On the stability of laminar flows between plates

Consider a two-dimensional laminar flow between two plates, so that $(x_1,x_2)\in \mathbb{R}\times [-1,1]$, given by ${\mathbf v}(x_1,x_2)=(U(x_2),0)$,where$U\in C^4([-1,1])$satisfies$U^\prime\neq0$in$[-1,1]$. We prove that the flow is linearly stable in the large Reynolds number limit, in two different cases: $\bullet$ $\sup_{x\in[-1,1]} |U"(x)| + \sup_{x\in[-1,1]} |U"(x)| \ll \min_{x\in[-1,1]}|U^\prime(x)|$(nearlyCouetteflows),$\bullet$ $U^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\ne }0$ in $[-1,1]$. We assume either no-slip or fixed traction force conditions on the plates, and an arbitrary large (but much smaller than the Reynolds number) period in the $x_1$ direction.