## Final Report Summary - MATKIT (Mathematical Aspects of Kinetic Theory)

Mathematical Kinetic Theory is a field at a cross-road between Partial Differential Equations (PDEs) and probability. It is strongly connected to theoretical physics in fluid mechanics and the foundation of statistical mechanics. This is a recent mathematical field but it has grown steadily in the last two decades and is flourishing. The goals of this project were focused on the understanding of regularity and asymptotic behaviours of collision processes, transport dynamics, and how they mix (hypocoercivity), and finally the derivation of the corresponding PDEs from particle systems. This has been unquestionably successful and several progresses have been made on these questions.

The most important progress made has been on this last axis (the derivation) with a 150 pages paper written by the PI and Mischler in Inventiones Mathematicae answering several conjectures made by Kac about the propagation of chaos for a class of many-particle jump processes that yield the space-independent Boltzmann equation in the mean-field limit. In particular this paper obtained the first uniform-in-time estimates of propagation of chaos. In another paper of the PI, Mischler and Wennberg and accepted in a top probability journal (PTRF), the new approach is extended to the derivation of Vlasov dynamics.

The second highlight is the long preprint submitted by the PI, Gualdani and Mischler introducing a general method of factorisation of non-symmetric operators to obtain quantitative decay rate on semigroups in Banach spaces, and then – at the price of many technical difficulties – using it to answer a conjecture by Villani (2008) on the mathematical justification of the H-theorem of Boltzmann. This preprint has already had a clear influence in the field: the abstract method has been used already by two other research groups and two PhD students.

The last highlight is the recent preprint of the PI with Bedrossian and Masmoudi that gives a simplified proof and new results about Landau damping for Vlasov dynamics. This is likely to open the path for further research in this field by providing more robust and more classical tools in the study of this damping induced by phase mixing. Inspired by the original paper of the PI and Villani on Landau damping, and also several scientific interactions with the PI, Bedrossian and Masmoudi gave the first result of inviscid damping for the 2D Euler equation. This should also open the way for many further research in this field in involving the research group of the PI, and confirms that the choice of research projects for this ERC were timely and fruitful.

And indeed in the last period of the grant, we have posted online and submitted a new preprint developing this new method of proof to give the first damping results in the whole space.

Finally many significant results about the regularity of the Boltzmann and Landau equation, hydrodynamical limit, fractional diffusive limit, and propagation of chaos have been obtained by the many PhD students and postdocs in the research group created by the PI. An international conference in kinetic theory was organised by the PI in Cambridge in June 2013 and beforehand a summer school in kinetic theory was co-organised by the PI with Filbet in Lyon in April 2012. They both attracted some of the best mathematicians working on kinetic theory. Among the many works of the junior members of my group, I can highlight for instance the beautiful paper by Mikaela Iacobelli recently posted on the arXiv and completed during her time in Cambridge in the ERC research group, about new subtle stability estimates in weak topology applied to the understanding of the quasi-neutral limit in plasma physics. Finally a last international conference in kinetic theory was organized by the research group of the ERC grant in May 2016, with more than 70 participants from many countries. The research group funded by the ERC has also seen an intense flow of academic visitors, from Europe, USA and Asia.

All this shows clearly the great success of the project, both in terms of new mathematical results, and in terms of the construction of the world-class research group.

The most important progress made has been on this last axis (the derivation) with a 150 pages paper written by the PI and Mischler in Inventiones Mathematicae answering several conjectures made by Kac about the propagation of chaos for a class of many-particle jump processes that yield the space-independent Boltzmann equation in the mean-field limit. In particular this paper obtained the first uniform-in-time estimates of propagation of chaos. In another paper of the PI, Mischler and Wennberg and accepted in a top probability journal (PTRF), the new approach is extended to the derivation of Vlasov dynamics.

The second highlight is the long preprint submitted by the PI, Gualdani and Mischler introducing a general method of factorisation of non-symmetric operators to obtain quantitative decay rate on semigroups in Banach spaces, and then – at the price of many technical difficulties – using it to answer a conjecture by Villani (2008) on the mathematical justification of the H-theorem of Boltzmann. This preprint has already had a clear influence in the field: the abstract method has been used already by two other research groups and two PhD students.

The last highlight is the recent preprint of the PI with Bedrossian and Masmoudi that gives a simplified proof and new results about Landau damping for Vlasov dynamics. This is likely to open the path for further research in this field by providing more robust and more classical tools in the study of this damping induced by phase mixing. Inspired by the original paper of the PI and Villani on Landau damping, and also several scientific interactions with the PI, Bedrossian and Masmoudi gave the first result of inviscid damping for the 2D Euler equation. This should also open the way for many further research in this field in involving the research group of the PI, and confirms that the choice of research projects for this ERC were timely and fruitful.

And indeed in the last period of the grant, we have posted online and submitted a new preprint developing this new method of proof to give the first damping results in the whole space.

Finally many significant results about the regularity of the Boltzmann and Landau equation, hydrodynamical limit, fractional diffusive limit, and propagation of chaos have been obtained by the many PhD students and postdocs in the research group created by the PI. An international conference in kinetic theory was organised by the PI in Cambridge in June 2013 and beforehand a summer school in kinetic theory was co-organised by the PI with Filbet in Lyon in April 2012. They both attracted some of the best mathematicians working on kinetic theory. Among the many works of the junior members of my group, I can highlight for instance the beautiful paper by Mikaela Iacobelli recently posted on the arXiv and completed during her time in Cambridge in the ERC research group, about new subtle stability estimates in weak topology applied to the understanding of the quasi-neutral limit in plasma physics. Finally a last international conference in kinetic theory was organized by the research group of the ERC grant in May 2016, with more than 70 participants from many countries. The research group funded by the ERC has also seen an intense flow of academic visitors, from Europe, USA and Asia.

All this shows clearly the great success of the project, both in terms of new mathematical results, and in terms of the construction of the world-class research group.