PDE-Applied Math Archives for Academic Year 2015

An example of hyperbolic relaxation toward a scalar conservation law with spatial heterogeneity

When: Thu, September 25, 2014 - 3:30pm
Where: Math 3206
Speaker: Magali Tournus (Pennsylvania State University) - https://www.ljll.math.upmc.fr/~tournus/

Parallelizable Block Iterative Methods for Stochastic Processes

When: Thu, October 2, 2014 - 3:30pm
Where: Math 3206
Speaker: Gil Ariel (Bar Ilan University) -
Abstract: In many applications involving large systems of stochastic differential equations, the states space can be partitioned into groups which are only weakly interacting. For example, molecular dynamics simulations of large molecules undergoing Langevin dynamics may be divided into smaller components, each at equilibrium. If the components are decoupled, then the equilibrium distribution of the entire system is a product of the marginals and can be computed in parallel. However, taking interactions into account, the entire state of the system must be considered as a whole and naïve parallelization is not possible. We propose an iterative method along the lines of the wave-form relaxation approach for calculating all component marginals. The method allows some parallelization between conditionally independent components, depending on the minimal coloring of the graph describing their mutual interactions. Joint work with Ben Leimkuhler and Matthias Sachs (University of Edinburgh).

Mixtures In Compressible Navier-Stokes Systems

When: Thu, October 30, 2014 - 3:30pm
Where: Math 3206
Speaker: Didier Bresch (University of Savoie, France) -

Analysis of 2+1 Diffusive-Dispersive PDE Arising in River Braiding

When: Thu, November 6, 2014 - 3:30pm
Where: Math 3206
Speaker: Charis Tsikkou (West Virginia University) -
Abstract: In the context of a weakly nonlinear study of bar instabilities in a sediment carrying river, P. Hall introduced an evolution equation for the deposited depth which is dispersive in one spatial direction, while being diffusive in the other. In this talk, we present local existence and uniqueness results using a contraction mapping argument in a Bourgain-type space. We also show that the energy and cumulative dissipation are globally controlled in time. This is joint work with Saleh Tanveer.

Kinetics of particles with short-range interactions

When: Thu, November 20, 2014 - 3:30pm
Where: Math 3206
Speaker: Miranda Holmes-Cerfon (Courant Institute) -
Abstract: Particles in soft-matter systems, such as colloids, tend to have very short-range interactions compared to their size. Because of this, traditional theories, that assume the energy landscape is smooth enough, will struggle to capture their dynamics. We propose a new framework to look at such particles, based on taking the limit as the range of the interaction goes to zero. In this limit, the energy landscape is a set of geometrical manifolds plus a single control parameter, while the dynamics on top of the manifolds are given by a hierarchy of Fokker-Planck equations coupled by "sticky" boundary conditions. We show how to compute dynamical quantities such as transition rates between clusters of hard spheres, and then show this agrees quantitatively with experiments on colloids. Finally, we show how dynamical ideas can be used to solve the mathematical problem of enumerating all the nonlinearly rigid packings of hard spheres.

Propagators for the wave equation and geometric microlocal analysis

When: Thu, December 4, 2014 - 3:30pm
Where: Math 3206
Speaker: Jesse Gell-Redman (Johns Hopkins University) - http://www.math.jhu.edu/~jgell/
Abstract: This lecture will be part research talk and part introduction to a multi-part course on microlocal analysis I will give next semester in the informal geometric analysis seminar. I will describe, in a non-technical fashion, recent work on the wave equation on asymptotically Minkowski space-times with Haber and Vasy, related to recent work on quasilinear wave equations due to Hintz and Vasy. The tools we use come from geometric microlocal analysis, which I will describe in broad terms and indicate how they are particularly useful in non-compact settings like Minkowski space.

Microlocal analysis is also useful in elliptic settings, particularly so on non-compact and singular spaces with certain structures thanks to work of Melrose and many others. If time permits I will describe common features connecting the elliptic and hyperbolic (wave equation) from the microlocal perspective.
Joint work with Nick Haber and Andras Vasy.

Different roles of short and long ranged forces in simple (e.g. argon) and networked (e.g. water) liquids

When: Thu, December 11, 2014 - 3:30pm
Where: Math 3206
Speaker: John D. Weeks (UMD) -

Stochastic PDEs and Turbulence

When: Thu, February 12, 2015 - 3:30pm
Where: Math 3206
Speaker: Nathan Glatt-Holtz (Virginia Tech) -

A Vector Field Method For Radiating Black Hole Space-times

When: Thu, March 26, 2015 - 3:30pm
Where: Math 3206
Speaker: Jesus Oliver (UC San Diego) -

Abstract: We study the global decay properties of solutions to the Wave equation in 3+1 dimensions on time-dependent, weakly asymptotically flat black hole space-times (M, g αβ ). Assuming a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy with any number of scalings, rotations, and translation derivatives applied to the solution. As a non-linear application, we also show global existence and decay for small data solutions of the Wave maps system
M → N , with N a Riemannian manifold. Joint work with J. Sterbenz.-

From molecular dynamics to kinetic theory and fluid mechanics

When: Thu, April 9, 2015 - 2:00pm
Where: Math 3206
Speaker: Prof. Laure Saint-Raymond (Harvard and MIT on leave from Ecole Normale Superieure) - Douglis Lecture, joint with CSCAMM and KI-Net
Abstract: In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium. This control of the collision process can be obtained in fluctuation regimes [1, 2]. In [2], we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.
Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models
[1] T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, to appear in Invent. Math. (2015).
[2] T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad limit, in preparation.

Characterization of Measures in the dual of BV

When: Thu, April 30, 2015 - 3:30pm
Where: Math 3206
Speaker: Monica Torres (Purdue University) -
Abstract: http://www2.cscamm.umd.edu/~jabin/TorresMaryland2015.pdf

High-dimensional Numerical schemes and Dimension Reduction techniques for Uncertainty Quantification based on Probability Density Functions

When: Thu, May 7, 2015 - 3:30pm
Where: Math 1308 (Note Room Change)
Speaker: Heyrim Cho (Brown University) -
Abstract: Probability density functions (PDFs) provide the entire statistical structure of the solution to stochastic systems. In this talk, we introduce the joint response-excitation PDF approach that generalizes the existing PDF evolution equations and enables us to do stochastic simulations with random initial condition, coefficient, and forcing, involving non-Gaussian colored noise. We develop efficient numerical algorithms to solve this system from low- to high- dimensions. In particular, we develop high-dimensional numerical schemes by using ANOVA approximation and separated series expansion. Alternatively, we employ dimension reduction techniques such as Mori-Zwanzig approach and moment closures to obtain reduced order PDF equations. The effectiveness of our approach is demonstrated in various stochastic dynamical systems and stochastic PDEs, such as Duffing oscillator that reveals chaotic dynamics and Burgers equation yielding multiple interacting shock waves at random space-time locations.

Null Frames and the Cubic Dirac equation

When: Thu, May 14, 2015 - 3:30pm
Where: Math 3206
Speaker: Tim Candy (John Hopkins University) -
Abstract: We give an overview of recent work on the problem of small data global well-posedness for the cubic Dirac equation in 2+1 dimensions. The main obstruction is a lack of available Strichartz estimates in low dimensions. To get around this difficulty, there are two key ideas. The first is the observation, originally due to Tataru, is that it is possible to construct null frames in which key endpoint Strichartz estimates can be recovered. The second is the fact that there is a subtle cancellation (or null structure) in the cubic nonlinearity that removes the dangerous parallel interactions. This is joint work with Nikolaos Bournaveas.