Long time evolution of a pair of 2D viscous point vortices

Abstract

This paper studies the long-time evolution of two point vortices under the 2D Navier-Stokes tokes equations. Starting from initial data given by a pair of Dirac measures, we derive an asymptotic expansion for the vorticity over time scales significantly longer than the advection time, yet shorter than the diffusion time. Building on previous works \cite{GS24-1, DG24}, we construct suitable approximate solutions $\Omega_a$ and employ Arnold's method to define a nonlinear energy functional $E_\ve[\om]$, with respect to which the linearized operator $\Lambda^{E,\star}$ around $\Omega_a$ is nearly skew-adjoint. A key innovation in this work is the introduction of ``pseudo-momenta'': $\varrho^e_a, \varrho^o_a,\varrho^{te}_a, \varrho^{to}_a$, which correspond to eigenfunctions or other nontrivial elements in invariant subspaces of $\Lambda^E$, derived from the Lie structure of the 2D Euler equations.