Born to be alive

In my broad-audience book Théorème Vivant I have evoked the chaotic process of my collaboration with Clément Mouhot on the problem of nonlinear Landau damping, and the birth of the resulting Theorem. The book has done fine; translations in Italian, German and Serbian are available, while English, Romanian, Bulgarian, Japanese and Korean versions are under way.

I will come back in a later post about what I learnt from the writing of this book, but for now I will just comment on the title: a Theorem is indeed, to the eyes and hearts of mathematicians, similar to a living being, with a destiny that cannot be controlled. One of the reasons why it is so hard to assemble is because precisely, you have to think of its each and every vital functions, their connections, the tiny details without which it may collapse, and the soul which holds it together.

In the book, I was focusing on the part of the story which is usually hidden and untold, that is, the period going from initial fertilization (the first discussion which starts it all) to birth (which is publication!). Of course, one may also ask about the life of this theorem after its birth, and many readers have asked me about this.

After their publication Theorems are scrutinized, generalized, used, corrected, modified, applied; they inspire and lead to new theorems. The process may be very quick or very slow; sometimes progress is tantalizing and sometimes it never occurs. Such is the impredictibility of science.

So it is with great pride that we can now announce a happy event for our Theorem, which played a key role in the birth of a beautiful newborn. People responsible for this are two talented young mathematicians, Jacob Bedrossian and Nader Masmoudi.

Holding a Professor position in New York University, Nader is an old acquaintance of mine, since we were together in École Normale Supérieure in the mid-nineties (he was my junior, in academic age, by just two years). Coming from Tunisia, a country with a strong tradition of excellent mathematicians, Nader specialized in partial differential equations and did his PhD under the supervision of Pierre-Louis Lions, who was also my PhD advisor — so Nader is my academic brother. He was working on hydrodynamic equations while I was more focusing on kinetic equations.

From the start, Nader has been extraordinarily productive. He has written over 100 high-level research papers with nearly 50 different collaborators. He quickly gained a reputation for versatility, strong analytic power and ability to attack hard problems inspired by physics. He provided a number of strong contributions to the theory of partial differential equations. Some of the subjects which he studied are the hydrodynamic and incompressible limits, the qualitative properties of boundary layers, the behavior of Schrödinger equations and other dispersive systems, the theory of mixtures of fluids and particles, the long-time behavior of general semilinear problems, etc.

Jacob is a younger mathematician, with already a great record in the field of partial differential equations, especially for the Keller-Segel equations, which model chemotaxis (say, the motions of crowds of bacteria driven by chemicals). He worked on both theoretical and numerical points of view, in the team of Andrea Bertozzi at UCLA.

After his PhD, Jacob moved to the Courant Institute of Mathematical Sciences, New York, to do his postodctoral thesis with Nader. Great move! Among other subjects, Jacob and Nader decided to work on a problem which several other teams were trying to solve: the adaptation of our nonlinear Landau damping work to the problem of the inviscid damping for the two-dimensional Euler equation.

Physicists have known for a long time that the qualitative behavior of a plasma and that of a two-dimensional perfect fluid share many common features. In particular, Landau damping was expected to be similar in both models. Clément and I were well aware of this, and after our work in plasma physics, we had tried to treat an incompressible fluid as well. But we soon found out that the case of a fluid was much more nasty than expected! After some preliminary investigations in 2010, in which we had identified some of the key issues and problems, we left the problem dormant. But where we had stumbled, Jacob and Nader managed to break through, and solved the case!

Let me go a little bit more into the details. Clément and I had proven the following, roughly speaking: If one considers a smooth homogeneous plasma (think of an assembly of electrons, repelling each other) with, say, a Gaussian distribution of velocities, and if one disturbs the plasma by submitting it to a brief electric field, then the electric field generated by the plasma will spontaneously disappear, or rather become extremely small as time goes. This quick decay is the Landau damping, named after Lev Landau, a legendary Russian physicist who made seminal contributions to plasma physics (and to many other areas of physics) in the thirties and forties. But while Landau had proven this result only in the linearized approximation, we could prove it for the full nonlinear equation. This provided a new theoretical advance about Landau damping, which has been a cornerstone of plasma physics for more than 60 years.

Besides the result itself, three ingredients of our proof were particularly notable, both for their mathematical contents and for their physical interpretation. The first is that nonlinear Landau damping is driven by mixing and a strong notion of smoothness. (In technical terms, a Gevrey regularity.) The second one is that the main enemy threatening to prevent nonlinear Landau damping is the possibility of accumulation of nonlinear echoes, due to resonances of the various waves of the plasma. The third one is that the problem of nonlinear Landau damping shares much structure with the celebrated theory of Kolmogorov-Arnold-Moser about the stability of certain physical systems like an idealized Solar system. At the level of mathematical tools, we used certain tailor-made norms to capture the strong oscillations of the distribution, and introduced a number of technical tricks. All this is explained in my lecture notes, or in simpler terms in a review article which I wrote for an audience of physicists.

The model studied by Bedrossian and Masmoudi is not a plasma, but a perfect two-dimensional fluid, obeying the venerable equations of Euler. Assume that the flow is a Couette flow: that is, the velocity of the fluid is horizontal, and proportional to the vertical coordinate; this is the simplest of so-called shear flows, which are very much studied in hydrodynamics. Then make a small disturbance of this flow, so that the velocity is no longer perfectly horizontal, neither perfectly proportional to the altitude; and let the flow evolve by itself, through the classical equations of fluids. Will the flow become horizontal again, as time passes by? This problem was posed more than hundred years ago by Orr; it is simple to state but highly nontrivial to solve. This is the question which Bedrossian and Masmoudi just answered positively in a prepublication.

In the linear approximation, an inviscid fluid is much easier to analyze than a plasma. Then linear theory does exhibit the same mixing features which are observed in plasma physics, and one often calls this inviscid damping. Some of the greatest names of hydrodynamics have worked on this linearized approximation: Rayleigh, Kelvin, Reynolds, Sommerfeld… But if the nonlinearity of the equation is taken into account, the analysis is quite more difficult for the fluid, in part because the disappearance of the vertical component is much slower than the disappearance of the electric field in the plasma problem, and also b ecause the limit behavior of the fluid is not universal. Moreover, experiments on fluids are extremely difficult, to the point that it was not clear, up to the Bedrossian-Masmoudi result, whether the commonly observed instabilities of the Couette flow were due to the nonlinearity of the Euler equation, or to the small impurities and defects that inevitably affect an experiment.

To arrive at this remarkable theorem, Bedrossian and Masmoudi adapted our ideas and general strategy about mixing and smoothness, echoes, and stability analysis. But they also had to find new ideas, and apply highly sophisticated technical tools, such as the so-called paraproduct algorithm. The norms which we used were tricky, but their norms are even way more complicated!

While the authors have every reason to be proud of this result, we are also proud to have contributed to its inspiration. With it, the global picture of the long-time behavior of classical systems becomes clearer, as I exposed in the above-mentioned review article which I wrote for physicists (and mathematicians alike). As for our theorem, it can rejoice, and welcome a new member of its family!

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