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 : 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 : 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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Date/Time : 30 Mar 2022 16:00
Venue : via Zoom
Speaker : Zexing Li, University of Cambridge

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https://nus-sg.zoom.us/j/86113239460?pwd=anRLYzEvaGFQVGVLNVF3dGVRM1pOUT09

Meeting ID: 861 1323 9460
Passcode: 220330

Abstract: The Chern-Simons-Schr\”odinger equation is a nonrelativistic quantum model describing the dynamics of a large number of charged particles in the plane interacting both directly and via a self-generated electromagnetic field. Mathematically, it is a gauged $L^2$-critical Schr\”odinger equation has soliton behavior. Considering the equivariant and self-dual case, Liu-Smith has proved below threshold charge wellposedness via concentration compactness method. In this talk, I will present our result on characterizing dynamics at the threshold charge. This is a joint work with Baoping Liu.

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Date/Time : 20 Oct 2021 09:00
Venue : via Zoom
Speaker : Siqi He, Stony Brook University

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https://nus-sg.zoom.us/j/88336974987?pwd=SVVRVStaVTJ6STVEL1JGRSsxQjVFdz09

Meeting number：883 3697 4987
Password：132347

Abstract: Special Lagrangian submanifolds are a distinguished class of real calibrated submanifolds defined in a Calabi-Yau manifold, which are calibrated by the real part of the holomorphic volume form. Given a compact, smooth special Lagrangian submanifold, Mclean proved that the space of nearby special Lagrangian submanifolds of it could be parametrized by the harmonic 1-forms. In this talk, we will discuss some recent progress on generalizing Mclean’s result to the branched deformations. We will describe how to use multi-valued harmonic functions to construct branched nearby deformations.

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Date/Time : 26 May 2021 15:00
Venue : via Zoom
Speaker : Wen Deng, Chinese Academy of Sciences

Join Zoom Meeting https://nus-sg.zoom.us/j/83402872988?pwd=UWxhOEFKam9oYnk1N1FqT2JCdnFzdz09 Meeting ID: 834 0287 2988 Password: 471879 We consider the stability problem of the Couette flow for the 2D Boussinesq system. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non self-adjoint operator in the perturbation equation that makes the stability possible. In this work, we prove the nonlinear stability of the Couette flow for 2D Boussinesq system with full dissipation and with only vertical dissipation in Sobolev spaces. This is a joint work with J. Wu and P. Zhang.

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Date/Time : 19 May 2021 15:00
Venue : via Zoom
Speaker : Yanlin Liu, Beijing Normal University

Join Zoom Meeting https://nus-sg.zoom.us/j/86563869492?pwd=MlRhL0k5VDhRbVFXTmFUVCtWZ0hsZz09 Meeting ID: 865 6386 9492 Passcode: 488917 We show that any unique global solution (here we do not require any smallness condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some scaling invariant spaces must eventually become a small solution. In particular, we show that the limits of $\|\omega^\theta(t)/r\|_{L^1}$ and $\|u^\theta(t)/\sqrt r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can refine some decay estimates for the axisymmetric solutions.

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Date/Time : 28 Apr 2021 15:00
Venue : via Zoom
Speaker : Chao Liu, Huazhong University of Science and Technology

Join Zoom Meeting https://nus-sg.zoom.us/j/81653239587?pwd=U2lCTS9qV01IejN3VDJ2eDB4dml2UT09 Meeting ID: 816 5323 9587 Password: 918779 Motivated by the astrophysical problems of star formations from molecular clouds, we make the first step on the possible behaviors of certain molecular clouds. This talk (1) establishes the diffuse boundary problem of Euler-Poisson system for describing the evolution of molecular clouds; (2) proves the local existence, uniqueness and continuation principle of the classical solution to the diffuse boundary problem; (3) proves the classical solution (without any symmetry condition) to the diffuse problem blows up at finite time if there is no the first class of global solution and the data is admissible (large scale, irregularly-shaped, expanding and rotational molecular clouds); (4) proves certain singularities can be removed from the boundary if the data is strongly admissible. This result partially answers Makino’s conjecture (1992) on the finite blowup of any tame solution without symmetries and gives the possibilities of star formations, fragmentation and possibilities of formations of shocks and physical vacuum boundary in perfect fluids with Newtonian self-gravity. This talk is based on the article arXiv:2102.11550.

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Date/Time : 27 Apr 2021 15:00
Venue : via Zoom
Speaker : Jinxin Xue, Tsinghua University

Join Zoom Meeting https://nus-sg.zoom.us/j/86384771548?pwd=RnMzcXJrcUtERmFrbHVXMUFrejVsZz09 Meeting ID: 863 8477 1548 Password: 244323 In this talk, we study the dynamics of geodesic flows in perturbed blackhole backgrounds using the techniques of Hamiltonian dynamics. The central object in our work is the blackhole shadow. It gives a normally hyperbolic invariant submanifold in the phase space and is robust under perturbations. We find a gap structure within the blackhole shadow and also use it to construct long chaotic orbit with large oscillation of angular momentum. Moreover, similar methodology applies to the event horizon of Kerr to give chaotic orbit skirting around the event horizon.

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