When solving numerically approximations of the Vlasov-Maxwell equations, the source terms in Maxwell's equations coming from the numerical solution of the Vlasov equation do generally not satisfy the continuity equation which is necessary for Maxwell's equations to be well-posed. Hence it is necessary to introduce generalized Maxwell's equations which are still well-posed when there are errors in the sources. Different such formulations have been introduced previously. The aim of this paper is to perform their mathematical analysis and verify the existence and uniqueness of the solution.