We present the first applications of a new time-dependent multi-zone jet radiation transfer code to the study the multiwavelength emission of the TeV Blazar Mrk 421. The code couples Fokker-Planck and Monte Carlo methods. For the first time all light travel time effects are fully considered as well as proper self-consistent treatment of Compton cooling, which depends on them. The first tests focus on the March 2001 observations of Mrk 421, still one of the best datasets available for phenomenology and X-ray/TeV data coverage. We summarize the results of scenarios of variability induced by injection of relativistic electrons in a blob encountering a shock, and with different combinations with a second component, either co-spatial or independent from the active region.

Longterm continuous X-ray observations of blazars with MAXI are reported. Thanks to its unprecedentedly high sensitivity as an all sky X-ray monitor, MAXI is an ideal observatory to investigate variability of blazars, which should give a clue to particle acceleration in their jet, as well as the jet dynamics. Actually, since it started its operation in the summer of 2009, MAXI has successfully alerted two strong X-ray flares from the BL Lac object Mrk 421. Especially, in one of these flares, the X-ray flux of the object was found to become the highest in history. By closely examining the MAXI data, the physical quantities associated with the flares were estimated. These results clearly demonstrate the potential of MAXI for the variability of blazars.