The APDE seminar on Monday, 3/17, will be given by Rana Badreddine (UCLA) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Robert Schippa ().

Title: The Calogero-Moser derivative NLS equation

Abstract: We consider a type of nonlocal nonlinear derivative Schrödinger equation on the torus, called the Calogero-Sutherland DNLS equation.We derive an explicit formula to the solution of this nonlinear PDE. Moreover, using the integrability tools, we establish the global well-posedness of this equation in all the Hardy-Sobolev spaces \(H^s_+(\mathbb{T}), s\geq0\) down to the critical regularity space, and under a mass assumption on the initial data for the focusing equation, and for arbitrary initial data for the defocusing equation. Finally, a sketch of the proof for extending the flow to the critical regularity \(L^2_+(\mathbb{T})\) will be presented.

The APDE seminar on Monday, 3/3, will be given by Gustav Holzegel (University of Münster) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Sung-Jin Oh ().

Title: Quasi-linear wave equations on black hole backgrounds

Abstract: After a brief introduction to the geometric features of black holes, I will discuss recent joint work with Dafermos, Rodnianski and Taylor introducing a new scheme to prove small data global existence results for quasilinear wave equations on sub-extremal Kerr black hole backgrounds. In the slowly rotating case (see arXiv:2212.14093), a key ingredient is a new purely physical, highly degenerate, spacetime estimate at the top order which avoids understanding the detailed structure of trapped geodesics. In the full sub-extremal case (see arXiv:2410.03639) this estimate needs to be tailored to a finite number of wave packets defined by an appropriate frequency decomposition in the azimuthal and stationary frequencies. The relation to the stability problem for black holes will also be discussed.

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.