Long-wave instability of periodic shear flows for the 2D Navier-Stokes equations

Abstract

In 1959, Kolmogorov proposed to study the instability of the shear flow $(\sin (y),0)$ in the vanishing viscosity regime in tori ${\mathbb{T}}_{\hat{I}}\pm \times \mathbb{T}$. 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 $\Vert {\partial }_y^{-1}U{\Vert }_{L^2}>\hat{I}½$ and $\hat{I}\pm \ll \hat{I}½$, with $\hat{I}½$ 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 $\hat{I}\pm$ approaches $0$.

Fixed some typos