  Mergers of neutron stars (NSs) and black holes (BHs) are nowadays observed
routinely thanks to gravitational-wave (GW) astronomy. In the isolated
binary-evolution channel, a common-envelope (CE) phase of a red supergiant
(RSG) and a compact object is crucial to sufficiently shrink the orbit and
thereby enable a merger via GW emission. Here, we use the outcomes of two
three-dimensional (3D) magneto-hydrodynamic CE simulations of an initially 10.0
solar-mass RSG with a 5.0 solar-mass BH and a 1.4 solar-mass NS, respectively,
to explore the further evolution and final fate of the post-CE binaries.
Notably, the 3D simulations reveal that the post-CE binaries are likely
surrounded by circumbinary disks (CBDs), which contain substantial mass and
angular momentum to influence the subsequent evolution. The binary systems in
MESA modelling undergo another phase of mass transfer (MT) and we find that
most donor stars do not explode in ultra-stripped supernovae (SNe), but rather
in Type Ib/c SNe. The final orbits of our models with the BH companion are too
wide, and NS kicks are actually required to sufficiently perturb the orbit and
thus facilitate a merger via GW emission. Moreover, by exploring the influence
of CBDs, we find that mass accretion from the disk widens the binary orbit,
while CBD-binary resonant interactions can shrink the separation and increase
the eccentricity depending on the disk mass and lifetime. Efficient resonant
contractions may even enable NS/BH to merge with the remnant He stars before a
second SN explosion, which may be observed as gamma-ray burst-like transients,
luminous fast blue optical transients and Thorne-\.Zytkow objects. For the
surviving post-CE binaries, the CBD-binary interactions may significantly
increase the GW-induced double compact merger fraction. We conclude that
accounting for CBD may be crucial to better understand observed GW mergers.

High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics;General Relativity and Quantum Cosmology