Mathematics Calendar

In this lecture we present new model system for the incompressible fluid flows. After brief introduction to the Euler/Navier-Stokes equations, 2D surface quasi-strophic equations and the Hall-MHD system, we introduce a new active vector system, which 'unifies' them all. We may view the new system as a singularized version for the 3D Euler equations, and a part of Hall-MHD system(E-MHD) correspond to the order two more singular one than the 3D Euler equations. The generalized surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one previously used for (gSQG). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.