What happens if two vortices collide? Do they merge? Bounce off each other conserving angular/linear momentum? Pass thru? Mutually destruct?
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There's plenty of special-case solutions to simple regimes of Navier-Stokes and the entire CFD industry is based on numerical solutions.
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The Clay prize I think is for finding general existence conditions, not solving in practice. I thin vortex collisions might be tractable.
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Galaxies do this often, I think the answer is: it depends ;)


