Presentation of work on (asymptotic) analysis of non-linear time-dependent PDEs modelling fast, self-interacting charged fermions in relativistic quantum mechanics, from the Dirac-Maxwell to Vlasov/Euler-Poisson equations. I focus on intermediate first and second order in 1/c models, such as the Pauli-Poisswell and Euler-Darwin equations, which are useful e.g. in plasma physics.
In a recent work, R. Carmona, Q. Cormier and M. Soner proposed a mean field game based on the classical Kuramoto model, originally motivated by systems of chemical and biological oscillators. Such MFG model exhibits several stationary equilibria, and the question of their ability to capture long time limits of dynamic equilibria is largely open. I will discuss in the talk how to show that, up to translations, there are two possible stationary equilibria only - the incoherent and the synchronised one - provided that the interaction parameter is large enough. Finally, I will present some local stability properties of the synchronised equilibrium.
In this talk, we present new findings on the regularity of first-order mean field games systems with a local coupling. We focus on systems where the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function for small densities. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, when f(m) = mθ with θ > 0, we prove that the speed of propagation is finite. In this case, we establish that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.
The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth outside a set of singular points. Explicit examples show that the singular set could be, in general, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension n≤4. The aim of this talk is to give an overview of these results.
In between elliptic PDEs, which do not depend on time (think of the steady-state Stokes equations), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the instationary Stokes equations), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic.
I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right. This is highlighted by a direct method for showing a priori Lp estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems, for example the Stokes equations and boundary value problems of Agmon-Douglas-Nirenberg type. In the talk, I will present these ideas and show how they can be extended to the setting of weighted Lp estimates, which is advantageous for extrapolation techniques and rougher boundary data.
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