# Research

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Here is a brief presentation of my research. It only reviews a selected list of papers. All the rest of my research papers can be found in my publication page (as well as on arXiv and mathscinet).

My main interest is the study partial differential equations in kinetic theory, but more generally I am interested in partial differential equations, functional inequalities and stochastic processes motivated by physics.

Here is a selection of projects I have been working on:

### Factorization of non-symmetric operators and applications

The celebrated $H$-theorem of Boltzmann is an analytical version of
the Second Law of thermodynamics in the context of the Boltzmann equation on the distribution $f=f(t,x,v)$. It reads $dH(f_t)/dt \leq 0$ with $H(f)=\int f \log f$. A mathematical question which goes back to Boltzmann is to prove constructive exponential rate of relaxation, that is to say that the above $H$-functional (opposite of the physical entropy) decays exponentially fast towards its minimal value under the constraints of the conservation laws. In the nonlinear and far from equilibrium regime, several breakthroughs occur in the late 1990’s and early 2000’s, which culminate with the landmark paper by Desvillettes and Villani published in Inventiones Mathematicae in 2005. Assuming a priori Sobolev regularity and polynomial moments, this was the first truly nonlinear quantitative result of relaxation to equilibrium for the full Boltzmann equation. However it was not able to provide the physical timescale of relaxation, for various deep reasons related to the so-called Cercignani’s conjecture and several interpolation arguments involved.

During my PhD, I started a long-term research program, alone and then in several collaborations, to bridge this gap and solve this question. The first step was the obtaining of new constructive estimates on the spectral gaps of the linearized equation (see the coercivity section and hypocoercivity section below). I then developed a new method for extending the functional space on sectorial semigroups, based on factorization formulas at the level of the resolvent, a detailed splitting of the collision operator and some sectoriality estimates in

Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentialsComm. Math. Phys. (2006), vol. 261, pp 629-672.

This paper proves the exponential $H$-theorem for the spatially homogeneous Boltzmann equation for hard spheres or hard potentials with cutoff. Then together with M. Gualdani and S. Mischler we considered the more difficult non spatially homogeneous case, which involves degenerate non-selfadjoint non-sectorial (and even non hypoelliptic) operators in

(see also [N11] and [A26] for another application). In this paper we solve the question raised in the work of Desvillettes and Villani mentioned above of proving the exponential rate (with optimal rate) for the full Boltzmann equation with periodic boundary conditions (under a priori smoothness conditions) for hard spheres. This is based on the  connection of our new “enlarged” linearized theory together with Desvillettes-Villani’s Theorem far from equilibrium mentioned above. The new method in [A8,A28,N11,A26] is based on some spectral theory factorization approach, with an iterated decomposition of the resolvent operator, together with an abstract Plancherel formula and some hypoelliptic and hypocoercive functional inequalities. It allows to compute decay rates for a certain class of semigroups in Banach spaces or in Hilbert without self-adjointness.

### Kac’s program in Kinetic Theory and propagation of chaos for stochastic interacting particle systems

In a famous paper of 1956, M. Kac proposed, as a first step in the understanding of the derivation of continuum mechanics from the particle systems, the program of deriving the (nonlinear) spatially homogeneous Boltzmann equation. The starting point is a many-particle Markov jump processes in the velocity space which preserves the number of particles, their total kinetic energy and momentum. Kac then introduced a rigorous mathematical formulation of the so-called “molecular chaos” for the law $f^N$ of the process ($N$-particle distribution): for any fixed non-zero integer $k$, the $k$-marginal of the symmetric distribution $f^N$ converges to the $k$-fold tensor product of a one-particle distribution $f$. This reduces the derivation problem into the question of proving the propagation of this chaos property along time for the many-particle process. Kac’s main motivation was the general question of relating the large-time behavior of the Markov jump process with the large-time behavior of the limiting nonlinear PDE.

On the one hand the propagation of chaos for the so-called Maxwell molecules with angular cutoff was done by McKean soon afterwards, but it took almost 30 years before the propagation of chaos was proved by Sznitman for the case of hard spheres interaction in the 1980’s using involved non-constructive probability techniques. On the other hand the issue of the large-time behavior has motived a series of beautiful works by Janvresse, Maslen, and Carlen-Carvalho-Loss in the 2000’s on the so-called “Kac’s spectral gap problem”, but these works had not been able so far to answer the question of passing to the limit in the relaxation time scales, due to the use of a symmetric $L^2$ framework which degenerates in the many-particle limit. In the recent work

Kac’s Program in Kinetic Theory in collaboration with S. Mischler

we prove the propagation of chaos with constructive rate for the stochastic many-particle processes corresponding to two most important realistic interactions: hard spheres and long-ranged (without Grad’s angular cutoff) Maxwell molecules interaction. Most importantly, we answer the main question of Kac about the large-behavior as follows: we obtain the first uniform in time rates on the propagation of chaos, as well as rates of convergence to equilibrium for the marginals of the many-particle system which are uniform in the number of particles. We also answer the more precise question of deriving the famous H-theorem of Boltzmann in this setting by proving the convergence of the microscopic entropy of the particle-system towards the entropy of the limit nonlinear PDE.We prove that for any time $t$:

$\frac1N \int_{\mathcal S^N} f^N _t \log \frac{f^N _t }{\gamma^N} \xrightarrow[N \to +\infty]{} \int_{\mathbb R^3} f_t \log \frac{f_t }{\gamma}$

where $\mathcal S^N$ is the set of microscopic velocities determined by the conservation laws, $\gamma^N$ is the many-particle invariant measure, and $\gamma$ is the Maxwellien equilibrium.

This work introduces a new quantitative method for the study of many-particle systems and mean-field limits based on a stability-consistency semigroup functional framework. The latter reverses the traditional perspective on this problem: the many-particle system is seen as a perturbation of the limit equation. This requires to reframe the evolution problems for functions acting on probabibilies, and this undercovers of “differential stability estimates” on the semigroup of the limit equation in order to control the fluctuations around chaos. This approach has been successfully applied to (not too singular) Vlasov and McKean-Vlasov equations in

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, together with S. Mischler and B. Wennberg

and a work is in progress on developping this new approach for hydrodynamic limits together with D. Marahrens.

### Landau damping and relaxation with constant entropy

The so-called Landau damping, discovered by L. Landau in 1946,
describes the exponential damping of spatial waves in a confined
plasma, and therefore the relaxation of the plasma to electric
neutrality: the electric field also exponentially converges to
zero. The plasma is described by the Vlasov-Poisson equation
$\frac{\partial f}{\partial t} + v\cdot\nabla_x f + F[f]\cdot\nabla_v f = 0$
in the torus, where $f=f(t,x,v)$ is the distribution of electrons, and
$F[f](t,x) = -\int_{\mathbb R^3} \int_{\mathbb T^3} \nabla W(x-y)\, f(t,y,w)\,dw\,dy$
is the self-induced electric force, with the Coulomb microscopic
potential $W(x) = 1/|x|$. This is a reversible nonlinear partial differential equation. Later Lynden-Bell argued in the 1960’s that a similar damping phenomenon should occur for galaxies, also described by the Vlasov-Poisson equation, however with the repulsive electric force replaced by the gravitational attractive force with $W(x)=-1/|x|$.

Since the discovery of Landau, this effect has been widely studied and also debatted due to the reversible nature of the Vlasov-Poisson equation. But theoretical studies only covered the linear level (rigorously) or quasi-linear level (not rigorously). These regimes only deal with very short time scales. In a joint work with C. Villani

On Landau damping (with C. Villani), Acta Mathematica 207 (september 2011), 29-201.

(see also [N6,N7,N10,N12] in the list of publications for review and popularization papers), we have established for the first time the nonlinear Landau damping in infinite time, for Coulomb and Newton interactions (therefore also including the case of galactic dynamics).

The Vlasov-Poisson is one of the most fundamental equations in
statistical mechanics, obtained as a many-body limit of the Newton equations. The Landau damping is an example of emergence of statistical irreversibility for a confined system of particles which follow a reversible evolution, and this result is indeed the first mathematical proof of a relaxation towards equilibrium for a confined gas without collision. It nourishes a lively physical debate anddraw new links with the KAM theory in dynamical systems, and hopefully will open the way for many works in this direction.

### Haff law and self-similarity for granular gases

Granular gases are many-particle systems composed of “grains”, that is macroscopic kind of particles which dissipate energy during a binary collisions $(v,v_*) \longrightarrow (v',v'_*)$. Such particle systems appear for instance in the important physical issue of the formation of planetary rings. They are modeled by a dissipative modified version of the Boltzmann equation where the rules of the collision takes into account the loss of energy towards the internal degrees of freedom $|v'|^2 + |v'_*|^2 < |v|^2 + |v_*|^2$. The mathematical challenge and difficulty is that this modified equation is an open system which therefore do not enjoy a nice entropy structure. Several conjectures were proposed by physicists in the 1980s and 1990s, about the rate of cooling of such gas (Haff), and about the “dynamical scaling” properties of such gas, i.e., about the uniqueness and attractive behavior of a self-similar solution, often called “homogeneous cooling state” in this context (Ernst and Brito). Partial answers had been obtained so far for a simplified “baby” interaction between the grains (pseudo-Maxwell molecules). We have solved these two questions in several joint works [A6,A7,A16,A17] for the very nonlinear case of the so-called inelastic hard spheres interaction (with not too large inelasticity) as widely considered by physicists. This culminates with the work

Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres (with S. Mischler), Comm. Math. Phys. (2009), vol. 288, pp 431-502

These works include several new estimates for the Boltzmann equation, such as the appearance of stretched exponential moment tails, the propagation of Orlicz’s norms, and the extension of Lions regularity result on the gain collision operator to the inelastic case. They also identify the correct quasi-elastic limit, and the associated temperature. Furthermore the paper mentioned above is based on some abstract inverse mapping theorems for some very degenerate perturbations with the help of some quantitative spectral theory estimates.

### Spectral gap and coercivity estimates for non-local collision operators

The question of a spectral gap for the collision operators in kinetic theory goes back to Hilbert in his book…

Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials (with C. Baranger), Rev. Matem. Iberoam. (2005), vol. 21, pp 819-841

Spectral gap and coercivity estimates for the linearized Boltzmann collision operator without angular cutoff (with R. Strain), J. Maths Pures Appl. (2007), vol. 87, pp 515-535