Date/Time : 29 Nov 2023 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaemin Park, University of Basel

In this talk, we study stationary solutions to the 2D incompressible Euler equations in the whole plane. It is well-known that any radial vorticity is stationary. For compactly supported vorticity, it is more difficult to see whether a stationary solution has to be radial. In the case where the vorticity is non-negative, it has been shown that any stationary solution has be radial. By allowing the vorticity to change the sign, we prove that there exist non-radial stationary patch-type solutions. We construct patch-type solutions whose kinetic energy is infinite or finite. For the finite energy case, it turns out that a construction of a stationary solution with compactly supported velocity is possible. This is based on the joint work with Javier Gomez-Serrano and Jia Shi.

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Date/Time : 08 Nov 2023 14:00
Venue : S17 #05-11 (SR5)
Speaker : Xiaoqian Xu, Duke Kunshan University

In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of application is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation, such as the famous Keller-Segel equation for chemotaxis. I will also discuss about the examples of such flows in this talk.

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Date/Time : 19 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

The studies of Poincaré-Einstein manifolds originate from the AdS/CFT correspondence in mathematical physics, which relates quantum gravity theory and conformal field theory. On the mathematics side, numerous conformal invariants were successfully constructed via the microlocal analysis on Poincaré-Einstein spaces, so that it has become a central area in conformal geometry.

A primary object in this talk is the so-called fractional GJMS operators, which provide a uniform treatment of conformally covariant operators with different orders. From the analytic point of view, the degenerate and nonlocal equations arising in this setting are particularly interesting. We will mainly exhibit a series of sharp dimension bounds and Hausdorff measure estimates for the singular set of those solutions. The proof is based on the quantitative differentiation theory for degenerate elliptic operators and involves subtle regularity issues on the Poincaré-Einstein metrics.

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Date/Time : 11 Oct 2023 14:30
Venue : S17 #05-11 (SR5)
Speaker : Chen Shibing, University of Science and Technology of China

Optimal partial transportation is an important generalization of the classical optimal transport problem, and it arises in many applications, such as image retrieval. Mathematically, the problem was first studied by Caffarelli and McCann. They established the existence, uniqueness, and Holder regularity of the free normal and the map near the free boundary while leaving open the question of their higher regularity. In this talk, we will introduce our recent work in this direction.

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Date/Time : 04 Oct 2023 14:30
Venue : S17 #05-11 (SR5)
Speaker : Shiwu Yang, Peking University

It has been shown that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space decay like linear solutions. One hence can define the associated radiation field on the future null infinity. In this talk, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem to the MKG system with general large data by using gauge invariant vector field method. This is based on joint work with W. Dai, H. Mei and D. Wei.

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Date/Time : 17 Aug 2023 13:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Fei, Shanghai Jiaotong University.

This paper concerns the existence of global weak solutions {\it \`a la Leray} for compressible Navier–Stokes–Fourier system with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

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Date/Time : 16 Aug 2023 16:10
Venue : S17 #04-04 (SR3)
Speaker : Naqing Xie, Fudan University

We discuss a uniqueness result of the electro-vacuum Einstein-Maxwell equations near the null infinity. This talk is based on a recent joint work with Xiaokai He (Hunan First Normal, Changsha) and Xiaoning Wu (AMSS, CAS, Beijing).

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Date/Time : 16 Aug 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Chengbo Wang, Zhejiang University

In this talk, I will present a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable C^1 coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy’s inequality, fractional chain rule and fractional Leibniz rule) which are adapted for the problem, the well-posed result is proved by iteration. In addition, our argument yields almost global existence for n=3 and global existence for dimension 4 and higher, when the initial data are small, spherically symmetric with almost critical Sobolev regularity.

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Date/Time : 30 Jun 2023 22:00
Venue : via Zoom
Speaker : Dawei Shen, Sorbonne University

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https://nus-sg.zoom.us/j/83328843795?pwd=bUEvRU5UaVJad054UFUzV2RaUTZsQT09

Meeting ID: 833 2884 3795
Passcode: 706950

I will start with an introduction of the Einstein-vacuum equations and the history of stability problems of black holes. Then, I will talk about the double null foliations of spacetimes, which induce the null structure equations and Bianchi equations. Next, I will introduce the linearization procedure and the linearized equations. Finally, I will present my main result, which provides a new proof of Kerr stability in external regions, which was first obtained by Caciotta-Nicolò in 2010 using vectorfield method.

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