  Kelvin-Stuart vortices are classical mixing layer flows with many
applications in fluid mechanics, plasma physics and astrophysics. We prove that
the whole family of Kelvin-Stuart vortices is nonlinearly stable for
co-periodic perturbations, and linearly unstable for multi-periodic or
modulational perturbations. This verifies a long-standing conjecture since the
discovery of the Kelvin-Stuart cat's eyes flows in the 1960s. Kelvin-Stuart
cat's eyes also appear as magnetic islands which are magnetostatic equilibria
for the 2D ideal MHD equations in plasmas. We prove nonlinear stability of
Kelvin-Stuart magnetic islands for co-periodic perturbations, and give the
first rigorous proof of the coalescence instability, which is important for
magnetic reconnection.
