**General
context**

My research activities are 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 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, fluid mechanicians and most of Physicists in dynamical systems back towards the Laboratoire J.A.Dieudonné (originally math Lab) which became de facto pluridisciplinary. I am doing 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,
and at the same time I am also interested to solve old classical
problems, on which I can use successfully my methods (see the section
on the Couette-Taylor problem, and the chapter on water
waves).

Dynamics and bifurcations related to Navier-Stokes systems

**Hydrodynamic instabilities and bifurcations **[6,
11, 14], Books: [M1], [M2]

These papers 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).

The paper [14]
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)).

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], [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 in the parameter set.

Bifurcation into quasi-periodic patterns [82] (2009) With A.Rucklidge (Leeds), we study bifurcating quasi-periodic patterns as in the Faraday experiment, 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 give a criterion for the selection of a specific quasipattern on a fluid mechanics model for a thin horizontal layer of viscous fluid, periodically shaken vertically. Finally, temptativelyin [87] and finally in [89], with Braaksma and Stolovitch, we give the first mathematical proof of existence of bifurcating quasipatterns on the Swift-Hohenberg PDE.

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 (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). 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. 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. 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 main 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).

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

The research of "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. 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 classical hydrodynamical instability problems). It appears that these results explain many recent experimental results.

A series of lectures I made at University of Stuttgart lead to the small book [M4] (1992), written with M.Adelmeyer, reedited in 1999. Recently, with Mariana Haragus (Besançon), we wrote a new 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.

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).

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).

Water waves

I started to work on water waves in 1990, influenced by K.Kirchgässner (Stuttgart) and Frédéric Dias (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]

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, and *we introduce in addition normal forms theory*
for reversible systems. We could prove (1990) the existence of a new
type of solitary waves, with damping oscillations at infinity, which
was quite a surprise for specialists. In [46] 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). The paper [49] (1993) played an important
role in the field of reversible dynamical systems, and its results are
regularly cited even for studies on some buckling problem for long
elastic rods, and in conferences on Hamiltonian systems. 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. 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 (Nice) 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)).

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 [79, 128p.], [84], [85, 87p.]

*Three-dimensional
"short crested (diamond) waves"*.
With P.Plotnikov [79] (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] 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).

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.

A side result (see [84] (2009)), which is of interest for experimentalist 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.