Relativistic fluid dynamics and the theory of relativistic detonation fronts are used to estimate the space-time dynamics of the burning of the D-T fuel in Laser driven pellet fusion experiments. The initial High foot heating of the fuel makes the compressed target transparent to radiation, and then a rapid ignition pulse can penetrate and heat up the whole target to supercritical temperatures in a short time, so that most of the interior of the target ignites almost simultaneously and instabilities will have no time to develop. In these relativistic, radiation dominated processes both the interior, time-like burning front and the surrounding space-like part of the front will be stable against Rayleigh-Taylor instabilities. To achieve this rapid, volume ignition the pulse heating up the target to supercritical temperature should provide the required energy in less than ~ 10 ps.
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