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Speaker: José A. Carrillo, University of Oxford

Title: "The Landau equation: Particle Methods & Gradient Flow Structure"

Abstract: The Landau equation introduced by Landau in the 1930’s is an important partial differential equation in kinetic theory. It gives a description of colliding particles in plasma physics, and it can be formally derived as a limit of the Boltzmann equation where grazing collisions are dominant. The purpose of this talk is to propose a new perspective inspired from gradient flows for weak solutions of the Landau equation, which is in analogy with the relationship of the heat equation and the 2-Wasserstein metric gradient flow of the Boltzmann entropy. Moreover, we aim at using this interpretation to derive a deterministic particle method to solve efficiently the Landau equation. Our deterministic particle scheme preserves all the conserved quantities at the semidiscrete level for the regularized Landau equation and that is entropy decreasing. We will illustrate the performance of these schemes with > efficient computations using treecode approaches borrowed from multipole expansion methods for the 3D relevant Coulomb case. From the theoretical viewpoint, we use the theory of metric measure spaces for the Landau equation by introducing a bespoke Landau distance dL. Moreover, we show for a regularized version of the Landau equation that we can construct gradient > flow solutions, curves of maximal slope, via the corresponding variational scheme. The main result obtained for the Landau equation shows that the  chain rule can be rigorously proved for the grazing continuity equation, this implies that H-solutions with certain apriori estimates on moments and entropy dissipation are equivalent to gradient flow solutions of the Landau > equation. We crucially make use of estimates on Fisher information-like quantities in terms of the Landau entropy dissipation developed by Desvillettes.
 

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José A. Carrillo is currently Professor of the Analysis of Nonlinear Partial Differential Equations in the Mathematical Institute at the University of Oxford and Tutorial Fellow in Applied Mathematics at The Queen's College. He previously held academic positions at Imperial College London, Universitat Autònoma de Barcelona, and Universidad de Granada, where he did his PhD.

He works on kinetic equations, nonlinear nonlocal diffusion equations. He has contributed to the theoretical and numerical analysis of PDEs, and their simulation in different applications such as granular media, semiconductors and lately in collective behavior. His main scholarship contributions in Analysis of PDEs are in nonlinear Fokker-Planck type equations; the use of optimal transport techniques and entropy methods to analyse theoretically and numerically gradient-flow structures for PDEs and their singularities; the analysis of kinetic models for self-organization, and their implications in mathematical biology and global optimization.

He served as chair of the Applied Mathematics Committee of the European Mathematical Society 2014-2017 and chair of the 2018 Year of Mathematical Biology. He was the Program Director of the SIAM activity group in Analysis of PDE 2019-2020. He has been elected as member of the European Academy of Sciences, Section Mathematics, in 2018 and SIAM Fellow Class 2019. He is currently the head of the Division of Mathematics of the European Academy of Sciences.

He was recognized with the SEMA prize (2003) and the GAMM Richard Von-Mises prize (2006) for young researchers. He was a recipient of a Wolfson Research Merit Award by the Royal Society 2012. He was awarded the 2016 SACA award for best PhD supervision at Imperial College London. He has been Highly Cited Researcher 2015-2020 by Web of Science. He has been awarded an ERC Advanced Grant 2019 to pursue his investigations in complex particle dynamics: phase transitions, patterns, and synchronization.