Dynamics near Couette flow for the $\beta$-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 $\beta$-plane equation. Let $T>0$ be the horizontal period of the channel and $\alpha={2\pi\over T}$ be the wave number. We obtain a sharp region $O$ in the whole $(\alpha,\beta)$ half-plane such that non-parallel steadily traveling waves do not exist for $(\alpha,\beta)\in O$ and such traveling waves exist for $(\alpha,\beta)$ 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 $\beta_*>0$ such that if $|\beta|\leq \beta_*$, then non-parallel 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 $T\in \left[T_\beta,\infty\right)$ and do not exist for $T\in \left(0,T_\beta\right)$, 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 $\beta\neq0$ 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 $\beta$'s term. 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 $\mathbb{T}\times\mathbb{R}$.