Georgia Institute of Technology College of Sciences

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.

We prove that negative energy solutions to the modified Benjamin-Ono (mBO) equation, which is L^2 critical, with mass slightly above the ground state mass, blow-up in finite or infinite time.   These blow-up solutions lie adjacent to those constructed by Martel & Pilod (2017) that have mass exactly equal to the ground state mass.  The solutions that we construct, with mass slightly above the ground state mass, are numerically observable and expected to be stable.  This is joint work with Svetlana Roudenko and Kai Yang.

We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in (n+1) dimensions with n ≥ 4 even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension n poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.

I will begin by presenting our recent results on the spherically symmetric Coulomb waves. Specifically, we study the evolution operator of H= -\Delta+q/|x| where q>0. Utilizing a distorted Fourier transform adapted to H, we explicitly compute the evolution kernel. A detailed analysis of this kernel reveals that e^itH satisfies an L^1 \to L^{\infty} dispersive estimate with the natural decay rate t^{-3/2}. This work was conducted in collaboration with Adam Black, Bruno Vergara, and Jiahua Zhou. Following this, I will discuss our ongoing research on the nonlinear Schrödinger equation, where we apply the distorted Fourier transform developed for the Coulomb Hamiltonian. This work is being carried out in collaboration with Mengyi Xie.