Georgia Institute of Technology College of Sciences

We study the sharp asymptotics for a class of quasilinear wave equations satisfying a weak null condition but not the classical null condition in three spatial dimensions. We prove that the asymptotics are very different from those for the equations satisfying the classical null condition. In particular, at leading order, the solution displays a continuous superposition of decay rates.

Moreover, we show that any solution that decays faster than expected in a compact spatial region must vanish identically. The talk is based on joint work in progress with Jonathan Luk and Dongxiao Yu. 

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p > 1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^+$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of ``resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\varepsilon)$-size perturbations of his initial datum. We discover a uniqueness threshold $\varepsilon \sim \nu^{\kappa_{\rm c}}$, below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. Joint work with Maria Colombo and Giulia Mescolini (EPFL).

The aim of this talk is to discuss recent advances around the convergence problem in mean field control theory and the study of associated nonlinear PDEs.

We are interested in optimal control problems involving a large number of interacting particles and subject to independent Brownian noises. As the number of particles tends to infinity, the problem simplifies into a McKean-Vlasov type optimal control problem for a typical particle. I will present recent results concerning the quantitative analysis of this convergence. More precisely, I will discuss an approach based on the analysis of associated value functions. These functions are solutions of Hamilton-Jacobi equations in high dimension and the convergence problem translates into a stability problem for the limit equation which is posed on a space of probability measures.

I will also discuss the well-posedness of this limiting equation, the study of which seems to escape the usual techniques for Hamilton-Jacobi equations in infinite dimension.