Category Archives: Spring 2026

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.

Global strong well-posedness of the CAO-problem introduced by Lions, Temam and Wang

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

Speaker: Felix Brandt

Abstract: The CAO-problem concerns a system of two fluids described by two primitive equations coupled by nonlinear interface conditions. Lions, Temam and Wang proved in their pioneering work the existence of a weak solution to the CAO-system. Its uniqueness remained an open problem.

In this talk, we show that this coupled CAO-system is globally strongly well-posed for large data in critical Besov spaces. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are controlled by paraproduct methods.

This talk is based on joint work with Tim Binz, Matthias Hieber and Tarek Zöchling.