Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909805715652608 |
|---|---|
| author | Colombo, Maria Dolce, Michele Montalto, Riccardo Ventura, Paolo |
| author_facet | Colombo, Maria Dolce, Michele Montalto, Riccardo Ventura, Paolo |
| contents | In 1959, Kolmogorov proposed to study the instability of the shear flow $(\sin(y),0)$ in the vanishing viscosity regime in tori $\mathbb{T}_α\times \mathbb{T}$. This question was later resolved by Meshalkin and Sinai. We extend the problem to general shear flows $(U(y),0)$ and show that every $U(y)$ exhibits long-wave instability whenever $\|\partial_y^{-1} U\|_{L^2} > ν$ and $α\ll ν$, with $ν$ being the kinematic viscosity. This instability mechanism confirms previous findings by Yudovich in 1966, supported also by several numerical results, and is established through two independent approaches: one via the construction of Kato's isomorphism and one via normal forms. Unlike in many other applications of the latter methods, both proofs deal with the presence of a delicate term in the linearized operator that becomes singular as $α$ approaches $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations Colombo, Maria Dolce, Michele Montalto, Riccardo Ventura, Paolo Analysis of PDEs Fluid Dynamics 76D05, 76D33, 35Q30, 35Q35, 35P15 In 1959, Kolmogorov proposed to study the instability of the shear flow $(\sin(y),0)$ in the vanishing viscosity regime in tori $\mathbb{T}_α\times \mathbb{T}$. This question was later resolved by Meshalkin and Sinai. We extend the problem to general shear flows $(U(y),0)$ and show that every $U(y)$ exhibits long-wave instability whenever $\|\partial_y^{-1} U\|_{L^2} > ν$ and $α\ll ν$, with $ν$ being the kinematic viscosity. This instability mechanism confirms previous findings by Yudovich in 1966, supported also by several numerical results, and is established through two independent approaches: one via the construction of Kato's isomorphism and one via normal forms. Unlike in many other applications of the latter methods, both proofs deal with the presence of a delicate term in the linearized operator that becomes singular as $α$ approaches $0$. |
| title | Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations |
| topic | Analysis of PDEs Fluid Dynamics 76D05, 76D33, 35Q30, 35Q35, 35P15 |
| url | https://arxiv.org/abs/2509.18070 |