Periodic Reporting for period 4 - MAFRAN (Mathematical Frontiers in the Analysis of Many-particle Systems)
Reporting period: 2022-03-01 to 2023-08-31
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).
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).
The PI has published three papers in collaboration with Cyril Imbert and Luis Silvestre and one with Jessica Guerand 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 two papers in collaborations with Eric Carlen and Joel Lebowitz and one long seminal paper with Emeric Bouin along the Axis IV of the proposal (Challenges in the asymptotic behaviour of open systems), and has made important progresses in collaboration with J. Bedrossian and N. Masmoudi along Axis II (Challenges on phase mixing and nonlinear stability) and, towards the end of the grant, has developed a new theory of hydrodynamic limit for stochastic lattice systems, making thus important progresses alone along Axis III (Challenges in the propagation of chaos); a simpler form of this new approach was published in the last period of the grant, while several follow-up papers are now being written about it. More than ten papers were also written by the postdocs and grad students of the group during the duration of the grant, with a vibrant scientific activity. These works have also contributed progresses along the axis of the project, in particular the graduate student Dominic Wynter (hired on the grant) made key progress on the understanding of propagation of chaos for the system of non-signed vortices. Yuzhe Zhu has published new results on the regularity theory of open kinetic systems with boundary conditions.
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 are delayed and unpredictable in pure mathematics.