Georgia Institute of Technology College of Sciences

We prove a new Holder estimate for drift-(fractional)diffusion equations similar to the one recently obtained by Caffarelli and Vasseur, but for bounded drifts that are not necessarily divergence free. We use this estimate to study the regularity of solutions to either the Hamilton-Jacobi equation or conservation laws with critical fractional diffusion.

We prove an extension of the Wiener inversion theorem for convolution of summable series, allowing the terms to take values in a space of bounded linear operators. The resulting algebra is no longer commutative due to the composition of operators. Inversion theorems arise naturally in the context of proving dispersive estimates for the Schr\"odinger and wave equation and lead to scale-invariant conditions for the class of admissible potentials. All results are joint work with Marius Beceanu.

Fokker-Planck equation is a linear parabolic equation which describes the time evolution of of probability distribution of a stochastic process defined on a Euclidean space. Moreover, it is the gradient flow of free energy functional. We will present a Fokker-Planck equation which is a system of ordinary differential equations and describes the time evolution of probability distribution of a stochastic process on a graph with a finite number of vertices. It is shown that there is a strong connection but also substantial differences between the ordinary

In the simplest form, our result gives a characterization of bounded,divergence-free vector fields on the plane such that the Cauchyproblem for the associated continuity equation has a unique boundedsolution (in the sense of distribution).Unlike previous results in this directions (Di Perna-Lions, Ambrosio,etc.), the proof does not rely on regularization, but rather on adimension-reduction argument which allows us to prove uniqueness usingwell-known one-dimensional results (it is indeed a variant of theclassical method of characteristics).Note that our characterizatio

We consider the three dimensional Navier-Stokes equations with a large initial data and we prove the existence of a global smooth solution. The main feature of the initial data is that it varies slowly in the vertical direction and has a norm which blows up as the small parameter goes to zero. Using the language of geometrical optics, this type of initial data can be seen as the ``ill prepared" case.

Let $\mathbb{H}$ be a Hilbert space and $h: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}$ be such that $h(x, \cdot)$ is uniformly convex and grows superlinearly at infinity, uniformy in $x$. Suppose $U: \mathbb{H} \rightarrow \mathbb{R}$ is strictly convex and grows superlinearly at infinity. We assume that both $H$ and $U$ are smooth. If

Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed

I will discuss the intermediate and long time dynamics of solutions of the nonlinear Schroedinger - Gross Pitaevskii equation, governing nonlinear dispersive waves in a spatially non-homogeneous background. In particular, we present results (with B. Ilan) on solitons with frequencies near a spectral band edge associated with periodic potential, and results (with Z. Gang) on large time energy distribution in systems with multiple bound states. Finally, we discuss how such results can inform strategies

In this talk,we study weighted L^p-norm inequalities for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems follow from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels.