Isabelle Gallagher
The Medal of the Erwin Schrödinger Institute for Mathematics and Physics for the year 2023 is awarded to Isabelle Gallagher, Professor at the Department of Mathematics and its Applications of the École Normale Supérieure in Paris.
Isabelle Gallagher has made fundamental contributions to a number of fields in partial differential equations, from fluid mechanics to nonlinear wave equations, and of course kinetic theory. To analyze these equations, her technical palette is very broad, including energy methods, combinatorics, semiclassical analysis, probability theory, but above all classical carmonic analysis. Her most decisive contributions are:
- Navier-Stokes equation: breaking the scaling barrier for strong solutions
- Rotating fluids: mathematical foundations for fluid flows with strong rotation
- Boltzmann equation: beyond Lanford's result, fluctuations and large deviations
Navier-Stokes equation
The modern mathematical theory of the Navier-Stokes equation can be traced to the work of J. Leray in 1934, who first constructed weak solutions, anticipating on the theory of distributions. These weak solutions were a revolution in mathematical fluid mechanics, but it soon became apparent that they were not the final word: the question of their smoothness and uniqueness has remained opened to this day. An alternative framework is that of strong solutions, which are unique, but only available in a perturbative setting: locally in time, or globally in time for small data. The first result in this direction is due to Fujita and Kato in 1964, and this theory culminated with the work of H. Koch and D. Tataru, who found in 2001 the optimal framework for constructing strong solutions.
The picture resulting from the results which have been recalled is frustrating: strong solutions are unique and smooth, but only known in the small; while weak solutions are known in the large, but their actual properties remain mysterious. Would it be possible to bridge the gap between these two approaches? This is the task Isabelle Gallagher set herself. Her most influential line of research can be described, in the jargon of PDEs, as breaking the barrier of scaling. Namely, strong solutions always require smallness in a scale-invariant norm, there is a natural connection between scale invariance and strong solutions. However, together with J.Y Chemin [1] and then M. Paicu [2], Isabelle Gallagher showed that smallness in a scale invariant norm is not needed to construct global, smooth solutions. The examples constructed by I. Gallagher and her collaborators were very surprising to the community of mathematical fluid dynamics. Pushing this idea further is one of the possible avenues for progress on the Navier-Stokes problem.
Rotating fluids
How the Earth rotation affects the dynamics of fluid flows is one of the fundamental questions of atmosphere and ocean science. A mathematical theory of rotating fluids has been built up over the last decades by Isabelle Gallagher and her collaborators (in particular J.Y. Chemin, R. Danchin, E. Grenier, L. Saint-Raymond). The challenge is to find a mathematical framework that takes into account the relevant physical effects, in particular wave propagation, fluid convection, and viscous dissipation. The simplest model for rotating fluids consists of the Navier-Stokes equation, to which the Coriolis force is added. Isabelle Gallagher and her collaborators pioneered the study of this system, combining methods from viscous fluid mechanics with Strichartz estimates, which account for the wave dispersion [3]; boundary layers have distinct features which were considered in [4]. These articles and a few others were the basis of the monograph [5], which remains the basic reference in the field. After these foundational works, Isabelle Gallagher turned to systems providing a more accurate description of fluids on the Earth surface. One basic problem is that the effect of the Earth rotation on geophysical fluids varies depending on the latitude considered [6]; a second problem has to do with the effect of topography and currents on oceanic and atmospheric dynamics [7]. Wave propagation on variable background is best studied through semiclassical analysis, which becomes a key ingredient.
The Boltzmann equation
The Boltzmann equation is the cornerstone of out of equilibrium statistical physics. This equation was famously poorly received by physicists when it was first proposed by Boltzmann, because of a number of (apparent) paradoxes. Not unrelated are the challenges posed by a rigorous mathematical derivation (from the Newtonian N-body problem), which was first established rigorously by Lanford in 1975.
Lanford's result raised a number of questions: is it possible to go beyond the (small) time scale he reached? Furthermore, while this result is akin to a law of large numbers, is there a corresponding central limit theorem? And what about a large deviation principle?
The monumental task of answering these questions was undertaken by Isabelle Gallagher and her collaborators T. Bodineau, L. Saint-Raymond, S. Simonella, and B. Texier. The first step was to clarify some aspects of the proof of Lanford, which was very terse. This was accomplished in [8], which has become the standard reference for this derivation. Going beyond Lanford's time scale is a famously difficult problem, which is related to the propagation of chaos for the N-body problem. While Isabelle Gallagher and her collaborators were not able to solve this problem, they could follow the dynamics of a tagged particle over large time scales, and show that it converges to Brownian motion [9]. This is the first derivation of Brownian motion from classical dynamics of hard spheres, and its method of proof was very influential. Finally, and most importantly, they were able to describe the fluctuations in the derivation of the Boltzmann equation, but also to establish rigorously a large deviation principle [10]. The corresponding formulas had been conjectured decades ago by Spohn and Rezakhanlou respectively, but their rigorous derivation had remained a challenge for the community of mathematical physics.
[1] J.-Y. Chemin and I. Gallagher. On the global wellposedness of the 3-d navier–stokes equations with large initial data. In Annales Scientifiques de l'Ecole Normale Superieure, volume 39, pages 679–698. Elsevier, 2006.
[2] J.-Y. Chemin, I. Gallagher, and M. Paicu. Global regularity for some classes of large solutions to the Navier-Stokes equations. Annals of Mathematics, pages 983–1012, 2011.
[3] J. I. Chemin, B. Desjardins, I. Gallagher, and E. Grenier. Anisotropy and dispersion in rotating fluids. In Nonlinear partial differential equations and their applications: Coll`ege de France seminar. Volume XIV, pages 171–191. Elsevier, 2002.
[4] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier. Ekman boundary layers in rotating fluids. ESAIM: Control, Optimisation and Calculus of Variations, 8:441–466, 2002.
[5] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier. Mathematical geophysics: An introduction to rotating fluids and the Navier-Stokes equations, volume 32. Clarendon Press, 2006.
[6] I. Gallagher and L. Saint-Raymond. Mathematical study of the betaplane model: equatorial waves and convergence results. Number 107. Soci ́et ́e math ́ematique de France Paris, 2006.
[7] I. Gallagher, T. Paul, and L. Saint-Raymond. On the propagation of oceanic waves driven by a strong macroscopic flow. In Nonlinear Partial Differential Equations, pages 231–254. Springer, 2012.
[8] I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: hard spheres and short-range potentials. European Mathematical Society Zu ̈rich, Switzerland, 2013.
[9] T. Bodineau, I. Gallagher, and L. Saint-Raymond. The brownian motion as the limit of a deterministic system of hard-spheres. Inventiones mathematicae, 203(2):493–553, 2016.
[10] T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. Statistical dynamics of a hard sphere gas: fluctuating boltzmann equation and large deviations. arXiv preprint arXiv:2008.10403, 2020.