Stability of plane Couette flows with respect to small periodic perturbations

We consider the plane Couette flow v0=(xn,0,…,0)v0=(xn,0,…,0) in the infinite layer domain Ω=Rn−1×(−1,1), where n≥2n≥2 is an integer. The exponential stability of v0v0 in LnLn is shown under the condition that the initial perturbation is periodic in (x1,…,xn−1)(x1,…,xn−1) and sufficiently small in the LnLn-norm.