We investigate the possible origin of extended emissions (EEs) of short gamma-ray bursts with an isotropic energy of ~ 10^(50-51) erg and a duration of a few 10 s to ~ 100 s, based on a compact binary (neutron star (NS)-NS or NS-black hole (BH)) merger scenario. We analyze the evolution of magnetized neutrino-dominated accretion disks of mass ~ 0.1 M_sun around BHs formed after the mergers, and estimate the power of relativistic outflows via the Blandford-Znajek (BZ) process. We show that a rotation energy of the BH up to > 10^52 erg can be extracted with an observed time scale of > 30 (1+z) s with a relatively small disk viscosity parameter of alpha < 0.01. Such a BZ power dissipates by clashing with non-relativistic pre-ejected matter of mass M ~ 10^-(2-4) M_sun, and forms a mildly relativistic fireball. We show that the dissipative photospheric emissions from such fireballs are likely in the soft X-ray band (1-10 keV) for M ~ 10^-2 M_sun possibly in NS-NS mergers, and in the BAT band (15-150 keV) for M ~ 10^-4 M_sun possibly in NS-BH mergers. In the former case, such soft EEs can provide a good chance of ~ 6 yr^-1 for simultaneous detections of the gravitational waves with a ~ 0.1 deg angular resolution by soft X-ray survey facilities like Wide-Field MAXI.
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