Date/Time : 21 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Van Tien Nguyen, National Taiwan University

Advanced mathematical techniques have been developed in the last two decades for precise construction of blowup solutions within the class of finite energy. However, the question of a complete classification of blowup dynamics remains open. In this talk, we consider the energy-critical nonlinear heat equation in $\mathbb{R}^3$  as a model example to illustrate our technique to archive such a blowup classification. It consists of two parts: 1) a review of different techniques for constructing finite-energy blowup solutions, and 2) a rigidity result of the finite-energy blowup.  

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Date/Time : 14 Aug 2024 10:00
Venue : S17 #06-11 (SR6)
Speaker : Jinxin Xue, Tsinghua University

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.

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Date/Time : 01 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Armin Schikorra, University of Pittsburgh

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.

I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione.

Joint work with Dorian Martino

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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 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 : 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 : 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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