Speaker: Ruiwen Shu (University of Maryland)

Energy minimization problems of attractive-repulsive pairwise interactions are very important in the study of pattern formation in biological and social sciences. In this talk, I will discuss some recent progress (joint work with Jose Carrillo) on the study of Wasserstein-\(\infty\) local energy minimizers by using the method of linear interpolation convexity/concavity. In the first part, we prove the radial symmetry and uniqueness of local minimizers for interaction potentials satisfying the 'linear interpolation convexity', which generalizes the result of O. Lopes 17' for global minimizers. In the second part, we show that the failure of linear interpolation convexity could lead to the formation of small scales in the support of local minimizers, and construct interaction potentials whose local minimizers are supported on fractal sets. To our best knowledge, this is the first time people observe fractal sets as the support of local minimizers.
 

Time: Febrary 5, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

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Alexander Kiselev, and Changhui Tan

Annals of PDE, Volume 8, No. 2, Article 16, 69pp. (2022)

Abstract

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian \((-\Delta)^{1/2}\).

	doi:10.1007/s40818-022-00135-4
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

 Speaker: Dallas Albritton (New York University)

In PDE analyses of fluid models, often we may identify a so-called critical space that lives precisely at the borderline between well-posedness and ill-posedness. What happens at this borderline? We explore this question in two active scalar equations with critical dissipation. In the critical surface quasi-geostrophic equations, we investigate the connection between non-uniqueness and large self-similar solutions that was established by Jia, Sverak, and Guillod in the Navier-Stokes equations. This is joint work with Zachary Bradshaw. In the critical Burgers equation, and more generally in critical scalar conservation laws, the analogous self-similar solutions are unique, and we show that all front-like solutions converge to a self-similar solution at the diffusive rates. This is joint work with Raj Beekie.
 

Time: November 13, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

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 Speaker: Xiang Xu (Old Dominion University)

The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential.
 

Time: October 30, 2020 3:30pm-4:30pm
Location: Virtually via Zoom
Host: Changhui Tan

 Speaker: Li Wang (University of Minnesota)

We develop a novel semi-implicit scheme for the crystal surface evolution equation, which suffers from significant stiffness that prevents simulation on fine spatial grids. Our method leverages the formal structure of the equation as the gradient flow of the total variation energy, with respect to a weighted \(H^{-1}\) norm. Inspired by the classical minimizing movement scheme, we reformulate the semi-implicit time discretization into an optimization problem, and then use a primal-dual hybrid gradient (PDHG) method to compute the minimizer. In one dimension, we prove the convergence of PDHG to the semi-implicit scheme, and show that, at the discrete level, our PDHG method converges at a rate independent of the grid size. This is joint work with Katy Craig, Jian-Guo Liu, Jianfeng Lu and Jeremy Marzuola.
 

Time: October 23, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

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 Speaker: Barak Sober (Duke University)

The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold \(M\) of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter \(h\), state-of-the-art discrete methods yield \(O(h)\) provable approximation rates. In this work, we prove that the Riemannian metric of the Moving Least-Squares Manifold (Manifold-MLS) introduced by (Sober & Levin 19) has approximation rates of \(O(h^{k-1})\). In other words, the Manifold-MLS is nearly an isometry with high approximation order. We use this fact to devise an algorithm that computes geodesic distances between points on \(M\) with the same rates of convergence. Finally, we show the potential and the robustness to noise of the proposed method in some numerical simulations.
 

Time: October 16, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Wolfgang Dahmen

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 Speaker: Jiguang Sun (Michigan Technological University)

We propose a new finite element approach, which is different from the classic Babuska-Osborn theory, for some nonlinear eigenvalue problems. The eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Finite element methods are used for discretization. The convergence of eigenvalues/eigenvectors is proved using the abstract approximation theory for holomorphic operator functions. Then the spectral indicator method is extended to compute the eigenvalues/eigenvectors. Two nonlinear eigenvalue problems are treated using the proposed approach.
 

Time: October 9, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Qi Wang

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 Speaker: Trevor Leslie (University of Wisconsin, Madison)

We consider the Euler Alignment Model with smooth, slowly decaying interaction protocol. It has been known since the work of Carrillo, Choi, Tadmor and Tan in 2016 that a certain conserved quantity '\(e\)' governs the global-in-time existence or finite-time blowup of sufficiently regular solutions. We give an interpretation of the quantity e and use it to analyze the structure of the limiting density profile. We draw two striking conclusions: First, the singular support of the limiting density measure (where 'aggregation' occurs) is precisely the image of the initial zero set of \(e\), under the limiting flow map. This allows us to reverse-engineer mass concentration sets of a specified topological genus, for example. Second, the smoothness of \(e\) at time zero controls the size of mass concentration set: If \(e_0\) is \(C^k\), then the mass concentration set has Hausdorff dimension at most \(1/(k+1)\). We show that this bound is sharp by means of an explicit example. This is joint work with Lear, Shvydkoy, and Tadmor. If time allows, we will also discuss the role of e in the limiting dynamics for the case of strongly singular interaction protocols.
 

Time: March 25, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Changhui Tan

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Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 53, No 3, pp. 3040–3071 (2021).

Abstract

We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data. Compared with the one-dimensional system, a major difference in multidimensional Eulerian dynamics is the presence of the spectral gap, which is difficult to control in general. We propose a new pair of scalar quantities that provides significantly better control of the spectral gap. Two applications are presented: (i) the Euler-Poisson equations: we show a sharp threshold condition on initial data that distinguish global regularity and finite time blowup; (ii) the Euler-alignment equations: we show a large subcritical region of initial data that leads to global smooth solutions.

	doi:10.1137/20M1358682
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	This work is supported by NSF grant DMS #1853001

Li Chen, Changhui Tan, and Lining Tong

Methods and Applications of Analysis, Volume 28, No.2, pp. 155-174 (2021).

Dedicated to Professor Ling Hsiao's 80th birthday.

Abstract

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

	doi:10.4310/MAA.2021.v28.n2.a3
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	This work is supported by NSF grant DMS #1853001