  We show the $H^1$ stability of shear flows of Prandtl type: $U^\nu =
(U_s(y/\sqrt{\nu}),0)$, in the steady two-dimensional Navier-Stokes equations,
under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and
$U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which
at most Gevrey stability can be obtained, even under global monotonicity and
concavity hypotheses. It provides the first positive answer to the inviscid
limit problem in Sobolev regularity for a non-trivial class of steady
Navier-Stokes flows with no-slip boundary condition.
