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

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