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arXiv · 2609.34610

Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation

Wanrong Yang·Jiahong Wu·Xiaoping Zhai
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Abstract

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 $a$.

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Wanrong Yang, Jiahong Wu, Xiaoping Zhai. 2026-09-28. Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation. https://arxiv.org/abs/2609.34610

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