Activités de recherche
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__ I started my researches activities in 1969 with a thesis under the supervision of Jacques Louis Lions. Since that time they have been devoted to the study of partial differential equations (in short PDE). The fact that PDE is a “Carrefour” subject has been an essential motivation.
__ Ordinary differential equations or dynamical systems involve functions depending only on one variable  (very often according to intuition it is the time). On the other hand PDE are concerned by objects defined in distributed  media  (space or space time domain).
__ The problems come from Physic or engineering sciences and it is often desirable to obtain complete results including numerical computations with error estimates. Interaction with Numerical Analysis and Modeling  is an important issue. During the forty previous years conjunction of Functional Analysis and progress of computers has been an important stimulus for the blooming of the subject.
__ The possibility of realizing approximate computation   has given a new importance to qualitative study of the problems (without explicit or semi explicit solutions). Therefore uniqueness and stability theorem became more valuable. In return these theorems contribute to the design of numerical codes.
__ The variety of the problems implies that there is no preexisting methodology and no mathematical tool should be a priori excluded.
__ Eventually one observes that it is at the level of macroscopic problems that can be observed at human scale (therefore the illustration of my home page with  the rainbow and the water waves illustrating light propagation and fluid mechanic) that the most fructuous results of the theory are  obtained. However in the previous centuries the same equations were used to settle problems that were more fundamental than applied. What I have in mind is the propagation of heat which motivated the works of Fourier or the atomic hypothesis which led Boltzmann to the equation carrying his name.
__ In these surroundings and more precisely since 1980 I have devoted my activities to the following issues:
1. Microlocal analysis and theorem of propagation to contribute to the solution of practical problems.
2. Equations of fluid mechanic like Navier-Stokes or Euler equations.
3. Kinetic equations
4. The microlocal theory and its applications.
1. Limit of N particles systems

__It is in hyperbolic linear problems with the wave  equation as the basic example that the notion of propagation appears the most naturally. This is an old idea related o the duality between waves and rays. It was illustrated by the Heisenberg uncertainity principle. However it is since the 1970 International Congress of  Mathematicians under the influence of strong personalities like Lax, Hormander ou Nirenberg that the subject became very fashionable and attractive for “young “ French mathematicians and took the name of microlocal analsis. In the mean time the connection between theory and applications was somehow lost while a huge request for mathematical contributions existed in the engineering and physical community.

__ One of the essential motivation of microlocal analysis was the study of the eigenvalues of the Laplacian in bounded domain. This is a sequence of positive numbers which go to infinity. As a consequence of the “universality “ of the Laplacian these numbers appear in a huge variety of phenomena from  the harmonics of a music instrument (a trumpet, a violin or a drum)  to the vibrating frequencies of an atom. Eventually this is a subject in number theory (count the number of points with interger coordinates in a sphere centered at the origin in function of its radius ).
The leading term in the asymptotic behavior of these numbers, called the Weyl term is known since the beginning of the 20 century. On the other hand the rest of the expansion is not of algebraic nature. Oscillating terms do appear. As this was observed by Balian et Bloch [1] and   Keller and Rubinow  [2] the oscillations are related to closed rays of geometric optic and it is the rigorous proofs of such behavior which implies mircolocal analysis.

REFERENCES [1] Balian, R.; Bloch, C. Distribution of eigenfrequencies for the wave equation in a finite domain. I. Three-dimensional problem with smooth boundary surface.  Ann. Physics  60  1970 401--447.
[2] Keller, J. B. and Rubinow S. I.; Asymptotic solution of eigenvalue problems. Annals of Physics, Volume 9, Issue 1, January 1960, Pages 24-75 .

__ A second motivation corresponds to the situation where the domain is no more bounded but is the complementary of an obstacle. The amount of energy located near the obstacle always goes to zero. However it is the way such quantity decay which once more depends on the geometry of the rays. Furthermore appears complex numbers which describe the dispersion of the wave. Such  numbers are called resonance  frequencies. They are characteristic of the obstacle and well known by radar engineers.

__ Jointly with J. C. Guillot, J. Ralston, J. Rauch and G. Lebeau  I have studied  how to apply to this problem the tools of microlocal analysis previously developed for the interior problem [3], [4].

REFERENCES  [3] B.,C.; Guillot, J.-C.; Ralston, J. La rélation de Poisson pour l'équation des ondes dans un ouvert non borne. Application \`a la théorie de la diffusion. French J Commun. Partial Differ. Equations 7, 905-958 (1982).
[4]  B.,C.; Lebeau, G.; Rauch, J. Scattering frequencies and Gevrey 3 singularities English   Invent. Math. 90, No.1, 77-114 (1987 )
__ The idea of adapting to distributed systems the methodology of control and stabilization and the need of understanding problems related to the stabilization of large structures like satellite antennas lead J. L. Lions to an abstract theory of the control of hyperbolic equations. Here also (in a program realized with G. Lebeau and J. Rauch ) microlocal analysis provides necessary and sufficient conditions for control and stabilization. It  explains why modal control (control of the first few eigenmodes of an oscillating structure) may lead to “overshooting”: excitation of high order modes. The above geometric conditions are seldom realized in practical applications and in spite of that the “overshooting” excitation in not frequently observed. This paradox can be explained by the fact that the higher order modes carry few energy or are damped [5].

REFERENCES [5] B.,C.; Lebeau, G.; Rauch, J. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. English J SIAM J. Control Optimization 30, No.5, 1024-1065 (1992).

__ More recently I have observed that mircolocal analysis contribute to the explanation of the efficiency of the “time reversal method.” This principle has been studied by several research laboratories in particular the LAO (laboratoire d’ondes et d’acoustique) under the guidance of Mathias Fink. The reversal method has many applications in  medicine (lysostrophy), non destructive control, target identification and telecommunications. I have shown how one can use the Egoroff theorem (which by now is classical tool in quantum chaos) to explain that the ergodicity of the media contributes to the success of the experiments [6]. The same type of analysis can be adapted to the identification of an ergodic media under the  solicitation of  independent random sources[7].

REFERENCES [6] B. C. ; A mathematical deterministic analysis of the time-reversal mirror. English CA Uhlmann, Gunther (ed.), Inside out: Inverse problems and applications. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 47, 381-400 (2003)
[7]  B. C., ;  Garnier,  J.  Papanicolaou G. Green function identification   by cross-correlation of noisy signals:  A semi-classical approach in preparation.

2. The Euler and Navier-Stokes equations for fluid mechanic

__ The Euler and Navier-Stokes are not only fundamental tools for fluid mechanic but also are model problems for the pathology of non linear systems.. 
The contribute to the subject I have proven that the analyticity of the initial data persists  as long as the vorticity remain bounded (in particular for all time in 2 space dimension). The  precise description of  the domain of analyticity contributes to the evaluation of the accuracy for Galerkin type discretisations [8].
Then I have proven that analyticity is a sufficient condition for the well posedness of the Kelvin Helmholtz equations [9].

REFERENCES [8] B.,C.; Benachour, S. Domaine d'analyticité des solutions de l'équation d'Euler dans un ouvert de Rn French J Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4, 647-687 (1977).
[9] B. C Sulem, C.; Sulem, P.L.;   Frisch, U.  Finite time analyticity for the two and three dimensional Kelvin- Helmholtz instability. Commun. Math. Phys. 80, 485-516 (1981).

__ In this problem it was proven by Gilles Lebeau and S. Wuilles Lebeau et S. Wu [10], [11] that minimal hypothesis on the vortex sheet implies it analyticity. This results which seems academic has some importance for the understanding of instabilities of vertex sheet and interfaces (Kelvin Helmholtz, Rayleigh Taylor etc…) Since singularities do exist ( they are either mathematically constructed, observed or eventually computed numerically) a contradiction argument implies that in their neighborhood the interface is less regular (more singular) that  what would lead to analyticity.
__ Eventually I have contributed to the issue of the stability of stationary solutions of the 2 [12]. This problem is linked to the appearance and persistence of coherent structures as observed   in cyclones or in the Jupiter red spot.

REFERENCES [10] Lebeau, G. Régularité du problème de Kelvin-Helmholtz pour l'équation d'Euler 2d. (French) [Regularity of the Kelvin-Helmholtz problem for the 2d Euler equation] A tribute to J. L. Lions.  ESAIM Control Optim. Calc. Var.  8  (2002), 801--825
[11]  Wu, S. Mathematical analysis of vortex sheets.  Comm. Pure Appl. Math.  59  (2006),  no. 8, 1065--1206.
[12]  B.,C.; Guo, Y.; Strauss, W.  Stable and unstable ideal plane flows. Chin. Ann. Math., Ser. B 23, No.2, 149-164 (2002). 

3. The Kinetic equations

__ In 1872 Boltzmann introduced the equation which carries his name as fundamental tool to deduce and validate macroscopic equations from particles dynamic. A rigorous mathematical treatment of these ideas (including full proofs) was one of the problems proposed by Hilbert  at the International Congress of Mathematicians in Paris in 1900.

__ Since the work of Leray in 1934 we have the proof for the existence for any finite energy in initial data of a global in time weak solution of the incompressible Navier-Stokes equations. On the other hand Di Perna and Lions [13] established in  1990 the existence of so called renormalized solutions of the Boltzmann equation. Such solutions have for the Boltzmann equation a status very similar to the status of the Leray solutions for Navier-Stokes equations. [13].

REFERENCES [13] Di Perna R. et Lions P.L. On the Cauchy Problem for the Boltzmann equation : Ann. of Math., 130, 1990, pp. 321-366.

__ Therefore a natural contribution to the Hilbert program is the proof that, under a convenient scaling,  any sequence of renormalized solutions of the Boltzmann equation converge to a Leray solution of the Navier-Stokes equation.

__ Two determining parameters are the Knudsen numbers,  which describes the rarefaction of the gas and the Mach number which is the ratio  between the sound speed at infinity and the characteristic velocity of the fluid. In 1991, jointly with F. Golse and D. Levermore I have shown that (several stability assumptions were required) when the ratio  of the Knudsen and Mach number remains finite the solution of the Boltzmann equation converge to the solutions of the Navier-Stokes equation [14]. In agreement with the Von Karman relation this situation corresponds to a finite Reynolds number. Then we have given a serie of results aiming at removing the stability assumptions. And it is only recently that this program has been fully completed with the contributions of F. Golse and L. Saint Raymond [15].

REFERENCES [14] B.,C.; Golse, F.;  Levermore, D. Fluid Dynamic Limits of Kinetic Equations I. Formal derivations Journal of Statistical Physics   63  323-344 (1991). [15] Golse, F., Saint-Raymond, L. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels.  Invent. Math.  155  (2004),  no. 1, 81--161.

__ Besides their importance for fundamental (or mathematical) physic, the Kinetic equations play an important practical role. They have to be used for rarefied enough so that the particles  cannot reach thermal equilibrium. The most classical example is the reentry in the atmosphere of a space vehicle. However,   the same mathematical approach  applies to nuclear reaction in a critical media, to the computation of electric current in devices which are so small that the electron cannot reach thermal equilibrium. Eventually it is also used to model the ionization between a compact disk and the reading head.
At this level one of the essential issue is the effect of the boundary of the domain. This is treated by a boundary layer analysis, using the resolution of the Milne problem.  As a consequence a “slip boundary condition” is introduced at the macroscopic level [16] and [17].

REFERENCES [16] B. C.   Caflisch R. and  Nicolaenko B.  The Milne  and Kramers Problem for the Boltzmann Equation of a Hard Sphere Gas Comm. in Pure and Appl. Math.  39 (1986), 323-352.
[17] B. C. ; Golse, F.; Sone, Y. Half-space problems for the Boltzmann equation: a survey.  J. Stat. Phys.  124  (2006),  no. 2-4, 275--300.

4. The limit of system N particles.
__ The main difficulty (philosophical, physical and mathematical) in the derivation of macroscopic equations from the reversible evolution of particles comes from the appearance of irreversibility. It is often measured in term of increase of entropy. In the derivation of the Boltzmann equation from the dynamic of hard spheres, the non-linearity is due to the binary shocks between particles. With weak convergence this nonlinearity contribute to the understanding of the appearance of irreversibility. Such effect is absent in linear models for instance in the derivation by Lorentz of an equation describing the evolution of electrons in metal. Therefore this analysis may be more complicated for linear problem. In particular recent works of Bourgain Golse and Wennberg indicate that, in some cases, the introduction of randomness is compulsory.[18] [19] 

__ The contribute to the understanding of the subject with J. F. Colonna and F. Golse we have constructed a linear model inspired by the Arnold cat map. In this model the computation are fully explicit and lead to a complete understanding of the appearance of diffusion.

__ Eventually since 2000 an important part of my activities is devoted to study of quantum systems in a team,   based at the Wolfang Pauli Institute in Vienna and including B. Ducomet, F. Golse, A. Gottlieb, N. Mauser et S. Trabelsi.  With a mean field hypothesis, or with N large but fixed there is no appearance of irreversibility. We have obtained results for Hartree and Hartree Fock approximations [20], [21]. Now we study the Multiconfiguration time dependent methods. This seems more adapted to computations in molecular chemistry.
REFERENCES [20] Bardos, C.; Golse, F.  Mauser, Norbert J. Weak coupling limit of the  N particle Schrödinger equation. Cathleen Morawetz: a great mathematician.  Methods Appl. Anal.  7  (2000),  no. 2, 275--293.
[21] Bardos, C.; Golse, F.; Gottlieb, A. D.; Mauser, N. J. Mean field dynamics of fermions and the time-dependent Hartree-Fock equation.  J. Math. Pures Appl. (9)  82  (2003),  no. 6, 665--683.