  With a view of applications to the simulations of supernova explosion and
proto neutron star cooling, we derive the Boltzmann equations for the neutrino
transport with the flavor mixing based on the real time formalism of the
nonequilibrium field theory and the gradient expansion of the Green function.
The relativistic kinematics is properly taken into account. The advection terms
are derived in the mean field approximation for the neutrino self-energy whiles
the collision terms are obtained in the Born approximation. The resulting
equations take the familiar form of the Boltzmann equation with corrections due
to the mixing both in the advection part and in the collision part. These
corrections are essentially the same as those derived by Sirera et al. for the
advection terms and those by Raffelt et al. for the collision terms,
respectively, though the formalism employed here is different from theirs. The
derived equations will be easily implemented in numerical codes employed in the
simulations of supernova explosions and proto neutron star cooling.
