Vector potential formulation of a quasi-static EM induction problem: existence, uniqueness and stability of the weak solution

• Published in: GEM international journal on geomathematics Vol. 2; no. 2; pp. 265 - 279
• Main Authors: Souček, O., Martinec, Z., Velímský, J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.11.2011
• Subjects: Applications of Mathematics; Computational Science and Engineering; Earth Sciences; Mathematics; Mathematics and Statistics; Original Paper; 35A02 (Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness); Vector potential; 78M99 (Optics, Electromagnetic Theory None of the above, but in this section); Existence and uniqueness; Quasi-static EM induction; 35A01 (Existence problems: global existence, local existence, non-existence)
• ISSN: 1869-2672; 1869-2680
• DOI: 10.1007/s13137-011-0019-9

We deal with the electromagnetic induction in a conductor with 3D distribution of electric conductivity in quasi-static approximation with the focus on theoretical aspects related to the solvability of this problem. We formulate the initial, boundary-value problem of electromagnetic induction in terms of a magnetic vector potential only, first in differential and then in integral forms. We prove that the problem is well posed in the Hadamard sense, that a solution exists, is unique and continuously dependent on data. The fact that no electric scalar potential is employed in the formulation and no gauge condition is imposed on the magnetic vector potential makes the formulation attractive for numerical implementations.