Post-collision inviscid limit of Navier–Stokes desingularizations

Determine the inviscid limit, at a post-collision time, of two-dimensional Navier–Stokes solutions initialized by point-vortex measures whose corresponding three-point-vortex dynamics undergoes a collision.

Background

The paper considers point-vortex solutions that cease to be collision-free in finite time. For collision-free configurations, point vortices can be desingularized into solutions of the two-dimensional Euler equations and obtained as inviscid limits of Navier–Stokes solutions. The unresolved issue is what happens after a finite-time collision, when the point-vortex dynamics itself is no longer directly defined.

The authors explicitly identify the post-collision behavior of the Navier–Stokes desingularization as beyond the reach of existing methods and instead study perturbations and regularizations of the three-vortex system. Thus, the post-collision inviscid limit is posed as an unresolved question rather than answered by the paper.

References

For example, assuming that for appropriate circulations $\kappa_1,\dots,\kappa_N$ the initial configuration $a_1,\dots,a_N$ leads to a collision for the solution of the point-vortex system~PV1, what is the inviscid limit of the Navier-Stokes solutions with initial data~Intr3 at a post-collision time $t$? Motivated by this question (the analysis of which seems to be out of reach of existing methods), we study in this article various perturbations of colliding solutions of the 3-vortex system.

— The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions  (2609.10847 - Gallay et al., 9 Sep 2026) in Section 1, Introduction