| Abstract: |
| The Helmholtz Kirchhoff system governs the evolution of two counter rotating point vortices in a 2D inviscid fluid, resulting in constant translation at a constant speed. However, at large but finite Reynolds numbers, the size of the vortex cores grows because of diffusion. This means that the point vortex approximation is not valid over long times for the resulting viscous dipole. This talk aims to systematically define an asymptotic expansion accounting for streamlines deformation from vortex interactions and to understand the finite size effects on the dipole`s translation speed. We then prove that the exact solution remains close to our approximation over a very long time interval, extending unboundedly as the Reynolds number approaches infinity. The proof relies on energy estimates inspired by Arnold`s variational characterization of the steady states of the 2D Euler equation, as recently revised by Gallay and Sverak for viscous fluids. This work is a collaboration with T. Gallay. |
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