Date/Time : 26 Jul 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Antonio De Rosa, University of Maryland

Abstract: We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3. Joint work with Guido De Philippis.

Add to calendar :

Date/Time : 18 Apr 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Matias Delgadino, University of Texas at Austin

Abstract: Weakly interacting diffusions are a go to model for a wide array of phenomena. In this talk, we will study when this model can be accurately approximated by its mean field limit. Phase transitions in the system will emerge as the natural obstruction to uniform in time propagation of chaos.

Add to calendar :

Date/Time : 28 Mar 2022 10:00
Venue : via Zoom
Speaker : Guanglian Li, University of Hong Kong

Join Zoom Meeting
https://nus-sg.zoom.us/j/87674182782?pwd=dEtkOWxldGl4RFVFNkJpV2syalJhUT09

Meeting ID: 876 7418 2782
Passcode: 684943

Abstract: I will present in this talk the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm recently proposed in [Li and Hu, J. Comput. Phys., 2021] to efficiently solve subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the advantages of multiscale methods that can effectively deal with heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Compared with the previous work for parabolic problem, the main challenge in both the analysis and simulation arises from the nonlocality of the fractional derivative. To conquer this obstacle, an auxiliary problem is constructed on each coarse temporal subdomain to uncouple the temporal variable completely. In this manner, the approximation properties of the correction operator is proved. In addition, a new summation of exponential sums is derived to generate single-step time stepping scheme, with the number of terms of O(|log τ_f|) independent of final time. Here, τ_f is the fine-scale time step size. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size and the fine-scale time step size. Several numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly.

Add to calendar :

Date/Time : 21 Mar 2022 10:00
Venue : via Zoom
Speaker : Changhui Tan, University of South Carolina

Join Zoom Meeting
https://nus-sg.zoom.us/j/82703700860?pwd=ZFdLUldUMytjMVhCMVo3L0VMTlZHQT09

Meeting ID: 827 0370 0860
Passcode: 355647

Abstract: In this talk, I will discuss a family of traffic flow models. The classical Lighthill-Whitham-Richards model is known to have a finite time shock formation for all generic initial data, which represents the creation of traffic jams. I will introduce a family of nonlocal traffic flow models, with look-ahead interactions. These models can be derived from discrete cellular automata models.

We show an intriguing phenomenon that the nonlocal slowdown interactions prevent traffic jams, under suitable settings. This talk is based on joint works with Thomas Hamori, Yongki Lee and Yi Sun.

Add to calendar :

Date/Time : 14 Mar 2022 10:00
Venue : via Zoom
Speaker : Emanuel Indrei, Purdue University

Join Zoom Meeting
https://nus-sg.zoom.us/j/85357515356?pwd=ajE1TlB3RXJZMzBRVHB3VmFmYXhaZz09

Meeting ID: 853 5751 5356
Passcode: 934049

Abstract: In 1947, Polya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $\alpha>0$ and suggested that the same holds when $n \ge 5$. In $1951,$ Polya \& Szego discussed the possibility of counterexamples. Recently, I constructed explicit $2n-4$-dimensional polygonal manifolds $\mathcal{M}(n, \alpha)$ and proved the existence of a computable $N \ge 5$ such that for all $n \ge N$, the admissible $n$-gons are given via $\mathcal{M}(n, \alpha)$ and there exists an explicit non-empty set $ \mathcal{A}_{n} \subset \mathcal{M}(n,\alpha)$ such that $P_n$ has the smallest principal frequency among $n$-gons in $\mathcal{A}_{n}$. Inter-alia a formula is proved for the principal frequency of $P \in \mathcal{M}(n,\alpha)$ when $n \ge 3$ in terms of an equilateral $n$-gon $P_E$ with the same area. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}$ estimates. Last, an application is given in the context of electron bubbles.

Add to calendar :

Date/Time : 02 Mar 2022 16:00
Venue : via Zoom
Speaker : Federico Franceschini, ETH Zurich

Join Zoom Meeting
https://nus-sg.zoom.us/j/81974497199?pwd=S0h6QXU0TmFpbkxWeW4wSmIzMTNEdz09

Meeting ID: 819 7449 7199
Passcode: 422551

Abstract: The thin obstacle problem is a classical free boundary problem: it consists of an elliptic PDE which characterise minimisers of a certain variational problem. The boundary of the zero set of solutions of this PDE is called the “free boundary” and understanding its features is an active research field for people in PDE. In this talk I will first explain some known basic results (examples, existence and smoothness of solutions). Then I will present a recent joint work with Prof. Serra concerning the analysis of the free boundary.

Add to calendar :

Date/Time : 14 Feb 2022 10:00
Venue : via Zoom
Speaker : Minh-Binh Tran, Southern Methodist University

Join Zoom Meeting
https://nus-sg.zoom.us/j/86090852474?pwd=TmJ2eUNPbTNEZW5aT3ZPRyt1cEhLZz09

Meeting ID: 860 9085 2474
Passcode: 995447

Abstract: Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. Recent mathematical interests on wave turbulence theory have their roots from the works of Bourgain, Staffilani and Colliander-Keel-Staffilani-Takaoka-Tao. In this talk, I will present some of our recent results on wave turbulence theory. In the first part of the talk, I will discuss our rigorous derivation of wave turbulence equations. The second part of the talk is devoted to the analysis of wave turbulence equations as well as some numerical illustrations. The last part concerns some physical applications of wave turbulence theory. The talk is based on my joint work with Staffilani (MIT), Soffer (Rutgers) and Pomeau (ENS Paris).

Add to calendar :

Date/Time : 07 Feb 2022 10:00
Venue : via Zoom
Speaker : Xinliang An, National University of Singapore

Join Zoom Meeting
https://nus-sg.zoom.us/j/85439997818?pwd=d0xxKy9wZEVYU0dYNXlIRENXVE5GQT09

Meeting ID: 854 3999 7818
Passcode: 723884

Abstract: We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave-speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them we give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.

Add to calendar :

Date/Time : 24 Jan 2022 10:00
Venue : via Zoom
Speaker : Zhiwen Zhang, The University of Hong Kong

Join Zoom Meeting
https://nus-sg.zoom.us/j/83299433896?pwd=UnpRbldHK2pKZXN2QVl2bFcxUXJwQT09

Meeting ID: 832 9943 3896
Passcode: 783017

Abstract: We study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficient on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting method for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical method. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

Add to calendar :