We consider nonlocal effects that arise when radiation emitted at one
radius of an accretion disk either heats or cools gas at other radii t
hrough Compton scattering. We discuss three situations: 1. Radiation f
rom the inner regions of an advection-dominated flow Compton cooling g
as at intermediate radii and Compton heating gas at large radii. 2. So
ft radiation from an outer thin accretion disk Compton cooling a hot o
ne- or two-temperature flow on the inside. 3. Soft radiation from an i
nner thin accretion disk Compton cooling hot gas in a surrounding one-
temperature flow. We describe how previous results are modified by the
se nonlocal interactions. We find that Compton heating or cooling of t
he gas by the radiation emitted in the inner regions of a hot flow is
not important. Likewise, Compton cooling by the soft photons from an o
uter thin disk is negligible when the transition from a cold to a hot
flow occurs at a radius greater than some minimum R-tr,R-min. However,
if the hot how terminates at R < R-tr,R-min, nonlocal cooling is so s
trong that the hot gas is cooled to a thin disk configuration in a run
away process. In the case of a thin disk surrounded by a hot one-tempe
rature flow, we find that Compton cooling by soft radiation dominates
over local cooling in the hot gas for M greater than or similar to 10(
-3)alpha M-Edd and R less than or similar to 10(4)R(Schw). As a result
, the maximum accretion rate for which an advection-dominated one-temp
erature solution exists decreases by a factor of similar to 10 compare
d with the value computed under an assumption of local energy balance.