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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - MAFRAN (Mathematical Frontiers in the Analysis of Many-particle Systems)

Teaser

Summary

The recent growing mathematical activity around the partial differential equations of kinetic theory has led to deeper and deeper conceptual breakthroughs. This has opened new paths, and has created new frontiers with other cutting-edge fields of research.

These frontiers correspond to three combined levels: the dialogue with the world-leading research community; the uncovering of deep new connexions and methods through this interplay; the possibilities of making significant progresses on a fundamental open problem:
I. with the elliptic regularity community (regularisation for nonlocal collision operators, De Giorgi-Nash theory): the main challenge is the well-posedness of the Landau-Coulomb equation;
II. with the dispersive and fluid mechanics equations communities (nonlinear stability driven by phase mixing): the main challenge is the damping stability of non-spatially homogeneous structures;
III. with the dynamical system and probability communities (mean-field and Boltzmann-Grad limits): the main challenge is the rigorous derivation of the fundamental equations of statistical mechanics on macroscopic times;
IV. with the applications to biology, ecology and statistical physics (emerging collective phenomena for open many-particle systems): the main challenge is the understanding of steady or propagation front solutions and their stability outside the realm of the 2d principle of thermodynamics.

With my grant, my goal is to create a world-class research centre devoted to these frontiers, which can rapidly lead to key advances with potential impact in mathematical analysis and fundamental physics (plasma physics, statistical mechanics).

Work performed

The PI has published one paper and submitted three preprints in collaboration with Cyril Imbert and Luis Silvestre in the newest line of research (Axis I of the proposal: Challenges in the regularity theory and stability for long-range non-local kinetic equations), has published one paper and submitted a new preprint in collaborations (main collaborators Eric Carlen and Joel Lebowitz) along the Axis IV of the proposal (Challenges in the asymptotic behaviour of open systems), and has made progresses in collaboration with J. Bedrossian and N. Masmoudi along Axis II (Challenges on phase mixing and nonlinear stability) and some progresses alone along Axis III (Challenges in the propagation of chaos).

A PhD student in the group, Megan Griffin-Pickering (not paid from grant funds) has submitted several preprints, one of which solving an important long-standing question by constructing global smooth solutions to the Vlasov-Poisson system with massless electrons that is extensively used in plasma physics. Among the other (postdoctoral) members of the group: Jessica Guerand has published a paper providing the first constructive proof of De Giorgi intermediate value lemma in H^1 for parabolic equations, and Ivan Moyano has released four preprints on the Vlasov-Navier-Stokes system, the link between kinetic equations and fractional diffusion, and spectral inequalities for the SchrÃ¶dinger operator.

Final results

There were new theorems and answers to questions we did not have answer for before, hence progress beyond the state of the art. Socio-economic impact not relevant in pure mathematics.