Georgia Institute of Technology College of Sciences

Couette flows are shear flows with a linear velocity profile. Known by Orr in 1907, the vertical velocity of the linearized Euler equations at Couette flows is known to decay in time, for L^2 vorticity. It is interesting to know if the perturbed Euler flow near Couette tends to a nearby shear flow. Such problems of nonlinear inviscid damping also appear for other stable flows and are important to understand the appearance of coherent structures in 2D turbulence. With Chongchun Zeng, we constructed non-parallel steady flows arbitrarily near Couette flows in H^s (s<3/2) norm of vorticity. Therefore, the nonlinear inviscid damping is not true in (vorticity) H^s (s<3/2) norm. We also showed that in (vorticity) H^s (s>3/2) neighborhood of Couette flows, the only steady structures (including travelling waves) are stable shear flows. This suggests that the long time dynamics near Couette flows in (vorticity) H^s (s>3/2) space might be simpler. Similar results will also be discussed for the problem of nonlinear Landau damping in 1D electrostatic plasmas.