We briefly introduce the restriction theory in harmonic analysis and its connections with PDEs through Strichartz estimate.
We then discuss the Kakeya and multilinear Kakeya estimates, which naturally arise from restriction theory.
The main part of the talk will focus on Larry Guth’s proof of the multilinear Kakeya estimate via the induction on scales method.
I will discuss recent progress on the vanishing-viscosity limit of the two-dimensional Navier–Stokes equation. Our approach is Lagrangian and probabilistic:
1. We develop a stochastic counterpart of the DiPerna–Lions theory to construct and control stochastic Lagrangian flows for the viscous dynamics.
2. We also establish a large-deviation principle that quantifies convergence to the Euler dynamics.
This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.
