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 nonvanishing size for NaviorStokes equations in the inviscid limit.
