Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

The Liouville equation is a semi-linear elliptic equation of exponential non-linearity. Its non-local version is a steady state of the Keller-Segel equation representing the distribution of living cells, such as slime molds. I will represent an extension of this equation to multi-agent systems and discuss some associated critical phenomena, and recent results with Debabrata Karmakar on the parabolic Keller segel system and its asymptotics in both critical and non-critical cases.

We consider the problem of finding on a given bounded and smooth
Euclidean domain \Omega of dimension n ≥ 3 a complete conformally flat metric whose Schouten
curvature A satisfies some equation of the form  f(\lambda(-A)) =1. This generalizes a problem
considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of
locally Lipschitz solutions. We also show that the Lipschitz regularity is in general optimal.

We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation. It is proved using a Lyapunov functional built from energy and mass, Taylor expansions, spectral estimates, and estimates for the Hilbert transform.

In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).