Special DG - MP - PDE Seminar: Dongho Chae (Sungkyunkwan University, Korea)

In the first part of the talk we discuss some new observations on the blow-up problem in the 3D Euler equations. We consider the scenarios of  the self-similar blow-ups and the axisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for the self-similar Euler equations. For the axisymmetric case we show that some uniformity condition for the pressure is not consistent with the global regularity. In the second part we present Liouville type theorems for the steady Navier-Stokes equations for both of the incompressible and the compressible cases. In the time dependent case we prove that some pressure integrals have  definite sign unless the solution is trivial.