Properties of the Biot-Savart operator acting on surface currents
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912899529703424 |
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| author | Gerner, Wadim |
| author_facet | Gerner, Wadim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_03108 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Properties of the Biot-Savart operator acting on surface currents Gerner, Wadim Mathematical Physics 14J81, 41A35, 55P99, 78A30, 78A46, 78A55 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. |
| title | Properties of the Biot-Savart operator acting on surface currents |
| topic | Mathematical Physics 14J81, 41A35, 55P99, 78A30, 78A46, 78A55 |
| url | https://arxiv.org/abs/2311.03108 |