水平耗散二维Navier--Stokes方程中Kolmogorov流的尖锐线性稳定性与增强耗散的缺失
Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
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中文总结 AI 辅助
研究水平耗散二维Navier--Stokes方程中Kolmogorov流的线性稳定性，证明无增强耗散，并给出各模态的精确衰减率与不稳定条件。
中文摘要 AI 辅助
我们研究了具有水平耗散$\nu\partial_1^2$的二维不可压缩Navier--Stokes方程中Kolmogorov流$U^{(0)}=(a\sin x_2,0)$的线性化动力学。与完全耗散情形不同，该各向异性系统允许$U^{(0)}$作为精确的无外力稳态。在每个水平Fourier模态$k$上，耗散化为标量$-\nu k^2$并与平流项交换，因此线性化半群精确分解为$e^{-\nu k^2t}$乘以无粘Euler群。对于$|k|>1$，我们证明了双边界，表明衰减率恰好为$\nu k^2$，且对剪切振幅$a$一致成立；因此不产生增强耗散。这反映了根本的不匹配：剪切将涡量转移到高频垂直方向，而水平耗散无法检测到。在临界模态$|k|=1$处，无粘群按$\sqrt{2at}$增长，产生大小为$(a/e\nu)^{1/2}$的瞬态放大，随后以速率$\nu$衰减。水平无关模态构成无限维无阻尼核。对于$0<|k|<1$，粘性谱是无粘谱的精确平移，且该模态线性不稳定当且仅当$a\Lambda(k)>\nu k^2$。特别地，当$L>2\pi$时，对所有充分大的$a$，该流线性不稳定。
英文摘要
We study the linearized dynamics of the Kolmogorov flow $U^{(0)}=(a\sin x_2,0)$ for the two-dimensional incompressible Navier--Stokes equations with horizontal dissipation $ν\partial_1^2$. Unlike the fully dissipative case, this anisotropic system admits $U^{(0)}$ as an exact unforced steady state. On each horizontal Fourier mode $k$, the dissipation reduces to the scalar $-νk^2$ and commutes with the advection, so that the linearized semigroup factors exactly into $e^{-νk^2t}$ times the inviscid Euler group. For $|k|>1$, we prove two-sided bounds showing that the decay rate is exactly $νk^2$, uniformly in the shear amplitude $a$; hence no enhanced dissipation occurs. This reflects a fundamental mismatch: the shear transfers enstrophy to high vertical frequencies, which horizontal dissipation does not detect. At the critical modes $|k|=1$, the inviscid group grows like $\sqrt{2at}$, producing a transient amplification of size $(a/eν)^{1/2}$ before decay at the rate $ν$. The horizontally independent modes form an infinite-dimensional undamped kernel. For $0<|k|<1$, the viscous spectrum is an exact translate of the inviscid one, and the mode is linearly unstable if and only if $aΛ(k)>νk^2$. In particular, when $L>2π$, the flow is linearly unstable for all sufficiently large $a$.
发表机构
- North Minzu University（北方民族大学）
- University of Notre Dame（圣母大学）
- Guangdong University of Technology（广东工业大学）
机构由 AI 辅助整理，请以论文原文为准。
