  The model for pulsar wind nebulae (PWNe) as the result of the
magnetohydrodynamic (MHD) downstream flow from a shocked, relativistic pulsar
wind has been successful in reproducing many features of the nebulae observed
close to the central pulsars. However, observations of well-studied young
nebulae like the Crab Nebula, 3C 58, and G21.5--0.9 do not show the toroidal
magnetic field on a larger scale that might be expected in the MHD flow model;
in addition, the radial variation of spectral index due to synchrotron losses
is smoother than expected in the MHD flow model. We find that pure diffusion
models can reproduce the basic data on nebular size and spectral index
variation for the Crab, 3C 58, and G21.5--0.9. Most of our models use an energy
independent diffusion coefficient; power law variations of the coefficient with
energy are degenerate with variation in the input particle energy distribution
index in the steady state, transmitting boundary case. Energy dependent
diffusion is a possible reason for the smaller diffusion coefficient inferred
for the Crab. Monte Carlo simulations of the particle transport allowing for
advection and diffusion of particles suggest that diffusion dominates over much
of the total nebular volume of the Crab. Advection dominates close to the
pulsar and is likely to play a role in the X-ray half-light radius. The source
of diffusion and mixing of particles is uncertain, but may be related to the
Rayleigh-Taylor instability at the outer boundary of a young PWN or to
instabilities in the toroidal magnetic field structure.
