Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
Abstract
We study the linearized dynamics of the Kolmogorov flow U(0)=(asin x2,0) for the two-dimensional incompressible Navier--Stokes equations with horizontal dissipation ν∂12. 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 −νk2 and commutes with the advection, so that the linearized semigroup factors exactly into e−νk2t times the inviscid Euler group. For |k|>1, we prove two-sided bounds showing that the decay rate is exactly νk2, 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 √(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)>νk2. In particular, when L>2π, the flow is linearly unstable for all sufficiently large a.
Related
- Global Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation
- Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow
- Global well-posedness for the 2D primitive equations with subcritical horizontal dissipation
- Stability of 2D MHD with Only Horizontal Dissipation near a Background Magnetic Field on ℝ2
- Enhanced Dissipation by Transport Noise for Stochastic p(x)-Laplace Equations
BibTeX
@misc{yang2026sharp,
title = {{Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation}},
author = {Wanrong Yang and Jiahong Wu and Xiaoping Zhai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34610},
eprint = {2609.34610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34610},
}
RIS
TY - UNPB
TI - Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
AU - Wanrong Yang
AU - Jiahong Wu
AU - Xiaoping Zhai
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34610
UR - https://arxiv.org/abs/2609.34610
AB - We study the linearized dynamics of the Kolmogorov flow U^(0)=(asin x_2,0) for the two-dimensional incompressible Navier--Stokes equations with horizontal dissipation ν∂_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 √(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.
ER -