Day 3 (June 12)

 It is a classical problem in fluid dynamics about the stability and instability of different fluid patterns in various physical settings, especially in the high Reynolds number limit with no-slip boundary condition. In this talk, we will first review the background and some recent main progress on Prandtl boundary layer theory. Then we will discuss a recent work on the structural stability of the steady 3D Prandtl system. We will also discuss the high Reynolds number limit for the steady 2D compressible flow.

 An inclusion is said to be neutral to uniform fields if upon insertion into a homogeneous medium with a uniform field it does not perturb the uniform field at all. It is said to be weakly neutral if it perturbs the uniform field mildly. Such inclusions are of interest in relation to invisibility cloaking and effective medium theory. There have been some attempts lately to construct or to show existence of such inclusions in the form of coreshell structure of a single inclusion with the imperfect bonding parameter attached to its boundary. In this talk we review recent progress in such attempts. We also discuss the over-determined problem for confocal ellipsoids which is closely related with the neutral inclusion problem, and its equivalent formulation in terms of Newtonian potentials. We will also talk about recent applications to stress estimates on a biological body.