Georgia Institute of Technology College of Sciences

I will discuss a natural elliptic obstacle problem that arises in the study of the Abelian sandpile. The Abelian sandpile is a deterministic growth model from statistical physics which produces beautiful fractal-like images. In recent joint work with Wesley Pegden, we characterize the continuum limit of the sandpile processusing PDE techniques. In follow up work with Lionel Levine and Wesley Pegden, we partially describe the fractal structure of the stable sandpiles via a careful analysis of the limiting obstacle problem.

In this talk, I will show recent results on the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle for $L^p$-viscosity solutions of fully nonlinear, uniformly elliptic partial differential equations with unbounded inhomogeneous terms and coefficients. I will also discuss some cases when the PDE has superlinear terms in the first derivatives. This is a series of joint works with Andrzej Swiech.

The basic problem faced in geophysical fluid dynamics isthat a mathematical description based only on fundamental physicalprinciples, the so-called the ``Primitive Equations'', is oftenprohibitively expensive computationally, and hard to studyanalytically. In this talk I will survey the main obstacles inproving the global regularity for the three-dimensionalNavier-Stokes equations and their geophysical counterparts. Eventhough the Primitive Equations look as if they are more difficult tostudy analytically than the three-dimensional Navier-Stokesequations I will show in this talk that they have a unique global(in time) regular solution for all initial data.Inspired by this work I will also provide a new globalregularity criterion for the three-dimensional Navier-Stokesequations involving the pressure.This is a joint work with Chongsheng Cao.

The incompressible Navier-Stokes equations provide an adequate physical model of a variety of physical phenomena. However, when the fluid speeds are not too low, the equations possess very complicated solutions making both mathematical theory and numerical work challenging. If time is discretized by treating the inertial term explicitly, each time step of the solver is a linear boundary value problem. We show how to solve this linear boundary value problem using Green's functions, assuming the channel and plane Couette geometries. The advantage of using Green's functions is that numerical derivatives are replaced by numerical integrals. However, the mere use of Green's functions does not result in a good solver. Numerical derivatives can come in through the nonlinear inertial term or the incompressibility constraint, even if the linear boundary value problem is tackled using Green's functions. In addition, the boundary value problem will be singularly perturbed at high Reynolds numbers. We show how to eliminate all numerical derivatives in the wall-normal direction and to cast the integrals into a form that is robust in the singularly perturbed limit. [This talk is based on joint work with Tobasco].