Stability of laminar monotone shear flows in a channel for high Reynolds number

We consider the stability of a laminar flow $U\in C^4([-1,1])$ in the two-dimensional channel $\mathbb{R}\times [-1,1]$ in the large Reynolds number limit. Assuming that $U$ is strictly monotone but allowing $U^{\mathrm{\prime }\mathrm{\prime }}$ to vanish, we obtain that if the operator $${\mathcal{K}}_{\nu }=-\frac{d^2}{d{x}^2}+\frac{U^{\mathrm{\prime }\mathrm{\prime }}}{U-\nu },$$ is strictly positive for all $\nu \in \mathbb{R}$ for which $U^{\mathrm{\prime }\mathrm{\prime }}(U^{-1}(\nu ))=0$,then $U$ is stable for sufficiently large Reynolds number. This contribution generalizes previous results mostly by allowing long wave perturbations (but much shorter than the Reynolds number).