论文标题
$β$平面方程的COUETTE流程附近的动力学
Dynamics near Couette flow for the $β$-plane equation
论文作者
论文摘要
在本文中，我们研究了在sobolev $ \ mathbb {t} \ times [-1,1] $的sobolev空间中的平面轴向流动附近的固定结构，以及$ \ m m m mathbb {t} \ times \ mathbb {r r} $ for gevrey space in Gevrey space in Gevrey space in Gevrey space in gevrey space的渐近行为。令$ t> 0 $为通道的水平周期，$α= {2π\ the t} $是波数。我们在整个$（α，β）$半平面中获得了一个尖锐的区域$ o $，因此在O $中不存在$（α，β）\不存在的非平行行驶波，其余的$（α，β）$在其余区域内存在，$ h^{\ geq5} $ veliftions的$（α，β）$。该地区$ o $及其其余的边界线取决于单数Rayleigh-kuo运营商的主要特征值的两条曲线。 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\右）$，靠近couette Flow，以$ h^{\ geq5} $速度扰动。这种对比的动力学在研究COUETTE流动的长时间动力学和科里奥利效应方面起着重要作用。此外，对于任何$β\ neq0 $和$ t> 0 $，都不存在非并行的行进波，速度在$（-1,1）$中汇聚为$ h^{\ geq5} $速度扰动，与此相反，与此相比，我们构建了$ hearear satesary stastary Solutions $ h^unde $ h^^5 flof。扰动，这是对[22]中定理1的概括，但是由于$β$的术语，构造更加困难。最后，我们通过将[4]的方法扩展到$ \ Mathbb {t} \ times \ times \ mathbb {r} $上的$β$ - 平面方程，证明了某些Gevrey空间中的couette流的非线性无粘性阻尼。
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 $β$'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 $\mathbb{T}\times\mathbb{R}$.
