Regularity Theory of Kinetic Equations with Rough Coefficients - Part 1

This lecture is the first part of a series exploring the regularity theory of kinetic equations with rough coefficients, presented by Clément Mouhot from the University of Cambridge. Dive into the extension of De Giorgi (1958) and Nash (1959) theory, which solved Hilbert's 19th problem and was a major contribution to 20th century PDE analysis. Learn about Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients, and how this theory was developed by Moser (1960-1964) to include the Harnack inequality. Explore recent advances in extending these concepts to hypoelliptic PDEs in kinetic theory, with particular focus on the Kolmogorov equation with rough coefficient matrices in kinetic diffusion. Discover quantitative robust methods based on trajectory construction and their connections to control theory and hypocoercivity through Mouhot's collaborative work with Dieter, Hérau, Hutridurga, Niebel, and Zacher. This 1 hour 55 minute lecture is available on CARMIN.tv, a French video platform specializing in mathematics and interdisciplinary scientific content.