On linear inviscid damping around monotone shear flows and singularity structures near boundaries

The HADES seminar on Tuesday, May 11 will be given by Wenjie Lu via Zoom from 3:40 to 5 pm.

Speaker: Wenjie Lu (University of Minnesota)

Abstract: Hydrodynamic stability is one of the oldest problems studied in PDEs. In this talk, I will introduce results related to the linear stability of monotone shear flows with boundaries. If the vorticity vanishes near boundaries, one can obtain optimal decay estimates in Gevery spaces. However, the boundary effect is significant and can be an obstruction for the scattering of the vorticity in high regularity spaces. In order to understand the asymptotic behavior more clearly, we need to have a full picture of the singularity structure of the generalized eigenfunctions. It turns out that we can actually track singularities of arbitrary derivatives of the generalized eigenfunctions. With this, we can get arbitrary many terms in the asymptotic, not only the main term. This is a recent joint work with Hao Jia.

The shock formation problem: an overview

The HADES seminar on Tuesday, April 27 will be given by Federico Pasqualottovia Zoom from 3:40 to 5 pm.

Speaker: Federico Pasqualotto

Abstract: Shock waves are a fundamental phenomenon which appears in the context of compressible fluid flow.In this talk, we will review the problem of shock formation, focusing on various techniques which are suitable to study the problem in one and several space dimensions.

The instability of Anti-de Sitter spacetime for the Einstein–scalar field system

The HADES seminar on Tuesday, March 30 will be given by Georgios Moschidis via Zoom from 3:40 to 5 pm.

Speaker: Georgios Moschidis

Abstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. According to this conjecture, there exist arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting  boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.  In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric  Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. 

On the Stability of Self-Similar Blowup in the Strong-Field Skyrme Model

The HADES seminar on Tuesday, February 16 will be given by Michael McNulty via Zoom from 3:40 to 5 pm.

Speaker: Michael McNulty (UC Riverside)

Abstract: The strong-field Skyrme model is a particular limiting case of the Skyrme model, a geometric field theory from nuclear physics and a quasilinear modification of the nonlinear sigma model (wave maps). Singularity formation in these models is known to serve as great toy models for physically realistic situations like gravitational collapse in Einstein’s equation of general relativity. This limit restores the scale invariance of the equation of motion allowing for the existence of self-similar solutions, i.e., singularity formation. In this talk, we discuss work in progress toward establishing the nonlinear stability of an explicitly known self-similar solution to the equation of motion for the strong-field Skyrme model. In addition, we discuss the current strategy used for studying nonlinear stability of self-similar blowup in the broader context of energy supercritical wave equations and the new challenges one faces when applying this to the strong-field Skyrme model. 

Stability analysis of nonlinear fluid models around affine motions

The HADES seminar on Tuesday, February 9 was given by Calum Rickard via Zoom from 3:40 to 5 pm.

Speaker: Calum Rickard (USC)

Abstract: The compressible Euler equations describe the flow of an inviscid ideal gas. The global-in-time existence of strong solutions is proven for three distinct compressible Euler systems in the presence of vacuum states which describe different physical and mathematical situations. Our results are obtained through perturbations around various forms of expanding background affine motions. The particular properties of the different affine motions present new mathematical challenges to the stability analysis in each case.

A Nonnegative Version of Whitney’s Extension Problem

The HADES seminar on Tuesday, November 17th will be given by Kevin O’Neill via Zoom from 3:40 to 5 pm.

Speaker: Kevin O’Neill (UC Davis)

Abstract: Whitney’s Extension Problem asks the following: Given a compact set $E\subset\mathbb{R}^n$ and a function $E\to\mathbb{R}$, how can we tell if there exists $F\in C^m(\mathbb{R}^n)$ such that $F|_E=f$? The classical Whitney Extension theorem tells us that, given potential Taylor polynomials $P^x$ at each $x\in E$, there is such an extension $F$ if and only if the $P^x$’s are compatible under Taylor’s theorem. However, this leaves open the question of how to tell solely from $f$. A 2006 paper of Charles Fefferman answers this question. We explain some of the concepts of that paper, as well as recent work of the speaker, joint with Fushuai Jiang and Garving K. Luli, which establishes the analogous result when $f\geq 0$ and we require $F\geq 0$.

Network Control on Scattering Manifolds

The HADES seminar on Tuesday, October 20th will be given by Ruoyu Wang via Zoom from 3:40 to 5 pm.

Speaker: Ruoyu Wang (Northwestern)

Abstract: Lebeau (’93) suggested that on a compact manifold the damped waves decay logarithmically with merely some smooth damping inside a small open set. This phenomenon exploits the Carleman estimate establishing the exponentially weak observability. The natural generalisation of “small” sets to establish such exponential weak observability on noncompact manifolds is the Network Control Condition, formulated by Burq and Joly (’16), an condition requiring an upper bound of distance from the region of observability to any points on the manifold. We will show that this condition guarantees the exponentially weak observability on cylinders and scattering (asymptotically conic) manifolds, and henceforth derive a logarithmic decay for the damped waves in the high frequency regime, via a n-weight Carleman argument.

Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition

The HADES seminar on Tuesday, October 13th will be given by Dongxiao Yu via Zoom from 3:40 to 5 pm.

Speaker: Dongxiao Yu

Abstract: In this talk, I will discuss the long time dynamics of a scalar quasilinear wave equation in three space dimensions. This equation satisfies the weak null condition introduced by Lindblad and Rodnianski, and it admits small data global existence which was proved by Lindblad. I will present a proof of the existence of the modified wave operators for this quasilinear wave equation. This is accomplished in three steps. First, we derive a new reduced asymptotic system by modifying Hörmander’s method. Next, we construct an approximate solution to the quasilinear wave equation by solving the reduced system given some scattering data. Finally, we prove that the quasilinear wave equation has a global solution which agrees with the approximate solution at infinite time.

Fractal Weyl Laws for Scattering Resonances

The HADES seminar on Tuesday, October 6th will be given by Yonah Borns-Weilvia Zoom from 3:40 to 5 pm.

Speaker: Yonah Borns-Weil

Abstract:It is well-known that on a bounded domain, the number of eigenvalues of the stationary Schrödinger equation in a given interval follow asymptotics known as Weyl laws. In scattering theory however, we work in an unbounded domain, and such operators need no longer have any eigenvalues. Instead, they have complex resonances, which satisfy a general upper bound (but not necessarily a lower bound) due to Sjöstrand that is analogous to the Weyl law. We present this bound, and describe how the proof must change if we instead count the eigenvalues in an h-dependent region. Following this, we present a result due to Sjöstrand and Zworski, which says that the exponent in such a Weyl law can depend on the fractal dimension of a hyperbolic trapped set. At the end, we will discuss what can still be said when the trapped set is not hyperbolic. Along the way, we will attempt to point out many of the standard “tricks” that are commonly used in scattering theory.

Singular solutions to the Einstein equations

The HADES seminar on Tuesday, September 22nd will be given by Jonathan Luk via Zoom from 3:40 to 5 pm.

Speaker: Jonathan Luk, Stanford

Abstract: I will discuss the construction of a class of low-regularity (merely $W^{1,2}$) solutions to the Einstein vacuum equations which have the property that the solutions are foliated by $2$-spheres so that the metric is more regular along the tangential directions of the $2$-spheres. I will first discuss a model semi-linear problem, then introduce the relevant geometric setup and give a sketch of the proof. This type of singular solutions is relevant to the problems of impulsive gravitational waves, high-frequency limits, null dust shells and the formation of trapped surfaces in general relativity (discussed in the Analysis and PDE seminar on 9/21).