Paradifferential calculus, introduced by Bony, is a central tool for the study and treatment of nonlinear partial differential equations. It allows the decomposition of products and operators using Coifman and Meyer paraproducts, thus offering a refined approach to problems of regularity and propagation of singularities. At the interface of harmonic analysis and microlocal analysis, it connects the Littlewood-Paley decomposition and Sobolev, Zygmund, or Besov spaces to the symbolic frameworks developed by Kahn-Nirenberg and Hi:irmander. It provides a robust methodology for linearizing nonlinear operators, comparing them with pseudodifferential calculus, and constructing suitable operators such as the paracomposition operators introduced by Alinhac. In this course, we will present the foundations of this calculus and its applications, emphasizing its unifying role between harmonic analysis, the theory of nonlinear PDEs, and the theory of dynamical systems. We will thus address a wide range of applications: Coifman-Meyer type bilinear inequalities, the study of commutators and elliptic regularity, free boundary problems (notably via the Dirichlet-Neumann operator), as well as the analysis of the Euler and Schridinger equations. We will also see how, in KAM (Kolmogorov-Arnold-Moser) theory, para-differential reduction allows us to overcome small divisor problems. This course presents the foundations of paradifferential calculus as introduced by Bony, its connection to the paraproducts of Coifman and Meyer and the microlocal analysis of Kahn-Nirenberg and Hiermander, as well as its recent applications to nonlinear equations, conjugation theories (KAM theorems), and regularity problems in the theory of elliptic equations.
This course is an introduction to the mathematical analysis of the Boltzmann equation in the kinetic theory of gases. Hydrodynamic limits of the kinetic theory of gases are also discussed in detail, most notably the limits leading to the Euler, or to the Navier-Stokes equations used in gas dynamics.
