Wednesdays at 4:15 PM in 384H.
The Cucker-Smale systems is a popular model of collective behavior of interacting agents, used, in particular, to model bird flocking and fish swarming. The underlying premise is the local alignment of the bird (or fish, or any other agent) velocities. The Euler-Cucker-Smale system is an effective macroscopic PDE limit of such particle systems. It has the form of the pressureless Euler equations with a non-linear density-dependent alignment term. The alignment term is a non-linear version of the fractional Laplacian to a power $\alpha\in(0,1)$. It is known that the corresponding Burgers' equation with a linear dissipation of this type develops shocks in a finite time. We show that nonlinearity enhances the dissipation, and the solutions stay globally regular for all $\alpha\in(0,1)$: the dynamics is regularized due to the nonlinear nature of the alignment.
We first review G-equation (a level set Hamilton-Jacobi equation modeling interface motion) and front speeds in two dimensional cellular flow, then discuss front speeds in three-dimensional Arnold-Beltrami-Childress (ABC) flow. The ABC flow can be viewed as a perturbation of cellular flow, or a nearly integrable system when one of the ABC parameters is small. We show the existence of ballistic orbits where at least one component grows linearly in time, based on the Hamiltonian formulation and time reversal symmetries of the flows. The symmetry based construction works in the non-perturbative case when all three ABC parameters are equal to one. The linear growth law of front speeds of G-equation follows from the ballistic orbits. We end with open problems of front speeds in ABC flows for dissipative G-equations and reaction-diffusion-advection equations.
Given the solution u of an elliptic PDE, there is a naturally associated parabolic PDE for which u is a fixed point in time. We propose the application of parabolic techniques (Brownian motion via the stochastic interpretation) to study the elliptic equation; most of the talk will be spent on a variety of applications using this underlying philosophy and include a geometric inequality (improving on a classical result of Elliot Lieb), capacity interpretations of dimensionality reduction techniques with applications in big data, localization phenomena of Schrodinger operators and probabilistic techniques for bounding eigenvalues.
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