Georgia Institute of Technology College of Sciences

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].

In this talk, we consider the n-dimensional bipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. In 1-D case, when the difference between the initial electron mass and the initial hole mass is non-zero (switch-on case), the stability of nonlinear diffusion wave has been open for a long time. In order to overcome this difficulty, we ingeniously construct some new correction functions to delete the gaps between the original solutions and the diffusion waves in L^2-space, so that we can deal with the one dimensional case for general perturbations, and prove the L^\infty-stability of diffusion waves in 1-D case. The optimal convergence rates are also obtained. Furthermore, based on the results of one-dimension, we establish some crucial energy estimates and apply a new but key inequality to prove the stability of planar diffusion waves in n-D case, which is the first result for the multi-dimensional bipolar hydrodynamic model of semiconductors, as we know. This is a joint work with Feimin Huang and Yong Wang.

Mathematical modeling, analysis and numerical simulation, combined with imagingand experimental validation, provide a powerful tool for studying various aspects ofcardiovascular treatment and diagnosis. At the same time, problems motivated bycardiovascular applications give rise to mathematical problems whose studyrequires the development of sophisticated mathematical techniques. This talk willaddress two examples where such a synergy led to novel mathematical results anddirections. The first example concerns a mathematical study of the benchmarkproblem of fluid‐structure interaction (FSI) in blood flow. The resulting problem is anonlinear moving‐boundary problem coupling the flow of a viscous, incompressiblefluid with the motion of a linearly viscoelastic membrane/shell. An existence resultfor an effective, reduced model will be presented.The second example concerns a novel dimension reduction/multi‐scale approach tomodeling of endovascular stents as 3D meshes of 1D curved rods. The resultingmodel is in the form of a nonlinear hyperbolic network, for which no generalexistence results are available. The modeling background and the challenges relatedto the analysis of the solutions will be presented. An application to the study of themechanical properties of the currently available coronary stents on the US marketwill be shown.This talk will be accessible to a wide scientific audience.Collaborators include: Josip Tambaca (University of Zagreb, Croatia), Ando Mikelic(University of Lyon 1, France), Dr. David Paniagua (Texas Heart Institute), and Dr.Stephen Little (Methodist Hospital in Houston).