Holmgren theorem”, estimation of the error in the observation of a solution, exact controllability a...

Justifying Asymptotics for 3D Water-Waves, David Lannes.- Generalized Solutions of the Cauchy Problem for a Transport Equation with Discontinuous Coefficients, Evgenii Panov.- Irreducible Chapman-Enskog Projections and Navier-Stokes Approximations, Evgenii Radkevich.- Exponential Mixing for Randomly Forced Partial Differential Equations: Method of...

According to a theory of H. Spohn, the time-dependent Hartree (TDH) equation governs the 1-particle state in $N$-particle systems whose dynamics are prescribed by a non-relativistic Schrödinger equation with 2-particle interactions, in the limit $N$ tends to infinity while the strength of the 2-particle interaction potential is scaled by $1=N$. In...

We present an α-regularization of the Birkhoff–Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show that an initially smooth self-avoiding vortex sheet remains smooth for all times under the α-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve w...

This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.

Without Abstract

We are concerned with the global (in time) regularity properties of the Burgers MRCM equation, which arises in the theory of turbulence (with α = 1) \frac¶U¶t(t,x) = - \frac¶2 ¶x2 [U(t,0) - U(t,x)]2 - n( - \frac¶2 ¶x2 )a U(t,x)\frac{{\partial U}}{{\partial t}}(t,x) = - \frac{{\partial ^2 }}{{\partial x^2 }}[U(t,0) - U(t,x)]^2 - \nu ( - \frac{{\pa...

Without Abstract

This paper reviews recent mathematical results on the half-space problem for the Boltzmann equation. The case of a phase transition is discussed in detail.

Problems of observability, controlability, and stabilization lead to the search for the same type of inequalities for solutions of Partial Differential Equations. The problems discussed below concern the observation, control or stabilization of waves governed by hyperbolic partial differential equations. A necessary condition for any of the three p...

Shock Control is a type of inverse problem for which the most suited solution method seems to be least square and optimal control algorithms (see [8]). As control theory assumes differentiability, there are mathematical difficulties when the modelling uses a system of conservation laws like the shallow water or Euler equations (see [15] for example...

The transport theory of sound particles is applied to the sound field modeling in architectural acoustics. A theoretical description is proposed for empty enclosures with complex boundary conditions, including both specular and diffuse reflections. As an example, the model is applied to street canyons. Therefore, an asymptotic approach is proposed...

Data Assimilation is important in meteorology and oceanography, because it is a way to improve the models with newly measured data, statically or dynamically. It is a type of inverse problem for which the most popular solution method is least square with regularization and optimal control algorithms. As control theory assumes differentiability, the...

Non blow-up of the 3D incompressible Euler Equations is proven for a class of three- dimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving r...

We present and discuss derivations of nonlinear 1-particle equations from linear N-particle Schrödinger equations with pair interaction in the time dependent case. We regard both the “classical” limit of vanishing Planck constant ħ → 0 which leads to Vlasov type equations and the “weak coupling” limit 1/N → 0 which leads to nonlinear 1 particle equ...

This article concerns the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find that the TDHF approximation is accurate when there are sufficiently many particles and the initial many-particle state is any Gibbs equilibrium state for noninteracting fermions (with Slater determinants...

The Knudsen layer in rarefied gas dynamics is essentially described by a half-space boundary-value problem of the linearized Boltzmann equation, in which the incoming data are specified on the boundary and the solution is assumed to be bounded at infinity (Milne problem). This problem is considered for a binary mixture of hard-sphere gases, and the...

this article we consider the Hamiltonian dynamics of systems of fermions and derive the time-dependent Hartree-Fock equation in the mean eld limit. We follow the approach of Spohn, who derived a mean eld dynamical equation (the time-dependent Hartree equation) for mean eld systems of distinguishable particles, remarking at the time that \the conver...

This article examines the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find the TDHF approximation to be accurate when there are sufficiently many particles and the initial many-particle state is a Slater determinant, or any Gibbs equilibrium state for noninteracting fermions. A...

We give a mathematical analysis of the "time-reversal mirror", in what concerns phenomena described by the genuine acoustic equation with Dirichlet or impedance boundary conditions. An ideal situation is first considered, followed by the boundary-data, impedance and internal time-reversal methods. We explore the relationship between local decay of...

The mathematical analysis of the Navier Stokes equations involved the contributions of such diverse personalities as, Euler, Leray, Kolmogorov, Arnold and others and in view of the needs of practical applications in engineering sciences success have been limited. However the results that have been obtained contribute to our understanding of the phy...

In this paper we present a synthetic method to differentiate with respect to a parameter partial differential equations in divergence form with shocks. We show that the usual derivatives contain the differentiated interface conditions if interpreted by the theory of distributions. We apply the method to three problems: the Burgers equation, the sha...

In this study, we examine one of the approaches to the investigation and solution of the Stokes equations: an original problem is regarded as an inverse problem and then reduced to a problem of optimal control. Iteration algorithms are suggested and results of numerical experiments are presented.

Sensitivity of shocks to data is a key point for fluid-structure and flutter control, and even more for sonic boom reduction. The linearized equations of fluids have Dirac masses and so it is not clear that the standard tools of optimal control apply to these. We show here that indeed great care have to be applied to find the linearized equations b...

this article we consider the Hamiltonian dynamics of systems of fermions and derive the time-dependent Hartree-Fock equation in the mean eld limit. We follow the approach of Spohn, who derived a mean eld dynamical equation (the time-dependent Hartree equation) for mean eld systems of distinguishable particles, remarking at the time that he converge...

Let L be a parabolic second order differential operator on the domain (Pi) over bar = [0; T] x R. Given a function (u) over cap : R --> R and (x) over cap >0 such that the support of (u) over cap is contained in (-infinity, -(x) over cap], we let (y) over cap : (Pi) over bar --> R be the solution to the equation: L (y) over cap = 0, (y) over cap\ {...

We derive the time-dependent Schrödinger–Poisson equation as the weak coupling limit of the N-body linear Schrödinger equation with Coulomb potential. To cite this article: C. Bardos et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 515–520.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Justification de l'équation de...

In the present paper a mathematical analysis of the "time reversal mirror" (cf. [4,9,10]) is given. As a first step of a more detailed program, the emphasis is put on phenomena which are described by the genuine acoustic equation with Dirichlet or "impedance" boundary conditions. An ideal situation is first considered then relation between the ques...