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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 $\mathbb{T}\times[-1,1]$, and asymptotic behavior of Couette flow in Gevrey spaces on $\mathbb{T}\times\mathbb{R}$ for the $β$-plane equation. Let $T>0$ be the horizontal period of the channel and $α={2π\over T}$ be the wave number. We obtain a sharp region $O$ in the whole $(α,β)$ half-plane such that non-parallel steadily traveling waves do not exist for $(α,β)\in O$ and such traveling waves exist for $(α,β)$ in the remaining regions, near Couette flow for $H^{\geq5}$ 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 $|β|\leq β_*$, then non-parallel traveling waves do not exist for any $T>0$, while if $|β|>β_*$, then there exists a critical period $T_β>0$ so that such traveling waves exist for $T\in \left[T_β,\infty\right)$ and do not exist for $T\in \left(0,T_β\right)$, near Couette flow for $H^{\geq5}$ velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any $β\neq0$ and $T>0$, there exist no non-parallel traveling waves with speeds converging in $(-1,1)$ near Couette flow for $H^{\geq5}$ velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for $H^{<{5\over2}}$ velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the $β$&#39;s term. 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 $\mathbb{T}\times\mathbb{R}$.