In this talk, I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for regularizing and stabilizing certain evolution equations, such as the Euler, Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit enhanced dissipation mechanism due to large spatial average in the initial data--a phenomenon which is similar to the ``Landau-damping" effect.

This is a joint Analysis seminar and nonlinear PDE seminar