  Sommerfeld paradox roughly says that mathematically Couette linear shear is
linearly stable for all Reynolds number, but experimentally arbitrarily small
perturbations can induce the transition from the linear shear to turbulence
when the Reynolds number is large enough. The main idea of our resolution of
this paradox is to show that there is a sequence of linearly unstable shears
which approaches the linear shear in the kinetic energy norm but not in the
enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes
in the Fourier series of all the shears. In experiments, such linear
instabilities will manifest themselves as transient nonlinear growth leading to
the transition from the linear shear to turbulence no matter how small the
intitial perturbations to the linear shear are. Under the Euler dynamics, these
oscillatory shears are steady, and cat's eye structures bifurcate from them as
travelling waves. The 3D shears $U(y,z)$ in a neighborhood of these oscillatory
shears are linearly unstable too. Under the Navier-Stokes dynamics, these
oscillatory shears are not steady rather drifting slowly. When these
oscillatory shears are viewed as frozen, the corresponding Orr-Sommerfeld
operator has unstable eigenvalues which approach the corresponding inviscid
eigenvalues when the Reynolds number tends to infinity. All the linear
instabilities mentioned above offer a resolution to the Sommerfeld paradox, and
an initiator for the transition from the linear shear to turbulence.
