Georgia Institute of Technology College of Sciences

In this presentation the analytical background of nonlinear observers based on minimal energy estimation is discussed. Originally, such strategies were proposed for the reconstruction of the state of finite dimensional dynamical systems by means of a measured output where both the dynamics and the output are subject to white noise. Our work aims at lifting this concept to a class of partial differential equations featuring deterministic perturbations using the example of a wave equation with a cubic defocusing term in three space dimensions. In particular, we discuss local regularity of the corresponding value function and consider operator Riccati equations to characterize its second spatial derivative.

I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706

It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system exhibits global-in-time regularity as long as the hyper-diffusion exponent is greater or equal to 5/4.  One should note that at 5/4, the system is critical, i.e., the energy level and the scaling -invariant level coincide. What happens in the super-critical regime, the hyper-diffusion exponent being strictly between 1 and 5/4 remained a mystery. 

The goal of this talk is to demonstrate that as soon as the hyper-diffusion exponent is greater than 1, a class of monotone blow-up scenarios consistent with the analytic structure of the flow (prior to the possible singular time) can be ruled out (a sort of 'runaway train' scenario). The argument is in the spirit of the regularity theory of the 3D HD NS system in 'turbulent scenario' (in the super-critical regime) developed by Grujic and Xu, relying on 'dynamic interpolation' – however, it is much shorter, tailored to the class of blow-up profiles in view. This is a joint work with Aseel Farhat.

Sampling from the Gibbs distribution is a long-standing problem studied across various fields. Among many sampling algorithms, Langevin dynamics plays a crucial role, particularly for high-dimensional target distributions. In practical applications, accelerating sampling dynamics is always desirable. It has long been studied that adding an irreversible component to reversible dynamics, such as Langevin, can accelerate convergence. Concrete constructions of irreversible components have also been explored in specific scenarios. However, a general strategy for such construction is still elusive. In this talk, I will introduce the concept of leveraging irreversibility to accelerate general dynamics, along with the quantification of irreversible dynamics. Our theory is mainly based on designing a modified entropy functional originally developed for linear kinetic equations (Dolbeault et al., 2015).