  Strong gamma-ray flares from the Crab Nebula have been recently discovered by
AGILE and confirmed by Fermi-LAT. We study here the spectral evolution in the
gamma-ray energy range above 50 MeV of the September 2010 flare that was
simultaneously detected by AGILE and Fermi-LAT. We revisit the AGILE spectral
data, and present an emission model based on rapid (within 1 day) acceleration
followed by synchrotron cooling. We show that this model successfully explains
both the published AGILE and Fermi-LAT spectral data showing a rapid rise and a
decay within 2-3 days. Our analysis constrains the acceleration timescale and
mechanism, the properties of the particle distribution function, and the local
magnetic field. The combination of very rapid acceleration, emission well above
100 MeV, and the spectral evolution consistent with synchrotron cooling
contradicts the idealized scenario predicting an exponential cutoff at photon
energies above 100 MeV. We also consider a variation of our model based on even
shorter acceleration and decay timescales which can be consistent with the
published averaged properties.
