The generalized surface quasi-geostrophic (gSQG) equation is an active scalar equation with a nonlocal velocity field; in the vortex patch setting, V-states are patch solutions that rotate uniformly without changing their shape. Understanding the stability of such V-states is a classical and difficult problem: even in the closely related two-dimensional Euler equation, the main noncircular example understood in detail is the Kirchhoff ellipse, and very little has been known beyond this special case. In this talk, I will first discuss linear instability, where we construct the V-states quantitatively and track the precise spectral mechanism by which a pair of eigenvalues becomes unstable. I will then explain how this linear mechanism leads to nonlinear orbital instability, and why this instability occurs generically for an open dense set of perturbations.
