The Biot-Savart operator of a bounded domain

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.

26 pages

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