  We consider the stability of symmetric flows in a two-dimensional
  channel (including the Poiseuille flow). In 2015 Grenier, Guo, and
  Nguyen have established instability of these flows in a particular
  region of the parameter space, affirming formal asymptotics results
  from the 1940's. We prove that these flows are stable outside this
  region in parameter space. More precisely we show that the
  Orr-Sommerfeld operator
  $$ {\mathcal B}
=\Big(-\frac{d^2}{dx^2}+i\beta(U+i\lambda)\Big)\Big(\frac{d^2}{dx^2}-\alpha^2\Big)
-i\beta U^{\prime\prime}\,, $$
  which is defined on $$
  D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\,
u(1)=u^\prime(1)=0\}. $$ is bounded on the half-plane $\Re \lambda \geq 0$ for
$\alpha \gg \beta^{-1/10}$ or $\alpha \ll \beta^{-1/6}$.
