The APDE seminar on Monday, 2/10, will be given by Hannah Cairo (UC Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Anuj Kumar ().

Title: A Counterexample to the Mizohata-Takeuchi Conjecture

Abstract: We derive a family of $L^p$ estimates on the X-Ray transform of positive measures in $\mathbb{R}^d$, which we use to construct a $\log R$-loss counterexample to the Mizohata-Takeuchi conjecture for every $C^2$ hypersurface in $\mathbb{R}^d$ that does not lie in a hyperplane. In particular, endpoint multilinear restriction estimates cannot be derived from MT-type estimates.

The APDE seminar on Monday, 2/03, will be given by Sameer Iyer (UC Davis) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Anuj Kumar ().

Title: Beyond the Goldstein Singularity & Recent Advances in the Boundary Layer Theory

Abstract: The Prandtl system describes the flow in the boundary layer that forms near the boundary when taking the inviscid limit in the Navier-Stokes system. It was first derived in 1904 by Prandtl. Many important questions related to the Prandtl system and the inviscid limit are still open. We will review some recent advances in this direction. Reversal occurs after the separation point, and is characterized by regions in which the velocity changes sign. The classical point of view of treating the stationary Prandtl system as an evolution equation in the tangential variable x completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. This is a joint work with Nader Masmoudi.