The HADES seminar on Tuesday, December 2nd, will be at 3:30pm in Room 740.

Speaker: Xueying Yu

Abstract: Strichartz estimates are fundamental tools for understanding the dispersive behavior of solutions to Schrödinger equations. In particular, bilinear Strichartz estimates provide sharper information by capturing interactions between two waves with different frequencies, which play a key role in many nonlinear problems. In this talk, we will first review several classical bilinear Strichartz estimates for the Schr\”odinger equation, with an emphasis on the strategies used in their proofs. We then present a new bilinear estimate in the setting of Schr\”odinger equations on negatively curved spaces.

The HADES seminar on Wednesday, November 19th, will be at 4:00pm in Room 732.

Speaker: Serban Cicortas

Abstract: 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 $n\geq4$ even, which are determined by small scattering data in the distant past or the distant future. 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.

The HADES seminar on Tuesday, November 18th, will be at 3:30pm in Room 740.

Speaker: Xilu Zhu

Abstract: We explore the long-time behavior of multispeed Klein–Gordon systems in space dimension two. In terms of Klein–Gordon systems, the space dimension two is somehow considered as a critical threshold with possible transition from stability to instability. To illustrate this, we first prove a global well-posedness result when Klein–Gordon systems satisfy Ionescu–Pausader non-degeneracy conditions and the nonlinearity is assumed to be semilinear. Second, on the other hand, we construct a specific Klein–Gordon system such that one of the nondegeneracy conditions is violated and its solution has an infinite time blowup, which implies a type of ill-posedness.

The HADES seminar on Tuesday, November 4th, will be at 3:30pm in Room 740.

Speaker: Ryan Martinez

Abstract: We present work, still in progress, with Mihaela Ifrim and Daniel Tataru, which proves global well-posedness, global $L^6$ based Strichartz estimates, and global bilinear spacetime $L^2$ estimates for non-integrable 1D defocusing cubic NLS at the sharp regularity $H^{-1/2 + \epsilon}$ with mild regularity assumptions on the nonlinearity; taking for granted a suitable local well-posedness theory.

In $L^2$, this problem was well understood by Ifrim and Tataru, by using a modified energy method in a frequency localized setting. However, below $L^2$ there are several challenges. First, Christ, Colliander, and Tao show that the initial data-to-solution map fails to even be uniformly continuous locally in time below $L^2$. For the completely integrable problem Harrop-Griffiths, Killip, and Visan proved global (and local) well-posedness in the sense of continuous dependence and local smoothing estimates for the problem in the sharp space. Our work supplements their work by in addition providing global $L^6$ and bilinear $L^2$ estimates, but does not itself depend on complete integrability. To emphasize this, we prove the result for general nonlinearities, of course assuming the existence of a local theory, which at this time, seems out of reach.

The main challenge of this work is that the modified energy method used by Ifrim and Tataru at $L^2$ fails at high frequency below $s = -1/3$. To overcome this we use an infinite series of corrections.

The HADES seminar on Wednesday, October 22st, will be at 4:00pm in Room 732.

Speaker: Sanchit Chaturvedi

Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small-scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. We provide an asymptotic expansion in viscosity of the viscous solution via the help of matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. Furthermore, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. This is joint work with John Anderson and Cole Graham.