Dynamics near Couette flow for the β-plane equation

Abstract

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T×[−1,1], and asymptotic behavior of Couette flow in Gevrey spaces on T×R for the β-plane equation. Let T>0 be the horizontal period of the channel and α=2πT be the wave number. We obtain a sharp region O in the whole (α,β) half-plane such that non-shear traveling waves do not exist for (α,β)∈O and such traveling waves indeed exist for (α,β) in the remaining regions, near Couette flow for H≥5 velocity perturbation. The borderlines between the region O and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists β⁎>0 such that if |β|≤β⁎, then non-shear traveling waves do not exist for any T>0, while if |β|>β⁎, then there exists a critical period Tβ>0 so that such traveling waves exist for T∈[Tβ,∞) and do not exist for T∈(0,Tβ), near Couette flow for H≥5 velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any β≠0 and T>0, we prove that there exist no non-shear traveling waves with traveling speeds converging in (−1,1) near Couette flow for H≥5 velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for H<52 velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the Coriolis effects. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the β-plane equation on T×R.