  The presence of superconducting and superfluid components in the core of
mature neutron stars calls for the rethinking of a number of key
magnetohydrodynamical notions like resistivity, the induction equation,
magnetic energy and flux-freezing. Using a multi-fluid magnetohydrodynamics
formalism, we investigate how the magnetic field evolution is modified when
neutron star matter is composed of superfluid neutrons, type-II superconducting
protons and relativistic electrons. As an application of this framework, we
derive an induction equation where the resistive coupling originates from the
mutual friction between the electrons and the vortex/fluxtube arrays of the
neutron and proton condensates. The resulting induction equation allows the
identification of two timescales that are significantly different from those of
standard magnetohydrodynamics. The astrophysical implications of these results
are briefly discussed.
