Global Solutions for the Generalized SQG Patch Equation

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter α ∈ ( 1 , 2 ) . The cases α = 0 and α = 1 correspond to 2d Euler and SQG respectively, and our choice of the parameter α results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.