Unified theory on V-states structures for active scalar equations

Abstract

This paper revolves around the existence of V-states close to Rankine vortices for active scalar equations with completely monotone kernels. This allows to unify various results on this topic related to geophysical flows. A key ingredient is a new factorization formula for the spectrum using a universal function which is independent of the model. This function admits several interesting properties allowing to track the spectrum distribution.