  We consider the angular momentum exchange at the corotation resonance between
a two-dimensional gaseous disk and a uniformly rotating external potential,
assuming that the disk flow is adiabatic. We first consider the linear case for
an isolated resonance, for which we give an expression of the corotation torque
that involves the pressure perturbation, and which reduces to the usual
dependence on the vortensity gradient in the limit of a cold disk. Although
this expression requires the solution of the hydrodynamic equations, it
provides some insight into the dynamics of the corotation region. In the
general case, we find an additional dependence on the entropy gradient at
corotation. This dependence is associated to the advection of entropy
perturbations. These are not associated to pressure perturbations. They remain
confined to the corotation region, where they yield a singular contribution to
the corotation torque. In a second part, we check our torque expression by
means of customized two-dimensional hydrodynamical simulations. In a third
part, we contemplate the case of a planet embedded in a Keplerian disk, assumed
to be adiabatic. We find an excess of corotation torque that scales with the
entropy gradient, and we check that the contribution of the entropy
perturbation to the torque is in agreement with the expression obtained from
the linear analysis. We finally discuss some implications of the corotation
torque expression for the migration of low mass planets in the regions of
protoplanetary disks where the flow is radiatively inefficient on the timescale
of the horseshoe U-turns.
