Date/Time : 24 Feb 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jing An, Shanghai Jiaotong University

A classical problem in the study of the long-time behavior of advection–reaction–diffusion equations is to characterize the precise asymptotics of front locations. The strength of the advection term induces pushed-to-pulled front transitions, which can be captured by the so-called weighted Hopf–Cole transform. In this talk, I will revisit existing results on convergence to traveling waves in both the pulled and pushed regimes, and demonstrate how this transformation can be applied to the chemotaxis–FKPP system, an important nonlocal advection–reaction–diffusion model, to obtain refined front asymptotics for a wider range of parameters.

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Date/Time : 20 Feb 2026 15:00
Venue : S16 #03-06
Speaker : Chengyang Shao, Institut des Hautes Études Scientifiques

This is a mini-course with 3 lectures (1.5 hours each).   The time and venue are:
20 Feb (Friday) 3pm-4:30pm S16-03-06
23 Feb (Monday) 3pm-4:30pm S17-05-11
25 Feb (Wednesday) 3pm-4:30pm S17-05-11

This mini-course equips the audience with essential mathematical tools for the construction of stable and unstable invariant manifolds in hydrodynamical free boundary problems. In particular, it will be shown that the nonlinear instability for a moving capillary water jet follows precisely the Rayleigh-Plateau instability law. Being fully nonlinear and nonlocal in nature, such problems exhibit severe loss of derivatives and cannot be treated by standard methods. Therefore, it has remained largely open. Paradifferential calculus, however, provides a refined framework for analyzing nonlinearities, allowing them to be handled in a manner analogous to linear operators. It yields a unified framework addressing the construction of stable and unstable invariant manifolds for quasilinear PDEs or hydrodynamical free boundary problems.

The course consists of three lectures. The first lecture offers a concise introduction to classical partially hyperbolic dynamical systems and the Lyapunov-Perron method, followed by a review of its extensions to semilinear dispersive PDEs and Eulerian systems with fixed boundaries. The specific difficulties arising in quasilinear and free boundary problems are then discussed. The second lecture provides a brief introduction to the calculus of paradifferential operators, emphasizing its algebraic similarities with classical (pseudo)differential operators. In particular, it will be shown how this calculus applies to the analysis of Dirichlet-Neumann operator in free boundary problems. Finally, the third lecture demonstrates how these tools can be applied to construct stable and unstable invariant manifolds for hydrodynamical free boundary problems, thereby extending the classical Lyapunov-Perron method to this setting.

Some familiarity with the standard theory of hyperbolic PDEs, such as paraproduct decomposition and energy estimates, will be helpful. No prior background in dynamical systems is required.

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Date/Time : 19 Feb 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Alessandro Pigati, Bocconi University

In this talk we will introduce a PDE way to construct hypersurfaces which are critical for anisotropic integrands. Namely, we study energy concentration for rescalings of an anisotropic version of Allen-Cahn.

Besides a Gamma-convergence result, we will sketch a proof of the fact that energy of stable critical points (of the rescaled Allen-Cahn) concentrates along an integer rectifiable varifold, a weak notion of hypersurface, using stability (or finite Morse index) to compensate for the lack of monotonicity formulas. As for the latter difficulty, we will mention in passing some recent results towards an anisotropic version of the Michael-Simon inequality, obtained in collaboration with Guido De Philippis (revisiting previous work of Almgren).

Among the technical ingredients, we will see a generalization of Modica’s bound and a diffuse version of the stability inequality for hypersurfaces.

This is joint work with Antonio De Rosa (Bocconi University).

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Date/Time : 11 Feb 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Lu Yong, Nanjing University

We study the non-relativistic limit of the quadratic and the cubic Klein-Gordon equations. For both the quadratic and the cubic Klein-Gordon equations, we show the validation of Schrodinger approximation up to long time of order $\epsilon^{-1}$ in the non-relativistic limit $\epsilon \to 0$, where $\epsilon$ is proportional to the inverse of the speed of light. We show specific error estimates which coincide with the numerical results.

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Date/Time : 04 Feb 2026 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jiajun Tong, Peking University

We will report recent progress on the 2-D immersed boundary problem with the Navier-Stokes equation, which models coupled motion of a 1-D closed elastic string and ambient fluid in the entire plane. This is based on joint works with Dongyi Wei.

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Date/Time : 28 Jan 2026 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Taoran He, Institut des Hautes Études Scientifiques

In this talk, I will present our recent work on the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system in 1+3 dimensions. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. The inclusion of the Vlasov field introduces several new challenges. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this entire regime. Our results extend the work of Fournodavlos-Rodnianski-Speck from the Einstein-scalar field system to the physically more complex system with the Vlasov field. This is joint work with Xinliang An and Dawei Shen.

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Date/Time : 28 Jan 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jakub Skrzeczkowski, University of Oxford

We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. However, the question of how fast it converges is not well-explored in the literature. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. First, we reprove the recent result of Amassad and Zhou (2025) yielding the rate of √ε using a simpler technique based on the Evolutionary Variational Inequality for both nonlocal and local equations. Next, using numerical simulations, we observe that the rate of √ε is not optimal. Therefore, we obtain a new formula for the Wasserstein distance between two abstract gradient flows which, when applied to the problem, together with Aronson-Benilian estimates, yields the rate of ε, suggested to be optimal by numerics. This is joint work with J.A. Carrillo, S. Fronzoni (Oxford), C. Elbar (Lyon), P. Gwiazda (Warsaw).

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Date/Time : 24 Dec 2025 10:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Yi Zhou, Fudan University

We propose a new physical space approach to bilinear estimates for dispersive equations. We can use this method to give alternative proofs other than the Bourgain space method of low regularity well posedness for dispersive equations. We can also use this method to give alternative proofs of global existence with small initial data for nonlinear wave equations. Moreover, we get new results including the global existence for one dimensional periodic Schrodinger map flow and global well psodness in the critical Besov space of Skew mean curvature flow in high dimensions.

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Date/Time : 24 Dec 2025 11:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Hong Kiat Tan, University of California, Los Angeles

2 x 2 systems of hyperbolic conservation laws arise throughout the physical sciences, such as in isentropic gas dynamics (the p-system), vehicular traffic flow models, shallow water equations, and particle-laden flows equations. A fundamental question for such systems concerns the structural stability of Riemann solutions, that is, whether the qualitative structure of shock and rarefaction waves persists under perturbations of the flux function and initial data.

In this talk, I will present a proof of generic structural stability for Riemann solutions to 2 × 2 systems of hyperbolic conservation laws in one spatial dimension. Specifically, we show that for almost every left and right states, shocks and rarefaction solutions of the same wave type are preserved under perturbations of the flux functions and the initial data. The proof requires the standard assumptions of strict hyperbolicity and genuine nonlinearity, together with geometric conditions motivated by differential topology, such as the regular manifold assumption.

The proof proceeds in two stages. First, we establish that transversality of the wave curves (the Hugoniot loci and rarefaction curves) implies structural stability of the Riemann solution. Next, we invoke a variant of Thom’s parametric transversality theorem to show that this transversality condition holds generically in the left and right states. We conclude by discussing applications of the theorem to numerical analysis and approximation schemes for 2 × 2 systems of conservation laws.

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