Georgia Institute of Technology College of Sciences

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).

In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a weak solution of the Euler equations in the usual sense. In DiPerna's paper and the follow-up papers by other authors this approach was implemented for the system of the Euler equations with the artificial viscosity. The extension of this technique to the system of the Navier-Stokes equations is complicated because of the lack of uniform (with respect to the vanishing viscosity), pointwise estimates for the solutions. We will discuss how to obtain the Tartar-Murat commutator relation and to work out the reduction argument using only the standard energy estimates. This is a joint work with Gui-Qiang Chen (Oxford University and Northwestern University).

We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coe±cient, water column depth and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. This is based upon a joint work with Sze-Bi Hsu, National Tsing-Hua University.