Date/Time : 17 Jul 2024 15:00
Venue : via Zoom
Speaker : Feng Shao, Peking University

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https://nus-sg.zoom.us/j/82730991505?pwd=LEzOeoXXrYPflm2EVK5xKzpiQCi6Pt.1

Meeting ID: 827 3099 1505
Passcode: 008679

In this talk, I will discuss the joint work with Dongyi Wei and Zhifei Zhang on the construction of self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for all $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving long-standing open problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.

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Date/Time : 29 May 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaydeep Singh, Princeton University

An important class of finite-time blowup solutions to the Einstein-scalar field system was constructed by Christodoulou (’94), going by the name of k-self-similar naked singularities. In this talk we will motivate the study of these spacetimes from a PDE and general relativity perspective, and discuss the subtle issue of their (in-)stability. Our main results will show how the chief instability mechanism, a blue-shift along the past light-cone of the singularity, is highly sensitive to the space of perturbations considered.

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Date/Time : 02 May 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Pengyu Le, BIMSA

In this talk, we shall introduce the Lorentz polarisation in the Minkowski spacetime, which is the analogy of the polarisation in the Euclidean space. Similar to the application of the Euclidean polarisation to the classical isoperimetric inequality, we shall apply the Lorentz polarisation to study an analogous isoperimetric inequality for the volume of domain of dependence in Minkowski spacetime, and some other geometric variational problems.

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Date/Time : 02 May 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Xuecheng Wang, Tsinghua University

The 3D relativistic Vlasov-Maxwell system is one of the fundamental models in the plasma physics, which describes the dynamics of electron and ions under the electromagnetic field created by particles themselves. In this talk, we discuss the existence of global solution and the propagation of regularity of the 3D relativistic Vlasov-Maxwell system for both the small initial data case and the large initial data case.

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Date/Time : 24 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Arick Shao, Queen Mary University of London

We consider heat operators on a bounded domain, with a critically singular potential diverging as the inverse square of the distance to the boundary of the domain. We establish the first boundary control results for such operators in all spatial dimensions.

First, we prove null controllability for a range of parameters representing the strength of the singular potential, and with control imposed on the entire boundary. Next, we prove a weaker approximate control result, but for the full range of parameters with a viable well-posedness theory, and with the control localised to any open subset of the boundary.

This is joint work with Alberto Enciso (ICMAT) and Bruno Vergara (Brown).

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Date/Time : 17 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Feng Yu, Peking University

Mixing and dissipation enhancement are two closely related concepts in the study of incompressible fluids. They also possess broad applications across various disciplines or even daily lives. In this talk, I will first introduce the idea and recent development of these topics. Then, I will explain how to apply these ideas to studying the advective Cahn-Hillard equation (ACHE). Without the advection, the Cahn-Hillard equation models the phase separation of a binary alloy. Our main result shows that on two and three dimensional tori, if the given flow is sufficiently in mixing (quantified in terms of dissipation time), then phase separation does not occur. Instead, the solution to the ACHE converges to its spatial mean exponentially in the sense of L^2. At the end of the talk, I will outline some open problems in these fields.

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Date/Time : 10 Apr 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Maciej Dunajski, University of Cambridge

We prove that the intrinsic Riemannian geometry of compact cross-sections of any Einstein extremal horizon must admit a Killing vector field. This establishes the rigidity theorem for the extremal Kerr black hole horizon. This extremal horizon is a special case of a quasi-Einstein structure. We shall discuss other examples of quasi-Einstein structures on compact surfaces, including a global result relevant in projective metrizability.

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Date/Time : 27 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Shi-Zhuo Looi, California Institute of Technology

We present a framework and algorithm for deriving the sharp asymptotics of linear and non-linear wave equations on asymptotically flat spacetimes with even spatial dimensions, including non-stationary spacetimes without spherical symmetry assumptions.

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Date/Time : 20 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Shu, Beijing University of Technology

In this talk we discuss quasi-neutral limit problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics and review the known main progress on rigorously establishing quasi-neutrality theory from the view of mathematics. Quasi-neutrality is one basic assumption in physics such as semiconductors and plasma, which is firstly proposed by W.Van Roosbroech in Bell System Tech. J., 1950. Firstly, we introduce some multi-scaling structure stability theory on quasi-neutral limit and boundary layer, initial layer and mixed layer problem for drift-diffusion models in semiconductors. Then, we give some recent progresses on well-posedness theory and quasi-neutral limit problem on Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics.

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