Confinement of vorticity in two dimensional ideal incompressible exterior flow

In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like $\mathcal{O}(t^{(1/2)+\varepsilon})$, for any $\varepsilon$ $\vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $\mathcal{O}(t^{1/3})$. The purpose of the present article is to refine Marchioro's results. We will prove that, if the initial vorticity is even with respect to the origin, then the exponent for the exterior of the disk may be improved to $1/4$. For flows in the exterior of a smooth, connected, bounded domain we prove a confinement estimate with exponent $1/2$ (i.e. we remove the $\varepsilon$) and in certain cases, depending on the harmonic part of the flow, we establish a logarithmic improvement over the exponent $1/2$. The main new ingredients in our approach are: (1) a detailed asymptotic description of solutions to the exterior Poisson problem near infinity, obtained by the use of Riemann mappings; (2) renormalized energy estimates and bounds on logarithmic moments of vorticity and (3) a new a priori estimate on time derivatives of logarithmic perturbations of the moment of inertia.