Lefschetz Center for Dynamical Systems Seminar
Abstract: Suppose that x: (-(infinity), a) -> C is a bounded, complex-valued solution of an ``analytic functional differential equation'' (sometimes called an ``analytic differential-delay equation''). Typically, it is easy to prove that x is infinitely differentiable, but is x real analytic at every point of its domain? We shall describe some general positive results (real analyticity everywhere), but we shall also discuss some simple-looking examples which show that these questions can be extremely subtle. In particular, we shall discuss examples of the form (*) x'(t)=ax(t) - a(1-r'(t))x(t-r(t)), where r(t):=2(pi) -q(sin(t)). For suitable constants a and q, (*) has an infinitely differentiable, 2(pi)-periodic solution which is real analytic on a nonempty open subset of [0, 2(pi)}, but is also not real analytic at an uncountable set of points in [0,2(pi)].
Abstract: The Maslov idempotent calculus provides a framework for a variety of asymptotic problems, including large deviations for Markov diffusions described by stochastic differential equations. The asymptotic limit is described through a deterministic optimization problem. This limit retains a ``stochastic'' interpretation, in which expectations are linear with respect to ``max-plus'' addition and scalar multiplication. The first part of the lecture will discuss max-plus stochastic differential equations, with associated Hamilton-Jacobi PDEs and variational inequalities. The second part of the lecture is concerned with controlled max-plus stochastic differential equations and associated two-controller, zero sum differential games. As an example, the solution to the max-plus version of the classical Mertion optimal consumption problem is given.
Center for Fluid Mechanics, Division of Applied Mathematics Fluids, Thermal and Chemical Processes Group, School of Engineering Joint Seminar Series
Abstract: The interface between liquids provides a versatile platform for assembly of small particles for functional membranes, capsules, and other materials. Owing to the interfacial tension, nanometer-to-millimeter-sized particles readily adsorb at the interface and become confined there. Here I will focus on of the role of the interfacial curvature when spherical particles bind. While adsorption on planar interfaces is reasonably well understood (in the absence of electrostatic charge), adsorption on anisotropically curved interfaces leads to geometric frustration that has some surprising consequences. I will describe experimental measurements of the interactions among particles as well as theoretical calculations. Using our model, we predict that spherical particles on constant-mean-curvature interfaces are driven toward lower Gaussian curvature, i.e., toward saddle-shaped regions. This ``curvature-capillary'' interaction may help explain the stability of Pickering emulsions and point the way to new methods of directing particle assembly. This work was done in collaboration with Nesrin Senbil, Chuan Zeng and Benny Davidovitch. I acknowledge support from the NSF-funded UMass MRSEC on Polymers and NSF grant CBET-0967620.
BU/Brown PDE Seminar
Abstract: We first study the interaction between homogenization and diffusion approximation in the case where the mean free path is much smaller than the period of the periodic coefficient involved in the Boltzmann equation. Then I will present recent work of anomalous case in the diffusion approximation. This is joint work with G. Bal, N. Ben Abdallah, A. Mellet and M. Vogelius.
BU/Brown PDE Seminar
Abstract: Fluid mixing is a key process we must understand in order to make predictions about a variety of natural phenomena: the growth of phytoplankton, oceanic transport of salt and heat, and the development of planetary magnetic fields, for example. Our group is working to connect the global statistics of mixing to topology and coherent structures in the underlying flow. Our tool is particle tracking in laboratory fluid experiments. I will talk about two different ways to characterize mixing geometrically, either with shape statistics of three-particle clusters, or by separating flow deformation into stretching and folding (affine and non-affine parts). The crucial role of critical points is evident in either approach, and our deformation work generalizes another approach for locating coherent structures, Finite Time Lyapunov Exponents.
Applied Mathematics Colloquium
Abstract: We present a non-linear lower bound for the fractional Laplacian, when evaluated at extrema of a function. Applications to the global well-posedness of active scalar equations arising in fluid dynamics are discussed. This is joint work with Peter Constantin.
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