Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022

• On preservation of moduli of continuity by parabolic evolution

  Speaker: Hussain Ibdah (University of Maryland College Park ) - https://www.math.umd.edu/~hibdah/
  

  When: Thu, September 9, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: In this talk, we will investigate the propagation of Lipschitz regularity by solutions to various nonlinear, nonlocal parabolic equations. We will locally analyze models such as the Michelson-Sivashinsky equation, incompressible Navier-Stokes system, and advection diffusion problems that include the dissipative SQG equation. Depending on the model, we will either show global well-posedness, derive new regularity criteria, or provide different proofs to and generalize previously obtained results. In particular, we will show that for abstract drift-diffusion problems, it is possible to break certain, supercritical Holder-type barriers and get regularity, which is rather surprising.
  
• Finite time singularities for nematic liquid crystal flow and Landau-Lifshitz-Gilbert equation

  Speaker: Yifu Zhou (Johns Hopkins University) - https://math.jhu.edu/~yzhou173/
  

  When: Thu, September 23, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: In this talk, we consider nematic liquid crystal flow (NLCF) and Landau-Lifshitz-Gilbert equation (LLG) in \R^2. (NLCF) is a system strongly coupling the nonhomogeneous incompressible Navier-Stokes equation and the transported harmonic map heat flow, while (LLG) serves as the basic evolution equation for the spin fields in the continuum theory of ferromagnetism. We will investigate how parabolic gluing method can be developed and applied to construct finite time blow-up solutions. This talk is based on joint works with Chen-Chih Lai, Fanghua Lin, Changyou Wang, Juncheng Wei and Qidi Zhang.
  
• Weyl formulae for Schrodinger operators with critically singular potentials

  Speaker: Xiaoqi Huang (University of Maryland) - https://sites.google.com/view/xiaoqi-math/
  

  When: Thu, October 7, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: In this talk we shall discuss generalizations of classical versions of the Weyl formula involving Schr\"odinger
  operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular
  potentials $V$. In particular, we extend the classical results of Avakumovi\'{c}, Levitan and H\"ormander by obtaining $O(\lambda^{n-1})$ bounds for the error term
  in the Weyl formula in the universal case when we merely assume that $V$ belongs to the Kato class,
  ${\mathcal K}(M)$, which is the minimal assumption to ensure that $H_V$ is essentially self-adjoint and
  bounded from below or has favorable heat kernel bounds. We shall discuss both local point-wise and integral versions of Weyl formulae, and also improvements over the
  error term under certain geometric conditions. This is based on joint work with Christopher Sogge and Cheng Zhang.
  
• Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation

  Speaker: Nima Moini (Berkley) - https://math.berkeley.edu/~nima/
  

  When: Thu, October 14, 2021 - 3:30pm
  Where: https://umd.zoom.us/j/92329363134?pwd=KzBtRHo2dm5VUm5PU0hHWnNCSzJ1UT09
  

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  Abstract: In this talk we will highlight some recent discoveries in kinetic theory. We will develop an uncertainty principle, new a-priori bounds and the concept of a blind cone with respect to an observer for the evolution particles in the mesoscopic scale solely based on conservation laws. We will show that the energy within any bounded set of the spatial variable is integrable over time. We will also discuss a generalization of these results for the 2-particle interactions and establish analogies to Morawetz and interaction Morawetz estimates for the nonlinear Schrodinger equation. We will demonstrate some applications for these estimates by showing that the total mass of the particles and interactions concentrate within a specific collection of arbitrarily acute blind cones with respect to any observer. This implies that, as uncertainty inevitably increases, particles will move away in a radial manner from any fixed observer thereby erasing the angular component of momentum. These results are independent of the specific structure of interactions, therefore they are also true for the Boltzmann equation. We will end the talk with a discussion about the Boltzmann equation. We will show the existence and uniqueness of a specific class of classical solutions to the Boltzmann equation with small initial data near vacuum. These solutions scatter to linear states in the $L^{\infty}$ norm, uniformly within any compact set of the spatial variable. Furthermore, we will discuss the asymptotic completeness of this class of solutions and establish another connection with the case of the nonlinear Schrodinger equation. Notably, this shows that solutions of the Boltzmann equation do not necessarily converge to a Maxwellian but can scatter to linear states arbitrarily close to any prescribed linear state.
  
• Tools for multimodal sampling

  Speaker: Michael Lindsey (Courant Institute-NYU) - https://quantumtative.github.io
  

  When: Thu, October 21, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: The task of sampling from a probability distribution with known density arises almost ubiquitously in the mathematical sciences, from Bayesian inference to computational chemistry. The most generic and widely-used method for this task is Markov chain Monte Carlo (MCMC), though this method typically suffers from extremely long autocorrelation times when the target density has many modes that are separated by regions of low probability. We present several new methods for sampling that can be viewed as addressing this common problem, drawing on techniques from MCMC, graphical models, and tensor networks.
  
• On the 2D Peskin problem

  Speaker: Robert Strain (University of Pennsylvania) - https://www2.math.upenn.edu/~strain/
  

  When: Thu, October 28, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  
• Asymptotic stability of kinks in 1D nonlinear scalar field theories

  Speaker: Jonas Luhrmann (Texas A&M University) - https://www.math.tamu.edu/~luhrmann/
  

  When: Thu, November 4, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: Nonlinear scalar field theories on the line such as the phi^4 model or the sine-Gordon model feature soliton solutions called kinks. They are expected to form the building blocks of the long-time dynamics for these models. In this talk I will survey recent progress on the asymptotic stability problem for kinks and I will present a recent result (joint work with W. Schlag) on the asymptotic stability of the sine-Gordon kink under odd perturbations.
  
• Linear vortex symmetrization: the spectral density function approach and Gevrey regularity

  Speaker: Hao Jia (University of Minnesota) - https://www-users.cse.umn.edu/~jia/
  

  When: Thu, November 11, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: The two-dimensional incompressible Euler equation is globally well-posed but the long-time behavior is very difficult to understand due to the lack of global relaxation mechanism. Numerical simulations and physical experiments show that coherent vortices often become a dominant feature in two-dimensional fluid dynamics for a long time. The mathematical analysis of vortices, especially in connection to the so-called `vortex symmetrization` problem, has attracted a lot of attention in recent years. In this talk, after a quick review of recent developments in the study of nonlinear asymptotic stability of shear flows and the symmetrization problem for (the special case of) point vortices, we turn to the general vortex symmetrization problem and report a recent result with A. Ionescu for the linearized flow. The linearized problem has been analyzed before by Bedrossian-Coti Zelati-Vicol who obtained control on the profile of the vorticity field in Sobolev spaces with limited regularity. Our main new discovery is that in the vortex problem, unlike the shear flow case, it is no longer possible to obtain smooth control uniformly in time on a single modulated profile for the vorticity field. Rather, there are two such profiles. To address this issue (for future nonlinear applications), we propose instead to control a new object, the so-called `spectral density function`, which is naturally associated with the linearized flow and can be bounded, for the linearized flow at least, in the same Gevrey space as the initial data.
  
• On the emergence of a quantum Boltzmann equation at the presence of a Bose-Einstein condensate

  Speaker: Michael Hott (The University of Texas at Austin ) - https://web.ma.utexas.edu/users/mhott/
  

  When: Thu, November 18, 2021 - 3:30pm
  Where: https://umd.zoom.us/j/96903133168?pwd=ZG55WWhiMkdSTXJUcWMwc2FCMitHUT09
  

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  Abstract: The mathematically rigorous derivation of a nonlinear Boltzmann equation from first principles is an extremely active research area. In classical physical systems, this has been achieved in various models, based on a variety of fundamental works. In the quantum case, the problem has essentially remained open. I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics of a Bose-Einstein condensate, starting with the von Neumann equation for an interacting Boson gas. This is based on joint work with Thomas Chen.
  
• TBA

  When: Thu, December 2, 2021 - 3:30pm
  Where:
  
• Traveling wave solutions to the free boundary Navier-Stokes equations

  Speaker: Ian Tice (Carnegie Mellon Unversity ) - https://www.math.cmu.edu/~iantice/
  

  When: Thu, December 9, 2021 - 3:30pm
  Where: Kirwan Hall 3206
  

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  Abstract: Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.
  
• Dissipation enhancing flows and applications

  Speaker: Yuanyuan Feng (Penn State) -
  

  When: Thu, December 16, 2021 - 3:30pm
  Where: https://umd.zoom.us/j/97664034062?pwd=L1dObFJMN0tQVHpKQ3JWUDBRaVFOdz09
  

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  Abstract: In this talk, we would first introduce the dissipation enhancing flows. We would focus on the dissipation time of mixing flows, shear flows and the planar helical flows. Then we will apply these flows to Kuramoto Sivashinsky equations in 2d or 3d to get global existence of the solution.