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https://www.Q.com/What-did-C%C3%A9dric-Villani-prove-for-his-Fields-Medal
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Cedric Villani
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What did Cédric Villani prove for his Fields Medal?
Friends,
I have been asked to answer this question, so I will try to do it at the right level, but it often happens that one is not the best to talk about one's own work.
Most of my research work falls into three main fields: collisional kinetic equations (regularity, stability, asymptotic regimes, effect of long-range interactions); collisionless kinetic equations (stability); optimal transport theory (in particular its links to geometry, via the study of Ricci curvature and regularity).
Optimal transport is the field in which my work has been most quoted, most read, and most followed; I also wrote two quite popular research synthesis books on the subject, one of which received the Joseph L. Doob Prize for a mathematical research monograph. However, the Fields Medal was awarded for the contributions in kinetic equations, and more precisely the studies of stability. They are very different for collisional and for collisionless kinetic equations.
In the case when collisions play a key role (Boltzmann equation), the key stabilizing mechanism is entropy production: by producing entropy (disorder), the collisions take the system closer to equilibrium, which is typically a Gaussian distribution in velocity variable, homogeneous in the position variable. Obstacles in mathematically studying this mechanism are many: in particular the complexity and nonlinearity of the collision operator, the fact that collisions are only felt in the velocity variable (their immediate effect is only a modification in velocities, but not in positions, because the particles'size is neligible; in the language of partial differential equations, the equation is degenerate). In this context I solved a conjecture by Cercignani to the effect that at any position, the entropy production is at least (almost) proportional to the difference between the entropy of the velocity distribution, and the entropy of the corresponding Gaussian distribution; these works are partly in collaboration with Giuseppe Toscani. Then I spent much energy on the degeneracy issue, and devised some linear and some nonlinear methods to tackle this issue; these are the contents of several papers with Laurent Desvillettes, and a memoir called "Hypercoercivity". One of the results on this subject is that any solution of the Boltzmann equation, say periodic in the position variable, uniformly smooth in time, will converge to equilibrium at a rate which can be explicitly controlled from the physical parameters around. There is a whole mathematical world in the proof and its ingredients, and it relates to several other areas of mathematics, in particular hypoellipticity, which is the study of degenerate diffusion processes.
In the case when collisions hardly play any role (Vlasov equation, also called collisionless Boltzmann equation), like for electrons repelling each other in a plasma, or stars attracting each other in a galaxy, the stabilizing mechanism is in the mixing of trajectories, which has a tendency to wipe out inhomogeneities. This phenomenon, quite more subtle than entropy production, is called Landau damping in the perturbative regime and violent relaxation in the nonperturbative one. This is considered very important in both plasma physics and astrophysics. While violent relaxation remains a mystery, Landau damping is more prone to treatment, and thus has important theoretical interest. Landau damping was treated for linearized regime as early as the forties. Still, for the nonlinear perturbative regime, it was debated since 1960, when it was understood that there is some apparent self-inconsistency in the linear approximation. On this problem various mathematicians and physicists had varying answers. With Clement Mouhot, we resolved the riddle by showing that Landau damping does hold in the nonlinear perturbative regime: that is, starting from a plasma which is a perturbation of a linearly stable equilibrium, then one remains close to equilibrium for all times, and there is convergence to a homogeneous state. Besides the tricky nature of the (long) proof, it did bring to the understanding of the phenomenon by showing an unexpected parallel with two other famous "paradoxical" statements from classical mechanics of the XXth century. One was the Kolmogorov theory of perturbation of integrable classical dynamical systems, according to which a perturbed mechanical "integrable" system may behave in an ordered way forever, even if there is no physical law to constrain him to do so. The other was the plasma echo phenomenon, according to which a plasma which is excited twice in a row at two different spatial frequencies will emit a spontaneous electric reaction with some delay. Besides these new insights, the proof also showed the tricky role of regularity (that is, the study of how smooth the particle distribution is) in this problem, which is solved only for a very stringent notion of regularity ("Gevrey regularity" or analyticity) and for the periodic setting.
Our techniques were recently used and improved by Bedrossian and Masmoudi to solve a problem in fluid mechanics that had been open for more than a century: the stability of a 2-dimensional incompressible inviscid shear flow. I wrote on this on my Blog, together with some explanations of the Landau damping part, see
Born to be alive
Actually, maybe I should have referred to this post in the first place. Hope this helps!
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