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The second-order Fermi acceleration (Fermi-II) driven by turbulence may beresponsible for the electron acceleration in blazar jets. We test this modelwith time-dependent simulations. The hard electron spectrum predicted by theFermi-II process agrees with the hard photon spectrum of 1ES 1101-232. Forother blazars that show softer spectra, the Fermi-II model requires radialevolution of the electron injection rate and/or diffusion coefficient in theoutflow. Such evolutions can yield a curved electron spectrum, which canreproduce the synchrotron spectrum of Mrk 421 from the radio to the X-rayregime.The photon spectrum in the GeV energy range of Mrk 421 is hard to fitwith a synchrotron self-Compton model. However, if we introduce an externalradio photon field with a luminosity of $4.9 \times10^{38}~\mbox{erg}~\mbox{s}^{-1}$, GeV photons are successfully produced viainverse Compton scattering. The temporal variability of the diffusioncoefficient or injection rate causes flare emission. The observed synchronicityof X-ray and TeV flares implies a decrease of the magnetic field in the flaringsource region.

Time-Dependent Models for Blazar Emissions with the Second-Order Fermi Acceleration