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Kelvin-Stuart vortices are classical mixing layer flows with manyapplications in fluid mechanics, plasma physics and astrophysics. We prove thatthe whole family of Kelvin-Stuart vortices is nonlinearly stable forco-periodic perturbations, and linearly unstable for multi-periodic ormodulational perturbations. This verifies a long-standing conjecture since thediscovery of the Kelvin-Stuart cat's eyes flows in the 1960s. Kelvin-Stuartcat's eyes also appear as magnetic islands which are magnetostatic equilibriafor the 2D ideal MHD equations in plasmas. We prove nonlinear stability ofKelvin-Stuart magnetic islands for co-periodic perturbations, and give thefirst rigorous proof of the coalescence instability, which is important formagnetic reconnection.

On the stability and instability of Kelvin-Stuart cat's eyes flows