  We derive the evolution equations describing a thin axisymmetric disk of gas
and stars with an arbitrary rotation curve that is kept in a state of marginal
gravitational instability and energy equilibrium due to the balance between
energy released by accretion and energy lost due to decay of turbulence. Rather
than adopt a parameterized alpha prescription, we instead use the condition of
marginal gravitational instability to self-consistently determine the position-
and time-dependent transport rates. We show that there is a steady-state
configuration for disks dominated by gravitational instability, and that this
steady state persists even when star formation is taken into account if the
accretion rate is sufficiently large. For disks in this state we analytically
determine the velocity dispersion, surface density, and rates of mass and
angular momentum transport as a function of the gas mass fraction, the rotation
curve, and the rate of external accretion onto the disk edge. We show that
disks that are initially out of steady state will evolve into it on the viscous
timescale of the disk, which is comparable to the orbital period if the
accretion rate is high. Finally, we discuss the implications of these results
for the structure of disks in a broad range of environments, including high
redshift galaxies, the outer gaseous disks of local galaxies, and accretion
disks around protostars.
