The Navier-Stokes equations in $\mathbb R^2_+$ with point vortex initial data: construction of the solution

TL;DR

This paper proves the existence and uniqueness of solutions to the 2D Navier-Stokes equations with point vortex initial data in a half-space, removing previous smallness restrictions by analyzing the linearized operator.

Contribution

It introduces a new functional framework to handle the solution's behavior near the vortex and boundary, extending prior results without smallness constraints.

Findings

Established existence and uniqueness of solutions with point vortex data.

Developed a tailored functional framework for boundary and vortex regions.

Removed the smallness assumption on total mass in the analysis.

Abstract

This is the first of two papers concerning the asymptotic behavior of the incompressible Navier-Stokes equations in a half-space at high Reynolds numbers, with initial data given by a point vortex. In the present work, we establish the existence and uniqueness of solutions subject to the non-slip boundary condition. This result was established in \cite{Ken} under the condition that the total mass is sufficiently small. Here, we eliminate the smallness assumption by analyzing the linearized operator near the point vortex and constructing a tailored functional framework-one designed to capture the distinct behaviors of the solution in the vicinity of the point vortex and the boundary, respectively.