The Biot-Savart operator of a bounded domain

TL;DR

This paper develops an integral representation of the velocity field for incompressible fluids in bounded domains, extending the classical Biot-Savart law with a new kernel that accounts for boundary conditions.

Contribution

It introduces a Biot-Savart operator for bounded domains, providing an explicit integral kernel with a singularity, which generalizes the classical law to bounded geometries.

Findings

Constructed the Biot-Savart integral kernel for bounded domains.

Proved the kernel has an inverse-square singularity on the diagonal.

Established the velocity-vorticity relation in bounded settings.

Abstract

We construct the analog of the Biot-Savart integral for bounded domains. Specifically, we show that the velocity field of an incompressible fluid with tangency boundary conditions on a bounded domain can be written in terms of its vorticity using an integral kernel $K_\Omega(x,y)$ that has an inverse-square singularity on the diagonal.