Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
September 21-25, 2009

CSIC Building (#406), Seminar Room 4122.

Dynamical structures near Couette flows

Dr. Zhiwu Lin

Georgia Institute of Technology

Abstract:  Plane Couette flows are known to be linearly stable for any Reynolds number but become turbulent for large Reynolds numbers. In recent years, there have been lots of work using ideas from dynamical systems to explain the turbulent structures near Couette flows, mainly numerically. I will describe two related mathematical results. First, (with Charles Li) we proved that Couette flows are structurally unstable, more precisely, there exist unstable shear flows arbitrarily close to Couette flows in the energy norm, for both inviscid and slightly viscous settings. It is also shown that there exist inviscid traveling wave solutions arbitrarily near Couette flows. Second, (with Chongchun Zeng) we showed the existence of stable and unstable manifolds for linearly unstable shear flows in the inviscid case. This result might suggest the existence of unstable manifolds of non-vanishing size for Navior-Stokes equations in the inviscid limit.