
In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.
In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudospectral algorithms were developed for inverting the LaplaceBeltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, wellconditioned solvers for the Maxwell equations will rely on these algorithms.
Fast advection asymptotics for a stochastic reactiondiffusionadvection equation in a twodimensional bounded domain will be discussed. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.
We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phasefield model, consists of the AllenCahn equation for the scalar phase field function coupled with a vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms and (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime.
The second model type is a nonlinear free boundary problem for a KellerSegel type system of PDEs in 2D with area preservation and curvature entering the boundary conditions. We find an analytic oneparameter family of radially symmetric standing wave solutions (corresponding to a resting cell) as solutions to a Liouville type equation. Using topological tools, traveling wave solutions (describing steady motion) with noncircular shape are shown to bifurcate from the standing waves at a critical value of the parameter. Our bifurcation analysis explains, how varying a single (physical) parameter allows the cell to switch from rest to motion.
We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L^{1} norm of the velocity field. Existing results in the literature use an L^{p} norm with p>1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.
I will review recent advances concerning the Prandtl's conjecture: slightly viscous flows can be decomposed into inviscid flows, plus a Prandtl's layer near solid boundary, in the inviscid limit.
Coupled oscillators arise in contexts as diverse as the brain, synchronized flashing of fireflies, coupled Josephson junctions, or unstable modes of the Millennium bridge in London. Generally, such systems are either studied for a small number of oscillators or in the infinite oscillator, mean field limit. The dynamics of large but finite networks of oscillators is largely unknown. Kinetic theory was developed by Boltzmann and Maxwell to show how microscopic Hamiltonian dynamics of particles could account for the thermodynamic properties of gases. Here, I will show how concepts of kinetic theory and statistical field theory can be applied to deterministic coupled oscillator and neural systems to compute dynamical finite system size effects.
The leastaction problem for geodesic distance on the `manifold' of fluidblob shapes exhibits instability due to microdroplet formation. This reflects a striking connection between Arnold's leastaction principle for incompressible Euler flows and geodesic paths for Wasserstein distance. A connection with fluid mixture models via a variant of Brenier's relaxed leastaction principle for generalized Euler flows will be outlined also. This is joint work with JianGuo Liu and Dejan Slepcev.
