Decay Asymptotics of the Viscous Camassa–Holm Equations in the Plane

Abstract

We consider the vorticity formulation of the two-dimensional viscous Camassa–Holm equations in the whole space. We establish global existence for solutions corresponding to initial data in $L^1$ and describe the large time behavior of solutions with sufficiently small and localized initial data. We calculate the rate at which such solutions approach an “unfiltered” Oseen vortex by computing the rate at which the solution of a scaled vorticity problem approaches the solution to a corresponding linearized equation.