Properties of the Biot–Savart Operator Acting on Surface Currents

• Published in: SIAM journal on mathematical analysis 2024-10, Vol.56 (5), p.6446-6482
• Main Author: Gerner, Wadim
• Format: Article
• Language: English
• Subjects: Mathematics; Physics
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/23M1615693

We investigate properties of the image and kernel of the Biot-Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot-Savart operator is L^2-dense in the space of square-integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighbourhood of the underlying plasma domain can in fact be approximated in any C^k-norm by elements of the image of the Biot-Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot-Savart operator which in particular implies that the dimension of the kernel of the Biot-Savart operator coincides with the genus of the coil winding surface and hence turns out to be a homotopy invariant among regular domains in 3-space. Lastly, we provide an iterative scheme which we show converges weakly in W^{−\frac{1}{2},2}-topology to elements of the kernel of the Biot-Savart operator.