An analysis of the Darwin model of approximation to Maxwell’s equations.

*(English)*Zbl 0755.35137Summary: This paper is concerned with the approximation of the Maxwell equations by the Darwin model, with appears as a correction of the quasi- electrostatic model, including the electric fields generated by magnetic induction. This model is obtained by neglecting the solenoïdal part of the displacement current in the Maxwell equations, and exhibits an elliptic character with infinite propagation speed. In this paper, we study appropriate decompositions of vector fields which give rise to the well-posedness of the Darwin model. Then, we show that the approximation properties of the Darwin model can be analyzed in terms of the dimensionless parameter \(\eta=v/c\), where \(v\) is the characteristic velocity associated with the data, and \(c\) is the speed of light. We show that the Darwin model approximates the Maxwell system up to the second order for the magnetic field, and to the third order for the electric field, with respect to \(\eta\).

##### MSC:

35Q60 | PDEs in connection with optics and electromagnetic theory |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |

35J50 | Variational methods for elliptic systems |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

78A25 | Electromagnetic theory, general |

78A30 | Electro- and magnetostatics |