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Souček, O.; Martinec, Z.; Velímský, J.

Vector potential formulation of a quasi-static EM induction problem: existence, uniqueness and stability of the weak solution. (English) Zbl 1255.35006

GEM. Int. J. Geomath. 2, No. 2, 265-279 (2011).
Summary: 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.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B35 Stability in context of PDEs
35D30 Weak solutions to PDEs
78A30 Electro- and magnetostatics

Keywords:

well-posedness in the Hadamard sense; initial-boundary value problem; quasi-static approximation
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References:

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