The Navier-Stokes equations in $\mathbb R^2_+$ with point vortex initial data: Zero-viscosity limit

TL;DR

This paper rigorously analyzes the zero-viscosity limit of Navier-Stokes solutions with point vortex initial data in a half-plane, using matched asymptotic expansions to connect viscous and inviscid behaviors.

Contribution

It develops a precise boundary-layer analysis and decomposition of vorticity, establishing convergence to the Lamb-Oseen vortex away from the boundary and to Prandtl boundary layers near it.

Findings

Proves convergence to the Lamb-Oseen vortex in the interior.

Shows solutions approach Prandtl boundary-layer system near the boundary.

Provides refined estimates for vorticity components in different regions.

Abstract

This is the second of two papers devoted to the asymptotic behavior of solutions to the incompressible Navier-Stokes equations in a half-space with point vortex initial data. A major difficulty stems from the interaction between the point vortex initial data and the boundary, which complicates the derivation of a valid asymptotic expansion. To overcome this, we carry out a precise matching between the point vortex and boundary-layer profiles to accurately capture the correct viscous behavior of the vortex in the half-plane. Based on this matched asymptotic analysis, we decompose the vorticity into three components: vorticity near the point vortex, vorticity near the boundary, and vorticity in the transition layer. A key point is that each component must be analyzed in its own distinct region. On this basis, we establish refined estimates and thereby achieve the inviscid limit for the point vortex. Finally, we rigorously prove that solutions to the Navier-Stokes equations converge to the Lamb-Oseen vortex away from the boundary, while approaching the Prandtl boundary-layer system in the near-boundary region.