Georgia Institute of Technology College of Sciences

We consider some recent models from stochastic or optimal control involving a very large number of agents. The goal is to derive mean field limits when the number of agents increases to infinity. This presents some new unique difficulties; the corresponding master equation is a non linear Hamilton-Jacobi equation for instance instead of the linear transport equations that are more typical in the usual mean field limits. We can nevertheless pass to the limit by looking at the problem from an optimization point of view and by using an appropriate kinetic

The Euler-Maxwell system describes the interaction between a compressible fluid of electrons over a background of fixed ions and the self-consistent electromagnetic field created by the motion.We show that small irrotational perturbations of a constant equilibrium lead to solutions which remain globally smooth and return to equilibrium. This is in sharp contrast with the case of neutral fluids where shock creation happens even for very nice initial data.Mathematically, this is a quasilinear dispersive system and we show a small data-global

This is a special PDE seminar in Skiles 005. In this talk, we will introduce the compactness framework for approximate solutions to sonic-subsonic flows governed by the irrotational steady compressible Euler equations in arbitrary dimension. After that, similar results will be presented for the isentropic case. As a direct application, we establish several existence theorems for multidimensional sonic-subsonic Euler flows. Also, we will show the recent progress on the incompressible limits.

As the cornerstone of two-fluid models in plasma theory, Euler-Maxwell (Euler-Poisson) system describes the dynamics of compressible ion and electron fluids interacting with their own self-consistent electromagnetic field. It is also the origin of many famous dispersive PDE such as KdV, NLS, Zakharov, ...etc. The electromagnetic interaction produces plasma frequencies which enhance the dispersive effect, so that smooth initial data with small amplitude will persist forever for the Euler-Maxwell system, suppressing any possible shock

Cubic focusing and defocusing Nonlinear Schroedinger Equations admit spatially (and temporally) periodic standing wave solutions given explicitly by elliptic functions. A natural question to ask is: are they stable in some sense (spectrally/linearly, orbitally, asymptotically,...), against some class of perturbations (same-period, multiple-period, general...)? Recent efforts have slightly enlarged our understanding of such issues. I'll give a short survey, and describe an elementary proof of the linear stability of some of these waves. Partly

In this talk we examine the cubic nonlinear wave and Schrodinger equations. In three dimensions, each of these equations is H^{1/2} critical. It has been showed that such equations are well-posed and scattering when the H^{1/2} norm is bounded, however, there is no known quantity that controls the H^{1/2} norm. In this talk we use the I-method to prove global well posedness for data in H^{s}, s > 1/2.

Find the abstract at the link: http://people.math.gatech.edu/~gchen73/research/GA_Tech.pdf

In this talk, we shall study the stability of the Prandtl boundary layer equations in three space variables. First, we obtain a well-posedness result of the three-dimensional Prandtl equations under some constraint on its flow structure. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the complicated secondary flow in the three-dimensional Prandtl boundary layers. Second, we give an instability criterion for the Prandtl equations in three space

Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. De Lellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations, which are designed to reduce the error term in solving the equation. In this talk, I will discuss an

The main focus of this talk is a class of asymptotic methods called averaging. These methods approximate complicated differential equations that contain multiple scales by much simpler equations. Such approximations oftentimes facilitate both analysis and computation. The discussion will be motivated by simple examples such as bridge and swing, and it will remain intuitive rather than fully rigorous. If time permits, I will also mention some related projects of mine, possibly including circuits, molecules, and planets.