General context

My research activities were strongly influenced by the courses (DEA 1966-67) given by Jean-Pierre Guiraud, my future thesis adviser, specially on the subject of hydrodynamic instabilities and bifurcations. That is why I worked in the field of theoretical fluid dynamics, a mixture between mathematical analysis and nonlinear dynamics, with the additional constraint to deal with fluid mechanics equations (Navier-Stokes or Euler equations). This explains why I was "invited speaker" at the ICM Congress of Berlin (1998) which was adressed to mathematicians (quite "pure" in general in ICM’s) (see [C31]), while I was presiding the Comité National Français de Mécanique for 8 years, whose purpose is in particular to prepare the french representation at the ICTAM Mechanics Congress. I created in 1980, with U.Frisch (Nice) a DEA (a year course for graduate students to prepare them for research) on Turbulence and Dynamical Systems, and I was the Director of this DEA for 15 years, which allowed in particular to catch good students and constitute a nice little performing team working on these topics. As a result was the creation with P.Coullet (nonlinear Physicist, famous for his discovery with C.Tresser of period doubling renormalization) of the Institut Non Linéaire de Nice (INLN) in 1991, which was pluridisciplinary for a long time, mixing  together mathematicians, mechanicians, and physicists (theoreticians and experimentalists). I was Director of INLN from 1991 to 1994 (Adjoint Director 2002-2003). The recent evolution of this Laboratory towards Optics, following the will of CNRS, has motivated the movement of mathematicians and fluid mechanicians backwards to the Laboratoire J.A.Dieudonné (originally math Lab). I am doing again my research in this laboratory since jan. 2007, now as an Emeritus Professor.  I received in 2008 the Prize Ampère of the Académie des Sciences de Paris, for all my works.

To sum up my research activities, let us say that I am trying to apply my favorite tools to explain in a mathematical way, i.e. at a fundamental level, certain physical phenomena, often with the help of discussions with experimentalists. At the same time I am also interested to solve old classical problems (considered as too difficult at that time), on which I could use successfully my methods (see the section on the Couette-Taylor problem, the Bénard-Rayleigh convection and the chapter on water waves). 
 

Dynamics and bifurcations related to Navier-Stokes systems

  Hydrodynamic instabilities and bifurcations, Books: [M1], [M2], [M5], [M6]

These works consider Navier-Stokes equations as a differential equation in a suitable Hilbert space, for being able to use ODE techniques and work on solutions continuous in time, with values in the domain of the nonlinear operator of the infinite dimensional vector field. The paper [6] (1971) is axed on functional analysis, while [11] (1972) gives one of the first proofs of the Hopf bifurcation theory on Navier-Stokes equations, in the same time and independently of D.Sattinger (Minnesota), and V.Yudovich (Rostov).  In [18] (already in 1978) we give the first concrete example of a Hopf bifurcation occuring on Navier-Stokes equations, example of interest in Meteodynamics (quite popular nowadays). More recently (2023) I studied again the Bénard-Rayleigh convection in the case of free-free boundaries, allowing explicit computations of the domain of stability of the various bifurcating solutions, in function of the Prandtl number ([97]). I could show that the only possibly stable solutions are rolls (classical ones) and, in contradiction with the usual thinking,  triangular pattern (see figure below).

The paper [14] (1977) provides a simple proof (the first proof is due to Kato and Fujita) of analyticity in time of the solutions of the Cauchy problem for Navier-Stokes equations near a basic solution (few years before another proof by Foias-Temam). This is useful, for example for building a Poincaré map near a closed orbit (periodic solution) and center manifold reduction for maps could then be applied for further bifurcations like into quasi-periodic flows (see the books [M1] and [M2] (undergraduate level)). Notice that the book [M2] on elementary bifurcation theory, with D.Joseph (1980),  has exercises for the 8 first chapters, is translated into russian and chinese, and has a second edition in 1990. Books [M5] with Mariana Haragus, and a part of [M6] (in french) give a "modern" point of view of useful techniques in instability and bifurcation problems ruled by EDPs originated from Physics (see below).

                   Bifurcation into invariant tori for maps  [20, 21, 33]  The aim was to understand for example some experimental results on Rayleigh-Bénard convection in a small aspect ratio apparatus where several frequencies appear after a succession of bifurcations. In the two first papers [20] (90p.), [21] (1979) with A.Chenciner (Paris), we give sufficient conditions on a family of mappings having an invariant torus, for getting a bifurcation to a higher dimensional invariant torus. These conditions bear on the linearized operator, including diophantine ones on the rotation number and on eigenvalues, so they seemed quite restrictive at that time. However, few years later, we could show [33] (1988) with J.Los (Marseille) that these bifurcations may appear quasi-generically, in the sense that an additional parameter provides a description of such bifurcations appearing on a Cantor set of parameters. 

                   Bifurcation into quasi-periodic patterns  [82] (2009) With A.Rucklidge (Leeds), we study bifurcating quasi-periodic patterns as in the Faraday experiment (horizontal viscous fluid layer periodically shaked vertically), and in Bénard-Rayleigh steady convection. Using Gevrey estimates, we could prove the existence of bifurcating quasi-periodic solutions of the Swift-Hohenberg model equation, up to an exponentially small remainder. In [C35] (2009) I show that this result extends to the steady Bénard-Rayleigh convection bifurcating flow. In [86] with M.Argentina (Nice) we consider the "real" Faraday problem and give a criterion for the selection of a specific quasipattern in starting from an instability where oscillations at half the forcing frequency are critical at the same time as the natural ones. In [89], with Braaksma and Stolovitch, we give the first mathematical proof of existence of bifurcating quasipatterns on the Swift-Hohenberg PDE (small divisor difficulty). On the same PDE the method also applies in [92] for proving that the superposition of two hexagonal patterns leads in general to a quasipattern of dodecagonal structure, see  the work [95] with A.Rucklidge for general results (see figure below). With B.Braaksma we prove in [90] that all this may be adapted for proving the existence of quasipatterns in Bénard-Rayleigh steady convection.

 The Couette Taylor problem  [27, 28, 38, 42, 44], Book: [M3] 

These papers and the book [M3] (1994) written with P.Chossat, deal with the Couette-Taylor problem (more than 100 years old problem) of an incompressible viscous flow between rotating coaxial cylinders, on which my team and myself worked for about 10 years (theoretically and numerically) in concertation with american experimentalists (H.Swinney, R.Tagg, D.Andereck) allowing discoveries, comparisons, and confirmations even quantitatives. We were able to predict for example the structure in "ribbons" of the flow when the Couette flow looses its stability while the cylinders are counter-rotating. This flow was later observed experimentally (R.Tagg, Colorado) and corresponds to standing waves in the axis direction, rotating in the azimuthal direction. In fact we systematically use the center manifold reduction and the symmetries broken by the critical eigenmodes. Moreover, [27] (1986) gives a (simple) proof of center manifold reduction near a non trivial group orbit of solutions (circle, torus,...), and then gives a simple explanation of the various physically observed flows occuring when the Taylor vortex flow looses its stability. In [28] (1987) we study the competition between two types of critical oscillating modes, which reduces to the study of a 8-dimensional ODE, and succeed in interpreting some physical results, of the team of D.Andereck (Ohio) and R.Tagg (Colorado) like the "interpenetrating spirals" regime. In [38, 42] (1989-90) we study (with A.Mielke, Stuttgart) the steady solutions in avoiding to impose an axial periodicity as before (axial coordinate plays the role of "time" in a reversible system). This leads to new solutions of Navier-Stokes equations, like spatially quasi-periodic ones (with J.Los), and solutions looking like Couette flow in the center, and like Taylor vortices at both infinities (case of corotating cylinders). In [44] (1991) we consider time-periodic solutions, still not imposing axial periodicity (case of counter-rotating cylinders), and we show for instance the existence of a "defect" solution connecting two symmetric helicoïdal flows (see figure above). This is precisely such type of flow which is first observed in certain ranges of parameter values.

The book [M3] constitutes an answer to most of the questions posed by R.Feynman on the Couette-Taylor problem in his famous physical course as "one of the most challenging problems in fluid mechanics" (end of the second volume on electromagnetism).

General Normal forms and applications

The section on the Couette-Taylor problem already uses normal form reduction for the study of the reduced amplitude equations on a center manifold. However this case is in general simplified by the occurence of symmetries, which are not present in more general problems. The normal form reduction is also used in water wave problems (see next chapter) in the frame of reversible systems. This section concerns general results and applications other than seen above.

Normal forms and reversible systems [29, 31, 40], [69], [74], Books: [M4], [M5], [M6]

The search for "normal forms" for simplifying the local study of vector fields near a singular situation, is an old subject, coming back to Poincaré and Birkhoff and, more recently, to V.Arnold. The paper [29] (1987, my most cited paper) is a result of a cooperation with physicists (P.Coullet et al), and allowed to provide a simple characterization of a normal form, only using elementary analytic tools (avoiding algebraic geometry). Part of our result was in fact included in a previous work (not really accessible) of Belitskii. Our result is now used as a classical one, including computing softwares (book by S.N.Chow). The paper [31] (1988) extends this type of result to time-periodic vector fields. In [40] (1990, which is my second most cited paper) with P.Coullet (Nice), we provide a list of 10 generic possible bifurcations of a steady periodic pattern in one direction, solution of a translation-invariant system (as for many classical hydrodynamical instability problems). For example, we prove that breaking the reflection symmetry for a periodic pattern, may induce a slow travelling wave. This is now a reference for numerous experimental observations. More recently I was interested in solving bifurcations for reversible dynamical systems (the vector field anticommutes with a symmetry), where the normal form is often integrable.  These systems are very important in Physics and Mechanics (see section on water-waves and on Bénard-Rayleigh defects in convection). My most cited work is [49] (1993) with M.C.Pérouème on 1:1 resonance, where we are able to prove the existence and give nearly explicitely homoclinic solutions, while people were only able to look for periodic solutions. These results are used for example in hamiltonian systems as in nonlinear optics, and also in bifurcations of elastic rods.

A series of lectures I made at University of Stuttgart lead to the small book [M4] (1992), written with M.Adelmeyer, reedited in 1999. With Mariana Haragus (Besançon), we wrote the book [M5] (329p., 2011) containing all proofs for center manifols and normal forms, and containing many exercises and problems (with solutions). Its originality is that it deals with infinite dimensional systems (PDE's, lattices,...) and in the detailed study of  normal forms, specially on how to use and concretely compute them, with a special emphasis on reversible systems. Lastly, with M.Haragus, we prove the existence of symmetric domain walls [93] in the Bénard-Rayleigh convection bifurcation problem. This uses normal forms for reversible systems for finding the heteroclinic which connects the two symmetric rolls regimes occuring at the bifurcation onset. These results are completed in [95] with free-rigid boundary conditions. Lastly, the defect pattern in rolls meeting orthogonally, where the rolls at infinities are no longer symmetric, is treated in collaboration with B.Buffoni in [96] by a variational method. These defects (domain walls) are now completely proved analytically (quite technical) in [98], [99] (2023).

In [69] and [74] (2005) with E.Lombardi, we extend for very general analytical vector fields the possibility to find an exponentially small remainder, in optimizing the number of terms in the normal form. This type of result was only known for specific hamiltonian vector fields, and is promised to many relevant physical applications, specially in the study of nonlinear oscillations of structures as explained in particular in [83]. Moreover an application, to center manifolds, analytic up to an exponentially small tail (see [83] (2009)), may be used to considerably simplify some existing proofs on the existence of exponentially small oscillatory tails (applied on water waves and Lattices, see below). We suggest applications in nonlinear vibration theory of large structures, where it is possible to justify mathematically certain techniques used by engineers under the name "normal nonlinear modes".

  Application of Center manifold reduction and Normal forms to Lattices [61], [62], [73], [78]

The paper [61] (2000) with K.Kirchgässner (Stuttgart) proves that we can use the same type of reduction for the local study of travelling waves in one-dimensional lattices (infinite chain of nonlinear oscillators, coupled with their nearest neighbors). The difficulty here is to show that the center manifold reduction applies despite of a non usual behavior of the spectrum of the linearized operator (however common for most lattices). Our results allow to prove the existence and reach analytically new solutions of physical interest of such systems, and introduce a new tool for the study of waves in lattices (see [62] (2000) for the Fermi-Pasta-Ulam lattice, and [73] (2005) for a review paper written with G.James, specially on travelling breathers.

The Bénard-Rayleigh convection problem [93], [95], [96], [97], [98], [99]

My thesis adviser, Jean-Pierre Guiraud introduced me to hydrodynamic instabilities, in particular the Couette-Taylor problem (see above) and also the Bénard-Rayleigh convection. It is the instabilities in an horizontal viscous fluid layer heated from below, the boundary conditions may be rigid-rigid, free-free, or free-rigid on horizontal planes.

Already in 1977 ([16] with R.Lozi), I was interested by the dynamo problem (generation of a magnetic field via a secondary Hopf bifurcation of a 3-dim convective regime). I worked also a little on convection in compressible fluids [43] (collaboration with CEA). In 2023 I considered again the classical bifurcation problem between planes with free-free boundary conditions, which has the advantage to give explicit calculations on the domain of stability of various bifurcating patterns, in function of the Prandtl number. I could show [97] (2023) that, contrary to generally admitted ideas, the only periodic solutions which can be stable are the rolls (the classical ones) and unexpectedly the triangular pattern (see figure above).

Finally, a series of quite thick papers is devoted to the study of certain defects in convection. With Mariana Haragus (Besançon), we first considered the case when two regimes of convective rolls meet symmetrically, making a certain angle [93] [95] (2020-21): this uses spatial dynamics (introduced by K.Kirchgässner in the eighties), a center manifold reduction to a 13-dimensional system and a normal form analysis. Then a truncated system of dimension 8 deserves an heteroclinic solution, the persistence of which needed to be proved, showing the existence of the defect connecting the two symmetric systems of rolls. Then, with M. Haragus and B.Buffoni (EPFL) we considered the defect where two systems of convective rolls meet orthogonally [96], (2023) and we could show how to obtain, via center manifold (again in 13 dimensions) to a truncated system in 8 dimensions (already introduced in a formal way by Pomeau-Manneville in 1983)  on which the existence of a (non symmetric here) heteroclinic solution, corresponding to the defect, is proved by a variationnal method (not allowing yet to conclude for the mathematical existence of the defect).  Then I was able to obtain analytically the avove heteroclinic [98]  (2023) and prove its persistence for the full Navier-Stokes-Boussinesq system [99] (2024). Notice that we obtain a one parameter family of defect-solutions, where the two systems of rolls at infinities have not necessarily the same wave number, adapting to a shift of rolls papallel to the wall.

Water waves

I started to work on water waves in 1990, influenced by K.Kirchgässner (Stuttgart) and Frédéric Dias (now in Dublin) (arriving in my Lab (Nice) at that time). The last results on Standing waves and 3D travelling waves (symmetric or nonsymmetrical with respect to the propagation direction) are exposed in thick papers (about 100p. each). This is a specificity of small divisor problems.

 Application of Center manifold reduction and Normal forms to Water Waves  [46, 49, 63] 

Matisse 1952

These papers deal with two-dimensional water wave problems (170 years old problem), in using the reversible dynamical systems techniques. We use the center manifold reduction, as was initiated by K.Kirchgässner in 1982 on elliptic problems in strips (the unbounded space coordinate is considered as a time and the problem is treated as an evolution problem even though the Cauchy problem is ill posed). We introduce in addition normal forms theory for reversible systems. We could prove [41](1990) and [49] (1993, already cited) the existence of a new type of solitary waves, with damping oscillations at infinity, which was quite a surprise for specialists, and which seems to be observable in particular situations (see ref. in [63]). In [46](1992) we also recover and complete the results obtained before, initiating for example an important work on bifurcating generalized solitary waves, with exponentially small oscillatory tails at infinity (see the Springer Lecture Notes in Maths 1741 of E.Lombardi). 
My works in this field were recognized internationally by a Max Planck - von Humboldt prize in 1993 (with K.Kirchgässner),  and by a prestigious invitation to make a sectional lecture at the International Mathematics Congress of Berlin in 1998.
The review paper [63] (2003) with F.Dias (Cachan) presents the ten years results obtained via reversible dynamical system theory applied on the water-wave problem (even for 3D problems). A very short version is the review [C31] for the Math Congress in Berlin (1998). 

  Water Waves and bifurcations from a continuous spectrum [52, 59, 65, 66, 75] 

The mathematical problem of the search of travelling waves is more difficult when one of the fluid layers is infinitely deep. The examination of numerical values of scales shows that this is physically relevant in most cases (if we wish a not too small domain of validity of our mathematical results). Once formulated as a dynamical system, the spectrum of the linearized operator contains the entire real line (essential spectrum). This prevents the use of a center manifold reduction as before.  I could show that this does not in general perturb the search for travelling waves [59] (1999), despite of the resonance due to  0 in the spectrum (we recover a result of Lyapunov-Devaney established in finite dimensions without resonance).
We had in particular to build a normal form theory in presence of this essential spectrum. In [52] (1996) (one layer with a free surface and surface tension) (with P.Kirmann, Stuttgart) we prove the existence of solitary waves with damping oscillations at infinity. Here the behavior at finite distance is close to the finite depth case, while near infinity the decay is polynomial instead of being exponential, due to the essential spectrum. In [65, 92p.] and [66] (2002-03) (two layers, one free surface, one free interface, no surface tension) (thick work with E.Lombardi (Toulouse) and S.M.Sun (Virginia)) we have a competition between a natural oscillation, and a slow polynomial decay at infinity. We give a new type of normal form, taking care of the effect of the essential spectrum at finite distance (the Benjamin-Ono equation appears here), coupled with the nonlinear natural oscillation, and we prove the existence of a family of generalized solitary waves, like a superposition of the Benjamin-Ono type of solitary wave (in the middle) with small periodic oscillations at infinity. Here the effect of the essential spectrum also lies at finite distance, contrary to [52]. In [66] it is shown that the small oscillations at infinity may be up to exponentially small (not 0). The generalization of the above study and generic assumptions which lead to the same type of bifurcating solutions is now made by M.Barrandon in his PHD thesis (dec. 2004). This relies strongly on a deep study of the resolvent of the reversibly symmetric linear operator near 0, when the essential spectrum fills the real axis (see the review paper [75] (2005) with M.Barrandon).

  Standing gravity waves in infinite depth ("le clapotis") [60, 64, 68 (112p.), 71, 72, 80]

The 2-dimensional standing gravity waves problem (no surface tension) at the free surface of a potential flow in an infinitly deep fluid layer, was a very old and challenging problem (Boussinesq, Rayleigh, ...), due to the infinitely many resonances, and to the occurence of derivatives in the nonlinear terms, of order higher than in the linear ones. The note [80] (2007) is in honor of J.Boussinesq, and presents his seminal work on the standing wave problem.
In [60] (1999) and [64] (2002) I simplify and improve the previous results of Toland and Amick (1987) on the possibility to compute a formal expansion in powers of the amplitude, without failing to satisfy the infinite set of compatibility conditions. Moreover, I showed a large and new set of multi-modal solutions. In [68] (2005) (112p., completed by [71] and [72]), with P.Plotnikov (Novosibirsk) and J.Toland (Bath), we finally prove the existence of the unimodal and multimodal standing waves for values of the amplitude lying in a Lebesgue set, near criticallity. This is a technical work , due to the necessity to overcome at the same time the difficulty of the complete resonance of the linearized operator at 0 and the loss of regularity of nonlinear terms larger than for linear ones. The necessity to invert the linearized operator at a non zero point (Nash-Moser theorem), leads to find a suitable choice of coordinates, and variables for inverting a second order nonlocal hyperbolic differential operator, with periodic coefficients, ...which gives a small divisor problem, and restricts the "good" set of amplitudes for which the standing waves ("clapotis" in french) exist.

  3D periodic travelling gravity waves [M6, 128p.], [84], [85, 87p.]  (with P.Plotnikov)

  Three-dimensional "short crested (diamond) waves". With P.Plotnikov [79]-[M7] (Memoirs AMS 2009, 128p.), we consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle 2ø between them. Denoting by µ the dimensionless bifurcation parameter (depending on the wave length along the direction of the travelling wave and on the velocity of the wave), bifurcation occurs for µ = cosø . For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" or "short crested waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the work [79]-[M6] is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates.  Among the obstacles we need to find a diffeomorphism of the torus which transforms the main coefficients into constants, and we need to improve a result of H.Weyl for the control of small divisors. Finally a quite general "descent method" allows to reduce the problem to invert a Fredholm operator. It is then shown that for almost all angles ø, the 3-dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter µ in a set having asymptotically a full measure near the bifurcation curve in the parameter plane (ø, µ). The infinite smoothness of these solutions is now proved by T.Alazard and G.Métivier, thanks to the use of paradifferential calculus (see the pedagogic paper in  la Gazette des Mathématiciens, oct 2010). Also notice that the spatial Fourier spectrum of the principal part of these symmetric waves fits exactly with the experimental results obtained in particular by D.Henderson (Penn State 2005) (see the ref in [M7]).

  Asymmetrical periodic travelling gravity waves (with P.Plotnikov) [85] (ARMA 2010, 87p.) The above result is now extended to asymmetrical periodic waves, for which the lattice of periods is no longer symmetric with respect to the critical propagation direction, and for which the propagation direction is in general not aligned with the critical propagation direction, given by the dispersion relation.  One difficulty here was to find a diffeomorphism of the torus with the properties described above, the additional difficulty being here that the rotation number of the horizontal projection of the velocity of fluid particles satisfies a diophantine condition. This diffeomeorphism is made part of the unknowns of the problem and we are able to obtain a similar result as above for the existence of such travelling waves with two artitrary angles (made by the basic wave vectors of the periodic lattice with the x-axis), satisfying non resonance conditions, instead of the above only angle ø, and two parameters: µ and a unit vector (giving the direction of propagation of the wave), which lie on a subset of asymptotic full measure at the bifurcation point (see figure below).

A side result (see [84] (2009)), which is of interest for experimentalist (but very difficult to manage) is that we show the phenomenon of "Directional Stokes drift" meaning that in the frame moving with the velocity of the waves, the horizontal projections, of the trajectories of fluid particles on the free surface, have an average direction making a small non zero angle with the direction of the waves (we give explicitely the value of this angle). This angle cancels for a specific value of the ratio of the amplitudes on each basic (non symmetric) wave vector.