BU/Brown PDE Seminar
BU/Brown PDE Seminar
Each month, a joint PDE seminar between the Departments of Mathematics at Boston University and Brown University and the Division of Applied Mathematics at Brown University will be held. The schedule and location for these events can be found below. For a list of all past events of the seminar please visit the BU/Brown PDE Seminar archive page
Panayotis Kevrekidis: Existence, Stability and Dynamics of Some Single- and Multi-Component Solitary Waves: From Theory to Experiments
In this talk, we will present an overview of recent theoretical, numerical
and experimental work concerning the static, stability, bifurcation and
dynamic properties of coherent structures that can emerge in one-
and higher-dimensional settings within Bose-Einstein condensates at
the coldest temperatures in the universe (i.e., at the nanoKelvin scale).
We will discuss how this ultracold quantum mechanical setting can
be approximated at a mean-field level by a deterministic PDE of
the nonlinear Schrodinger type and what the fundamental nonlinear
waves of the latter are, such as dark solitons and vortices. Then, we will try to
go to a further layer of simplified description via nonlinear ODEs encompassing the
dynamics of the waves within the traps that confine them, and the interactions
between them. Finally, we will attempt to compare the analytical
and numerical implementation of these reduced descriptions to
recent experimental results and speculate towards a number of
interesting future directions within this field.
Konstantinos Spiliopoulos: PDE's, Stochastic Processes in Narrow Domains and Wave Front Propagation
We will consider the second initial boundary problem in narrow domains of width ε << 1 for linear second order differential equations with nonlinear boundary conditions. Using probabilistic methods we show that the solution of such a problem converges as
ε ↓ 0 to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front. In addition, an important and interesting problem, which is related to the previous one, is the Wiener process with instantaneous reflection in a narrow tube which, in contrast to before, is assumed to be non-smooth asymptotically. In this case, the limiting process is a Markov process that can be described by its generator. I will report these results as well as recent results on large deviations for such limiting process and how they can be used to study front propagation for reaction-diffusion equations in non-smooth narrow domains.
Location: Brown University, Division of Applied Mathematics, Room 110, 182 George Street
Time: 3:30 - 5:30 pm
Jens Rademacher: The semi-strong regime in reaction-diffusion systems
Reaction-diffusion models frequently possess strongly differing diffusion coefficients. The corresponding asymptotic limit of partly vanishing diffusion coefficients is a singular perturbation with fast and slow spatio-temporal scales that is known as `semi-strong regime'. This regime is particularly interesting for the analysis of nonlinear waves as it allows for explicit analysis of existence, stability and interaction. This talk illustrates some recent results of this kind, in particular the different kinds of interaction laws between localized nonlinear waves that arise.
Justin Kao: The dynamics of heterogeneous coating flows
Fluid coatings occur in a wide variety of situations, from the manufacture
of semiconductors to the foam lacing left behind after drinking a pint of
beer. Sometimes coating uniformity is desirable, but in many cases, these
coatings are heterogeneous, with defects or spontaneous or deliberate
patterning. In this talk I will examine three specific problems: the
deposition of isolated bubbles in the Landau-Levich (dip coating) flow,
the self-assembly of particles in Landau-Levich flow of a suspension, and
the rupture of thin liquid films on chemically patterned substrates. These
phenomena are explained through a combination of modeling, experiment, and
analysis.
Location: Boston University, Mathematics and Computer Science (MCS) building, Room 148, 111 Cummington Street
Time: 3:30 - 5:30 pm
Aaron Hoffman: Orbital stability of the one-kink in the Sine-Gordon equation
Completely integrable differential equations enjoy a multitude of symmetries. It is well known
that this rich and rigid algebraic structure can be exploited to construct special solutions such
as multi-soliton solutions. Less well-known is the fact that, at least in some cases, this symmetry
can also be exploited to show that such special solutions are stable (in an appropriate sense of
stable depending on the particular problem). A convenient feature of stability is that, unlike
complete integrability, it is often robust with respect to perturbations. Thus in some cases stability
can be established for special solutions of near-integrable systems as well. In this talk I will present
work in progress with C.E. Wayne in which we exploit these ideas to establish orbital stability of the
one-kink in the Sine-Gordon equation. I will also mention earlier work with C.E. Wayne where we
use these ideas to provide an alternate proof of the stability of solitary waves in the Fermi-Pasta-Ulam lattice.
Thomas Powers: Mechanics of Swimming Microorganisms
At the small scale of a cell swimming in water, inertial
effects are unimportant. Therefore, the motion of the fluid is
governed by Stokes equations, which are linear. Nevertheless, there
are many situations in which nonlinear effects are important. In this
talk I will describe two such situations. The first is swimming in a
viscoelastic material, which is motivated by the fact that many
microorganisms move in non-Newtonian media such as mucus. I will
present a simple model that shows how fading memory affects swimming
speed. We will also present experimental results for a helix swimming in a
viscoelastic fluid. The second situation I will consider is the
synchronization of rotors via hydrodynamic interactions. This problem is motivated by the
observed coordination of nearby beating cilia.
Location: Brown University, Division of Applied Mathematics, Room 110, 182 George Street
Time: 3:30 - 5:30 pm
Jared Speck: The global stability of the Minkowski spacetime solution to the Einstein-nonlinear electromagnetic system in wave coordinates
Abstract: The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of
general relativity to a model of nonlinear electromagnetic fields. In this talk, I will discuss the
family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable
from a sufficiently regular Lagrangian, ii) they reduce to the linear Maxwell model in the weak-field
limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition.
I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a
proof of the global nonlinear stability of the 1 + 3 −dimensional Minkowski spacetime solution to
the coupled system. This stability result is a consequence of a small-data global existence result for
a reduced system of equations that is equivalent to the original system in a wave coordinate gauge.
The analysis of the spacetime metric components is based on a framework recently developed by
Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of
quasilinear wave equations with nonlinearities that satisfy the weak nul l condition. The analysis of
the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension
of a geometric energy-method framework developed by Christodoulou, together with a collection
of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I
work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic
potentials.
Michael Brenner: A failed attempt to find a singularity of the Euler Equation
Abstract: I will discuss our recent efforts to find a singular solution to the Euler equation, by considering initial conditions with interacting vortex filaments, with arbitrary circulations and artibrary initial shapes. Instead, we provide (asymptotically consistent) arguments that indicate that it is quite unlikely that singular solutions from such initial conditions can exist.
Location: Boston University, Mathematics and Computer Science (MCS) building, Room 144,111 Cummington Street
Time: 3:30 - 5:30 pm
Nathan Kutz: Application of dimensionality reduction methods
Dimensionality reduction is a common method for rendering tractable a host of problems arising in the physical, engineering and biological sciences. In recent years, methods from data analysis have started playing critical roles in more traditional applied mathematics problems typically analyzed with dynamical systems and PDE techniques. In this talk, three disparate examples will be considered from (i) image processing, (ii) PDE solution techniques and (iii) neuroscience. In each case, dimensionality reduction, typically achieved through a principal component analysis (PCA) or orthogonal mode decomposition (POD), i.e. a singular value decomposition, achieves remarkable success in providing a mathematical framework which is much more amenable to analysis, thus allowing for a better characterization of the physical, engineering or biological system of interest.
Christopher Chong: Justification of the KdV equation in periodic FPU chains
In this talk, we aim to show error estimates for the approximate description of macroscopic wave packets in infinite periodic chains of coupled oscillators (i.e. periodic FPU chains) based on a derived Korteweg–de Vries (KdV) equation. We utilize a discrete Bloch wave transform, a normal form transform, and a special chosen energy to obtain the desired estimates. Despite the complicated resonance structure arising in the system, the proof is relatively simple.
Location: Brown University, Division of Applied Mathematics, Room 110, 182 George Street
Time: 3:30 - 5:30 pm