Email: (nestor.g@nyu.edu)
Office: Warren Weaver Hall 1005B. Office hours: Tuesdays 3:15-4:15 and by appointment.
Lectures: Tuesdays and Thursdays, 2:00 to 3:15 pm, Warren Weaver Hall 201.
The Boltzmann equation and partial differential equations.
The Boltzmann equation is an integro-differential PDE that is a foundational part of modern statistical mechanics. The study of this equation has been the impetus for foundational developments in the theory of partial differential equations, functional and harmonic analysis, and stochastic processes.
This course is guided by the principle that recent breakthroughs in the study of the Boltzmann equation provide a helpful framework to introduce many important techniques in modern PDE.
A concrete objective of this course is to fully cover two recent results in particular. The first, by Cyril Imbert and Luis Silvestre (2019, Journal of the Euopean Mathematical Society) shows that any potential singularities for the Boltzmann equation must cause singularities detectable at the macroscopic scale of fluid dynamics. The second is a result by the instructor and Luis Silvestre (2025, Acta Mathematica) ruling out singularities altogether for the Landau equation in the spatially homogeneous regime -- later expanded to the Boltzmann equation by Cyril Imbert, Luis Silvestre, and Cedric Villani (2025, Inventiones Mathematicae).
An important tool in all of this will be regularity theory, particularly the works of De Giorgi, Nash, and Moser for elliptic and parabolic equations. De Giorgi-Nash-Moser theory has been crucial in developments throughout PDE beyond the Boltzmann equation, including in probability theory, geometric analysis, and the calculus of variations, and as such this course should be of benefit to students with a broad variety of interests.
The analysis of this equation has been progressing steadily for several decades, specially in recent years. The recent progress in kinetic analogues of De Giorgi, Nash, Moser together with the use of the Fisher information to derive bounds has opened many new research directions. Today kinetic equations is a highly active and lively research field, I hope the students will acquire a solid perspective of some the exciting directions in the field and are encouraged to join the ongoing efforts in this area.
Prerequisites. I will assume familiarity with the basics of real analysis and measure theory in , the basics of Sobolev spaces, and PDE at the level taught in an introductory graduate course.
Course materials. I will be posting weekly lecture notes throughout the seemster. However, the interested readers are encouraged to check out the following reference that I will be following/referring to throughout the semester. PDFs of these papers and lecture notes will be available in Brightspace.
Francesca Anceschi, Mirco Piccinini, Annalaura Rebucci, New perspectives on recent trends for Kolmogorov operators, NdAM: Kolmogorov Operators and their Applications Workshop (2022).
Nestor Guillen and Luis Silvestre, The Landau equation does not blow up, Acta Mathematica (2025).
Nestor Guillen and Luis Silvestre, The Landau equation and Fisher information, Notices of the AMS (2026).
Cyril Imbert , De Giorgi’s regularity theory for elliptic, parabolic and kinetic equations (lecture notes from a graduate course taught at the Simons-Laufer institute in Berkeley, draft).
Cyril Imbert and Luis Silvestre, The Harnack inequality for the Boltzmann equation without cutoff, Journal of the European Mathematical Society (2019).
Cyril Imbert, Luis Silvestre, and Cedric Villani, On the monotonicity of the Fisher information for the Boltzmann equation, Inventiones Mathematice (2025).
Russell Schwab and Luis Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Analysis & PDE (2016).
Luis Silvestre , A new regularization mechanism for the Boltzmann equation, Communications in Mathematics Physics (2016).
Cedric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics (2002).
Cedric Villani, Fisher information in kinetic theory, arXiv preprint arXiv:2501.00925 (2025).
This is a broad list of the topics we will cover, all in the tentative order in which we will cover them.