Georgia Institute of Technology College of Sciences

This talk gives a blowup criteria to the incompressible Navier-Stokes equations in BMO^{-s} on the whole space R^3, which implies the well-known BKM criteria and Serrin criteria. Using the result, we can get the norm of |u(t)|_{\dot{H}^{\frac{1}{2}}} is decreasing function. Our result can obtained by the compensated compactness and Hardy space result of [6] as well as [7].

In this talk I will present a new notion of Ricci curvature that applies to finite Markov chains and weighted graphs. It is defined using tools from optimal transport in terms of convexity properties of the Boltzmann entropy functional on the space of probability measures over the graph. I will also discuss consequences of lower curvature bounds in terms of functional inequalities. E.g. we will see that a positive lower bound implies a modified logarithmic Sobolev inequality. This is joint work with Jan Maas.

I will present a method for constructing periodic or quasi-periodic solutions for forced strongly dissipative systems. Our method applies to the varactor equation in electronic engineering and to the forced non-linear wave equation with a strong damping term proportional to the wave velocity. The strong damping leads to very few small divisors which allows to prove the existence by using a fixed point contraction theorem. The method also leads to efficient numerics. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.

Using metric derivative and local Lipschitz constant, we define action integral and Hamiltonian operator for a class of optimal control problem on curves in metric spaces. Main requirement on the space is a geodesic property (or more generally, length space property). Examples of such space includes space of probability measures in R^d, general Banach spaces, among others. A well-posedness theory is developed for first order Hamilton-Jacobi equation in this context. The main motivation for considering the above problem comes from variational formulation of compressible Euler type equations. Value function of the variation problem is described through a Hamilton-Jacobi equation in space of probability measures. Through the use of geometric tangent cone and other properties of mass transportation theory, we illustrate how the current approach uniquely describes the problem (and also why previous approaches missed). This is joint work with Luigi Ambrosio at Scuola Normale Superiore di Pisa.