Dynamics near Couette flow for the β-plane equation

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T x [-1,1], and asymptotic behavior of Couette flow in Gevrey spaces on T x R for the beta-plane equation. Let T > 0 be the horizontal period of the channel and alpha = 2 pi T be the wave number. We obtain a sharp region O in the whole (alpha, beta) half-plane such that non-shear traveling waves do not exist for (alpha, beta) E O and such traveling waves indeed exist for (alpha, beta) 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 beta. > 0 such that if |beta| < beta., then non-shear traveling waves do not exist for any T > 0, while if |beta| > beta., then there exists a critical period T beta > 0 so that such traveling waves exist for T E [T beta, oo) and do not exist for T E (0, T beta), 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 beta not equal 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< a 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 beta-plane equation on T x R. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the researchers investigate the stationary structures near the planar Couette flow and the asymptotic behavior of Couette flow for the beta-plane equation. By determining the boundaries between different regions, they conclude that the existence of non-shear traveling waves depends on specific conditions. This contrasting dynamics plays a significant role in studying the long time dynamics of Couette flow with Coriolis effects.