Journal Articles SIAM Journal on Mathematical Analysis Year : 2024

Properties of the Biot-Savart operator acting on surface currents

Abstract

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.
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Dates and versions

hal-04271879 , version 1 (06-11-2023)
hal-04271879 , version 2 (28-11-2024)

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Wadim Gerner. Properties of the Biot-Savart operator acting on surface currents. SIAM Journal on Mathematical Analysis, 2024, 56 (5), pp.6446-6482. ⟨10.1137/23M1615693⟩. ⟨hal-04271879v2⟩
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