We try to bridge the gap between the theory of linear density-velocity-gravitational perturbations in the early universe, and the relaxed galaxies we observe today. We succeed quantitatively for dark matter if dark matter is warm. The density runs of baryons and of dark matter of relaxed galaxies are well described by hydro-static equations. The evolution from initial linear perturbations to final relaxed galaxies is well described by hydro-dynamical equations. These equations necessarily include dark matter velocity dispersion. If the initial perturbation is large enough, the halo becomes self-gravitating. The adiabatic compression of the dark matter core determines the final core density, and provides a negative stabilizing feedback. The relaxed galaxy halo may form adiabatically if dark matter is warm. The galaxy halo radius continues to increase indefinitely, so has an ill-defined mass.
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