Unique maximal globally hyperbolic developments for the 3D compressible Euler equations in spherical symmetry

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The HADES seminar on Tuesday, April 21st, will be at 3:30pm in Room 740.

Speaker: Dongxiao Yu

Abstract: We study the 3D non-relativistic compressible Euler equations. We assume that the flow is irrotational and isentropic. We treat all equations of state except that of the Chaplygin gas, for which shocks are not expected to form. For an open set of initial data with tails at infinity, we provide a complete and precise description of the maximal globally hyperbolic development (MGHD). The boundary of this MGHD consists of an initial singularity known as the crease, a singular boundary where gradient blow-up occurs, and a Cauchy horizon emanating from the crease. The analysis involves delicate competition between dispersion and resonant nonlinear terms. Moreover, we prove that this MGHD is unique by applying a uniqueness theorem of Eperon-Reall-Sbierski. This is joint work with Leonardo Abbrescia and Jared Speck.

Inverse theorems in additive combinatorics

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The HADES seminar on Tuesday, April 14th, will be at 3:30pm in Room 740.

Speaker: James Leng

Abstract: Let $r_k(N)$ denote the largest subset of the integers from $1$ to $N$ without a $k$-term arithmetic progression. A famous open problem is to find optimal upper bounds on $r_k(N)$. In this talk, I will survey work leading up to the current best upper bounds. Of particular note is the inverse theory of Gowers norms, which I will motivate through both finitary combinatorics and ergodic theory. Time permitting, I will end off by describing my favorite open problem in this area.

Overdamped quasi-normal modes for Schwarzschild black holes

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The HADES seminar on Monday, May 4th, will be at 3:30pm in Room TBD.

Speaker: Tyler Guo

Abstract: Quasi-normal modes (QNM) of black holes are supposed to describe the ringdown of decaying gravitational waves. This ringdown is dominated by the modes closest to the real axis but the overdamped modes (deeper in the complex) remain of interest to physicists and mathematicians.

In this talk, I will give a rigorous definition of QNM as scattering resonances. I will then explain how, in the Schwarzschild case, the problem of finding QNM can be reduced to a study of eigenvalues of a family of non-self-adjoint 1D operators using the method of complex scaling. To explain the structure of the resulting spectral problems I will consider the simpler self-adjoint case and give a proof of the emergence of Bohr–Sommerfeld quantisation rules. If time permits, I will discuss the modifications needed to treat the non-self-adjoint case.

Late-time tails of the Einstein vacuum equation near Minkowski spacetime in wave gauges

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The HADES seminar on Tuesday, March 31th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: It has been expected that wave gauges lead to weak time decay of solutions of the Einstein vacuum equation near Minkowski spacetime, even though Lindblad-Rodnianski used the standard wave gauge to prove global nonlinear stability of Minkowski spacetime in their renowned work. Lindblad further observed the weak $\tau^{-1}$ decay by identifying the source of the time asymptotics. In this talk, I will introduce a new result that proves i) the late time tail; i.e. the leading term in the time asymptotic expansion, is indeed $\tau^{-1}$ in the standard wave gauge, and ii) by slightly modifying the wave gauge condition, one can achieve decay better than $\tau^{-1}$, hence better than expected before. The proof adapts the iteration scheme developed in Luk-Oh to the weak null system of wave equations under wave gauges in order to identify the tail, and chooses the wave gauge so that the zeroth spherical harmonic in the semilinear weak null structure that causes the weak decay is canceled.

On ODE blow-up surfaces for the focusing NLW

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The HADES seminar on Tuesday, March 17th, will be at 3:30pm in Room 740.

Speaker: Warren Li

Abstract: We consider the focusing wave equation for all powers in all dimensions. It is well-known that the equation admits spatially homogeneous blow-up solutions, often dubbed ODE blow-up, terminating in a singular hypersurface at {t=T}. In this talk, we show both that we can construct solutions that (locally) blow-up on an arbitrary spacelike hypersurface, unique up to the choice of a function we call auxiliary scattering data, and that such blow-up hypersurfaces and auxiliary scattering data is stable to perturbations away from the singularity. For instance, we show smooth perturbations of the ODE blow-up solution yields a smooth spacelike blow-up hypersurface. This is based on joint work with Isti Kadar (ETH).

Existence of weak solutions to a model of the geodynamo

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The HADES seminar on Tuesday, March 10th, will be at 3:30pm in Room 740.

Speaker: Tom Schang

Abstract: In this talk, I will discuss a model of the earth’s magnetic field that has previously been simulated numerically, but has not been shown to be well-posed. This model couples solid physics for the electrically conducting inner core with magnetohydrodynamic (MHD) equations in the liquid outer core, as well as the magnetic field outside of the core, which is taken to be non-conducting. I will define and prove existence of Leray-Hopf-type weak solutions for this system. Particular challenges include the transmission conditions of the magnetic field coming from non-constant physical parameters and extending the magnetic field to a non-conducting exterior. To address these problems, we must carefully define a function space and prove appropriate embeddings.

Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

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The HADES seminar on Tuesday, March 3rd, will be at 3:30pm in Room 740.

Speaker: Abishek Rajan

Abstract: We consider the space of smooth gradient expanding Ricci soliton structures on S^1\times \mathbb R^3 and S^2\times \mathbb R^2 which are invariant under the action of SO(3)\times SO(2). In the case of each topology, there exists a 2-parameter family of cohomogeneity one solitons asymptotic to cones over the link S^2\times S^1, as constructed by Nienhaus-Wink and Buzano-Dancer-Gallaugher-Wang. Analogous to work of Bamler and Chen, we define a notion of expander degree for these cohomogeneity one solitons through a properness result. We then proceed to calculate this cohomogeneity one expander degree in the cases of the specific topologies

Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

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The HADES seminar on Tuesday, February 24th, will be at 3:30pm in Room 740.

Speaker: Hongjing Huang

Abstract:

We consider one-dimensional scalar quasilinear Klein–Gordon equations with general nonlinearities, on both $\mathbb R$ and $\mathbb T$.
By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions.
Our main result asserts  that solutions with small initial data of size $\epsilon$ persist on the improved cubic timescale $|t| \lesssim \epsilon^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case 
of $\mathbb R$, we are further able 
to use dispersion in order to extend the lifespan to $\epsilon^{-4}$. This generalizes earlier results 
obtained by Delort, \cite{Delort1997_KG1D}  in the semilinear case.
This joint work with Mihaela Ifrim and Daniel Tataru.

Winning of inhomogeneous badly approximable vectors

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The HADES seminar on Tuesday, February 17th, will be at 3:30pm in Room 740.

Speaker: Liyang Shao

Abstract:

Badly approximable vectors are one of the central topics in Diophantine approximation. Though being null in Lebesgue measure, these vectors are known to have ‘thick’ structure, e.g. full Hausdorff dimension, or even stronger, the winning property that was first proven by Schmidt in the unweighted setup in the 1960s, and in the weighted setup recently by Beresnevich-Nesharim-Yang.

In this talk, we will first briefly introduce how the study of such vectors can be rooted in counting rational points, which is connected to subjects including harmonic analysis and homogeneous dynamics. Then we will describe how non-divergence estimates from homogeneous dynamics can give a winning property of inhomogeneous weighted badly approximable vectors. The second part is joint work with Shreyasi Datta.

Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation

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The HADES seminar on Tuesday, February 10th, will be at 3:30PM on Zoom.

Speaker: Yongming Li

Abstract: In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach, using the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.