Place a point vortex in an unbounded fluid and the solution is the Lamb-Oseen vortex — a Gaussian core that spreads by viscous diffusion. Place the same vortex near a wall with a no-slip boundary condition and the problem becomes fundamentally harder. The wall generates vorticity of its own, a boundary layer that interacts with the original vortex, and the coupled system has resisted rigorous mathematical treatment for large circulation.
The paper on viscous evolution of a point vortex in a half-plane (arXiv: 2603.21796) proves global existence and uniqueness for all Reynolds numbers — removing the smallness condition that previous results required. The key insight is a decomposition: separate the solution into the Lamb-Oseen vortex (which would exist in unbounded space) and a boundary layer correction (which accounts for the wall). This decomposition allows the techniques developed for the whole-plane problem to be applied to the vortex part, while the boundary layer part is handled by different methods suited to thin viscous layers.
The solution has finite energy for positive times — even though the initial point vortex has infinite energy — and converges to zero in energy norm as time goes to infinity. The vortex eventually diffuses and the boundary layer dissipates.
The through-claim: the mathematical difficulty of vortex-wall interaction is not in either piece separately — the vortex in free space is solved, and boundary layers in simple flows are solved. The difficulty is in their coupling. Decomposing the solution into the two pieces and handling each with the appropriate tools converts one hard problem into two tractable ones. The mathematical structure mirrors the physical structure: a vortex and a wall, interacting but separable.
2603.21796. Mathematical fluid dynamics / Navier-Stokes / point vortex / boundary layer / global existence.