Date/Time : 30 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Simon Eberle, Basque Center for Applied Mathematics, Spain

We will investigate an old problem posed by Isaac Newton on shapes that do not admit gravity in the cavity. We will see that it can be equivalently formulated as a question on the rigidity of global solutions of the obstacle problem, a question that recently has been answered in full generality.

This is joint work with A. Figalli, H. Shahgholian and G. S. Weiss.

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Date/Time : 02 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Wu Yifei, Tianjin University

In recent years, more and more attention has been paid to the low regularity numerical study based on the practical needs. In this talk, some Fourier integrators are proposed for solving the KdV equation and the nonlinear Schrodinger equation. The designation of the scheme is based on the exponential-type integration, Splitting methods and the Phase-Space analysis of the nonlinear dynamics. By the rigorous analysis, the new schemes provide the first-order or second-order accuracy in Sobolev spaces for rough data, and reduce the regularity requirement of existing methods so far for optimal convergence. Moreover, the conservation laws of the numerical solutions are considered.

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Date/Time : 23 Feb 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Vikas Krishnamurthy, Indian Institute of Technology Hyderabad

Point vortices are solutions of the Euler equation with a singular (point) distribution of vorticity. Point vortex equilibria have been studied in connection with various experimentally observed equilibrium configurations, and are also related to areas of mathematical physics. In this talk we will first discuss the background and importance of studying point vortex equilibria. We then describe a large class of new exact solutions of the two-dimensional steady incompressible Euler equation, called Liouville chains. They consist of point vortices embedded in a non-constant background vorticity field, which is described by the Liouville PDE, such that the whole arrangement is stationary. Liouville chains can be constructed as iterated solutions starting from a simple purely point vortex equilibrium (without a background field). Each iteration forms one link in the chain and can terminate after one step, a finite number of steps, or go on indefinitely. We describe several examples illustrating all these properties.

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Date/Time : 16 Feb 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Javier Gómez-Serrano, Brown University

In this talk I will present results on singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. These equations describe the motion of a compressible ideal gas, which is characterized by a parameter called the adiabatic constant. Finite time singularities for generic adiabatic constants were found in the recent breakthrough of Merle, Raphaël, Rodnianski and Szeftel. Our results allow us to drop the genericity assumption and construct smooth self-similar profiles for all values of the adiabatic constant. In particular, we will construct the first smooth self-similar profile for a monoatomic gas. Part of the proof is very delicate and requires a computer-assisted analysis. Joint work with Tristan Buckmaster and Gonzalo Cao-Labora.

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Date/Time : 02 Feb 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Jan Sbierski, University of Edinburgh

The strong cosmic censorship conjecture roughly states that Einstein’s theory of general relativity is generically a deterministic theory. Using the language of PDEs this can be paraphrased as stating that generically global uniqueness holds for the initial value problem for the Einstein equations. I will start out this talk by giving a non-technical introduction to this conjecture restricted to the interior of rotating black holes. Moreover, I will discuss the analytic as well as the geometric aspects of the conjecture and outline its current state.

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Date/Time : 26 Jan 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Matthew M. Lin, National Cheng Kung University

Gauging the distance between a mixed state and its nearest separable state is important but challenging in the quantum mechanical system. We, in this talk, propose a dynamical system approach to tackle low rank approximation of entangled bipartite systems, which has several advantages, including 1) A gradient dynamics in the complex space can be described in a fairly concise way; 2) The global convergence from any starting point to a local solution is guaranteed; 3) The requirement that the combination coefficients of pure states must be a probability distribution can be ensured; 4) The rank can be dynamically adjusted. The theory, algorithms, and some numerical experiments will be presented in this talk.

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Date/Time : 12 Jan 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Van Tien Nguyen, National Taiwan University

In this talk I will present constructive examples and numerical illustrations of blowup solutions to the Keller-Segel system in R^d.

• L^1-critical (d = 2): There exist finite time blowup solutions that are of Type II with finite mass. Blowup rates are quantized according to the spectrum of a linearized operator in the self-similar setting. There is also the case of multiple collapsing blowup solutions formed by a collision of single-solutions.

• L^1$-supercritical (d >= 3): We exhibit finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling wave of the 1D viscous Burgers equation. There also exist solutions that blow up in finite time with infinite mass. The solution is asymptotically self-similar with a logarithmic correction to its profile for d = 3,4. We found such an asymptotic profile can be either radial or completely non-radial.

The talk is based on results obtained in collaboration with C. Collot (Paris Cergy), T-E. Ghoul (NYU Abu Dhabi), N. Masmousdi (NYU), N. Nouaili (Paris Dauphine), H. Zaag (Paris Nord).

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Date/Time : 18 Nov 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Hoang-Hung Vo, Saigon University

In this talk, I will present some recent advances for the qualitative properties of the principal eigenvalue of nonlocal operators, which arise from many natural phenomena, especially in ecology and population dynamics. I shall first introduce several notions of generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan and then employ them to investigate the qualitative properties of the principal eigenvalue the asymptotic behavior with respect to the dispersal range and dispersal rate. Different limits are obtained and linked to different phenomena in ecology. We also obtain the long time dynamics of nonlocal Fisher-KPP equation assuming that the environment is unfavorable far away. By taking into account on the seasonal effect on the population, the qualitative properties of the principal eigenvalue of time-periodic nonlocal operators are also discussed, new limiting properties of the principal eigenvalue and long time dynamics nonlocal Fisher-KPP equation with time periodic coefficients are further investigated.

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Date/Time : 11 Nov 2022 16:00
Venue : via Zoom
Speaker : David Hilditch, The University of Lisbon

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https://nus-sg.zoom.us/j/84455764091?pwd=WDF6dWJFVHVFNHZSWHY3VlNWM0xodz09

Meeting ID: 844 5576 4091
Passcode: 171086

In my talk I will consider the initial value problem in general relativity, with smooth initial data tuned to the threshold of black hole formation. Starting in the early 1990s with the work of Choptuik such setups have been studied numerically with various simplifying assumptions. For instance spherical symmetry has often been assumed and combined with a scalar field matter model to drive the dynamics. In that context a remarkably simple phenomenology manifests near the threshold of collapse, with a naked singularity appearing in the limit of perfect tuning. I will review recent progress in the more subtle setting in which symmetry assumptions are dropped, so as to include the fundamental case of pure gravitational waves collapsing under their own self-gravity.

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