  In this work we explore the evolution of magnetic fields inside strongly
magnetized neutron stars in axisymmetry. We model numerically the coupled field
evolution in the core and the crust. Our code models the Hall drift and Ohmic
effects in the crust, the back-reaction on the field from magnetically-induced
elastic deformation of the crust, the magnetic twist exchange between the crust
and the core, and the drift of superconducting flux tubes inside the core. The
correct hydromagnetic equilibrium is enforced in the core. We find that i) The
Hall attractor found by Gourgouliatos and Cumming in the crust exists also for
configurations when the B-field penetrates into the core. However, the
evolution timescale for the core-penetrating fields is dramatically different
than that of the fields confined to the crust. ii) The combination of Jones'
flux tube drift and Ohmic dissipation in the crust can deplete the pulsar
magnetic fields on the timescale of $150$ Myr if the crust is hot
($T\sim2\times10^8$ K), but acts on much slower timescales for cold neutron
stars, such as recycled pulsars ($\sim 1.8 $ Gyr, depending on impurity
levels). iii) The outward motion of superfluid vortices during the rapid
spin-down of a young highly magnetized pulsar, can result in a partial
expulsion of flux from the core when $B\lesssim 10^{13}$ G. However for
$B\gtrsim2\times 10^{13}$ G, the combination of a stronger magnetic field and a
longer spin period implies that the core field cannot be expelled.
