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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Date/Time : 25 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Bjoern Bringmann, Princeton University

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

This is joint work with S. Cao.

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Date/Time : 19 Nov 2025 15:00
Venue : S16 #05-17
Speaker : Ely Sandine, University of California, Berkeley

The Euler-Poisson system describes the evolution of a self-gravitating compressible fluid. I will present recent work proving the existence of certain self-similar implosion profiles for this system. These solutions belong to a family numerically conjectured by Hunter. I will discuss related PDEs, previous results for this system and aspects of the proof.

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Date/Time : 13 Nov 2025 15:00
Venue : S16 #05-17
Speaker : Myoungjean Bae , Korea Advanced Institute of Science & Technology (KAIST)

In the study of inviscid compressible flows, the Mach number, defined by

M =local speed of a flow/sound speed,

plays a significant role in that not only it determines a physical feature of a flow, but also it determines a mathematical feature of the governing equations. Especially when $M=1$, it is directly related to the degeneracy of a component in the compressible Euler system (or the Euler-Poisson system). The degeneracy brings a great challenge to establish the well-posedness of a boundary value problem for a transonic flow containing a sonic interface.

In this talk, I present two examples of sonic interfaces:
1. a sonic arc occurring in a self-similar flow of the compressible Euler system with a pseudo-transonic transition;
2. a sonic interface occurring in an accelerated transonic flow of compressible Euler-Poisson system.
The most interesting aspect is that the first result shows weak discontinuity across which all the flow variable are continuous but their derivatives are not, while the second result shows regular interface across which all the flow variable are $C^1$. And, it leads us to the following question: What is a general criterion to determine the type of a sonic interface?

This talk is based on collaborations with G.-Q. Chen, B. Duan, M. Feldman, and C. Xie.

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Date/Time : 31 Oct 2025 15:00
Venue : S16 #04-35
Speaker : Vincenzo Vespri, University of Florence

We first consider Harnack-type inequalities for solutions to equations involving the doubly nonlinear operator, a natural bridge between the porous medium and the p-Laplacian operators. This class of degenerate and singular equations exhibits a rich interplay between diffusion and nonlinearity, leading to delicate regularity properties. The development of intrinsic scaling techniques has been essential in establishing local bounds and continuity estimates. After discussing the isotropic case, we turn to anisotropic operators, where diffusion acts differently along distinct spatial directions. This emerging field is still at the beginning of systematic investigation, posing numerous open questions concerning the geometry of intrinsic metrics, Harnack inequalities for general operators, and the role of directional degeneracies in regularity theory.

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Date/Time : 15 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Kai Dong, National University of Singapore

In this talk, we give a new proof of the past nonlinear stability of Kasner solutions for $3+1$-dimensional Einstein vacuum equations with polarized $U(1)$-symmetry. Our method, following Beyer, Oliynyk, and Arturo Olvera-Santamaria \cite{beyer2021fuchsian}, relies on energy estimates for a class of first order hyperbolic PDEs, namely Fuchsian systems.

We show that the perturbed solutions are asymptotically Kasner, geodesically incomplete and crushing at the big bang singularity. This is achieved by reducing the $3+1$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of propagation, we obtain global existence and precise asymptotic behavior of the solutions up to the singularities.

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Date/Time : 15 Oct 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Liang Zhao, National University of Singapore

In this talk, we will consider the simplification from the multi-dimensional (M-D) bipolar Euler–Poisson (2EP) equations to the unipolar Euler–Poisson (1EP) equations of electrons, which are both important models in plasma physics and semiconductors. Based on some rigorous convergence analysis of certain physical parameters, we are able to justify these simplifications for two classes of global-in-time solutions of the M-D 2EP system. The first is the smooth solution of the damped M-D 2EP system near the constant equilibrium state, and the second is the regular solution to the Cauchy problem for the undamped M-D 2EP system with far-field vacuum. The global-in-time error estimates between solutions to 2EP and 1EP are also established.

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Date/Time : 07 Oct 2025 13:00
Venue : S16 #05-18
Speaker : Jiakun Liu, The University of Sydney

In this talk, we will introduce the optimal transportation with some interesting applications in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric and Brunn-Minkowski inequalities in geometry; and an application in image recognition relating to a transport between hypercubes. We will also discuss our recent research on the regularity of free boundaries in optimal transportation.

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