  The second-order Fermi acceleration (Fermi-II) driven by turbulence may be
responsible for the electron acceleration in blazar jets. We test this model
with time-dependent simulations. The hard electron spectrum predicted by the
Fermi-II process agrees with the hard photon spectrum of 1ES 1101-232. For
other blazars that show softer spectra, the Fermi-II model requires radial
evolution of the electron injection rate and/or diffusion coefficient in the
outflow. Such evolutions can yield a curved electron spectrum, which can
reproduce the synchrotron spectrum of Mrk 421 from the radio to the X-ray
regime.The photon spectrum in the GeV energy range of Mrk 421 is hard to fit
with a synchrotron self-Compton model. However, if we introduce an external
radio photon field with a luminosity of $4.9 \times
10^{38}~\mbox{erg}~\mbox{s}^{-1}$, GeV photons are successfully produced via
inverse Compton scattering. The temporal variability of the diffusion
coefficient or injection rate causes flare emission. The observed synchronicity
of X-ray and TeV flares implies a decrease of the magnetic field in the flaring
source region.
