  Active galactic nuclei (AGN) drive fast winds in the interstellar medium of
their host galaxies. It is commonly assumed that the high ambient densities and
intense radiation fields in galactic nuclei imply short cooling times, thus
making the outflows momentum-conserving. We show that cooling of high-velocity,
shocked winds in AGN is in fact inefficient in a wide range of circumstances,
including conditions relevant to ultra-luminous infrared galaxies (ULIRGs),
resulting in energy-conserving outflows. We further show that fast
energy-conserving outflows can tolerate a large amount of mixing with cooler
gas before radiative losses become important. For winds with initial velocity
v_in>~10,000 km s^-1, as observed in ultra-violet and X-ray absorption, the
shocked wind develops a two-temperature structure. While most of the thermal
pressure support is provided by the protons, the cooling processes operate
directly only on the electrons. This significantly slows down inverse Compton
cooling, while free free cooling is negligible. Slower winds with v_in~1,000 km
s^-1, such as may be driven by radiation pressure on dust, can also experience
energy-conserving phases but under more restrictive conditions. During the
energy-conserving phase, the momentum flux of an outflow is boosted by a factor
~v_in/2v_s by work done by the hot post-shock gas, where v_s is the velocity of
the swept-up material. Energy-conserving outflows driven by fast AGN winds
(v_in~0.1c) may therefore explain the momentum fluxes Pdot>>L_AGN/c of
galaxy-scale outflows recently measured in luminous quasars and ULIRGs. Shocked
wind bubbles expanding normal to galactic disks may also explain the
large-scale bipolar structures observed in some systems, including around the
Galactic Center, and can produce significant radio, X-ray, and gamma-ray
emission. [Abridged]
