Date/Time : 26 Aug 2026 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Zach Turner, Duke University

Chemotaxis plays a crucial role in the facilitation of biological and ecological processes, often to significantly improve reaction rates between two or more interacting species. One important example is signaling in the immune system; immune cells release chemokines when they sense invaders, which mediates the chemotaxis of lymphocytes towards the site of infection. Another example is reproduction, in which spermatozoa follow the chemical gradient of a chemoattractant released by oocytes. In this talk reporting on a work in progress, we consider a system of two PDEs which describe the flux-limited chemotaxis of two reacting densities. We build on the results of An et al. J. Math. Biol. 92, 57 (2026)., by treating the three-dimensional case. We argue that whether significant reaction enhancement occurs is largely governed by the spatial scale of the initial concentration of the attracted density, distinguishing it from two-dimensional chemotaxis. In particular, we show that there are reasonable scenarios where three-dimensional chemotaxis could possibly fail to practically improve biological reaction rates.

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Date/Time : 26 Aug 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Shi-Zhuo Looi, California Institute of Technology

Dyadic shell models are infinite systems of ODEs that retain the quadratic nonlinearity, energy identity, and scaling of the 3D incompressible Navier-Stokes equations. Such models can be realized as averaged Navier-Stokes equations at the PDE level, and some of them blow up in finite time despite satisfying the energy identity, showing that these features do not preclude singularity formation. Among dyadic models with these features, the Obukhov model is distinguished by its regulated cascade mechanism: the inviscid version is globally regular. However, the inviscid proof relies on exact energy conservation, which viscosity destroys. We prove that the viscous Obukhov model is globally regular for all initial data in the critical Sobolev space and above, obtaining a sharp threshold exponent result. The proof introduces a critical rescaling that shows a viscous activation threshold for each shell, and establishes regularity by showing that two competing scales, viscous damping and nonlinear growth, become incompatible at high frequencies. Time permitting, we discuss a proof of global regularity for the Katz-Pavlovic dyadic model close to the sharp threshold exponent (above which blowup is known).

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Date/Time : 19 Aug 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Silu Yin, Hangzhou Normal University

In this talk, I will introduce a Morawetz type estimate for the wave equations in $\R^2$ and develop a non-negative Morawetz type energy. With this estimate in hand, an alternative proof of global existence for the Cauchy problem of quasilinear wave equation with small and compactly supported data is given. What is more, such weighted Morawetz type energy estimate also works for certain wave system with multiple speeds, which can be used to prove global existence of some admissible harmonic elastic wave system in $\R^2$.

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Date/Time : 17 Aug 2026 11:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : In-Jee Jeong, Korea Institute For Advanced Study

For the incompressible Euler equation in vorticity form in two and three dimensions, we consider the initial value problem with data in the critical Sobolev space, W^{d/p,p}(R^d). In the case 1<p<infinity, we give a simple proof of strong illposedness, based on the Hardy inequality and “Key Lemma” of Kiselev and Sverak, using vorticities having the form of “diadic bubbles”. Then, we discuss endpoint cases, namely p = 1 and p = infinity.

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Date/Time : 07 Aug 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Zijian Wang, Nankai University and NUS

In this talk, we will present several new results investigating the (in)stability properties of rotating drops and jets. 

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Date/Time : 04 Aug 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Giovanni Brigati, Institute of Science and Technology, Austria

In this talk, I am proposing a gradient-flow interpretation for a class of non-linear kinetic Fokker—Planck equation (among which the linear case of the Vlasov—Fokker—Planck). The challenge lies in the fact that such dynamics are driven by an interplay of conservative and dissipative effects. Namely, free transport, and diffusion in the velocity variables only. Thus, we rely on the geometry of “kinetic optimal transport”, which is based on Newton’s equations. In this framework, regular curves of measures can be rewritten as solutions to Vlasov’s equations, for a certain force field.

The first main result of the talk is a chain rule for a family of free energy functionals along Vlasov’s equations, which allows us to compute the slope of the energy. Curves of maximal slope correspond to solutions to the non-linear kinetic Fokker—Planck equation. I will then discuss convergence for JKO-like schemes, and further analytical properties of the non-linear kinetic equations, together with a few open questions. My presentation is based on joint works with Guillaume Carlier, Jean Dolbeault, Jan Maas, and Filippo Quattrocchi.

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Date/Time : 28 Jul 2026 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Chengyang Shao, Institut des Hautes Études Scientifiques

In this talk, I will present a mathematical proof for the existence of time-periodic oscillating capillary droplet. The major difficulty is that the non-flat underlying geometry makes toric Fourier techniques unavailable, while the quasilinear nature still requires delicate spectral analysis. The analysis starts with paraproduct on compact Lie groups based on invariant Littlewood-Paley decomposition, followed by a toolset of explicit formulas for paradifferential operators. In case of zonal symmetry on the 3-sphere, an effectively computable symbolic calculus is available. It will then be shown how this paradifferential toolset is used to construct time-periodic solution to the spherical capillary water waves equation, giving mathematical justification of Rayleigh’s formula for oscillating capillary droplet. The paradifferential method not only incorporates the underlying geometry and analysis, but also effectively balances the loss of regularity caused by small denominators, allowing a “fixed point” style proof instead of Nash-Moser/KAM iteration. The talk is based on a working paper co-authored with Haocheng Yang.

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Date/Time : 28 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Zhongkai Tao, Institut des Hautes Études Scientifiques

The initial data of the vacuum Einstein equations on a spacelike hypersurface satisfy the Einstein constraint equations. It is an underdetermined PDE system. We exploit the underdeterminedness to find nontrivial solutions of the Einstein constraint equations with interesting properties. A key ingredient is the construction of Bogovskii-type operators for underdetermined PDEs satisfying a robust symbol condition. The Bogovskii-type operators have favorable support and regularizing properties, allowing us to manipulate the solutions flexibly. This is based on joint work (partly in progress) with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Sung-Jin Oh (UC Berkeley)

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Date/Time : 23 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Xuantao Chen, Sorbonne Universite

We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on two-spheres. Our method can be applied to construct various types of initial data, including data with arbitrarily fast decay, borderline decay, and short-pulse type data. This is joint work with Sergiu Klainerman.

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