[en] Numerical simulations suggest that merging double white dwarfs (WDs) may produce a newborn neutron star surrounded by a fossil disk. We investigate the evolution of the fossil disk following the coalescence of double WDs. We demonstrate that the evolution can be mainly divided into four phases: the slim disk phase (with time ≲1 years), the inner slim plus outer thin disk phase ($\sim <!--\; ???\; -->10$–${10}^6$ years), the thin disk phase ($\sim <!--\; ???\; -->{10}^2$–${10}^7$ years), and the inner advection-dominated accretion flow plus outer thin disk phase, given the initial disk mass ∼0.05–0.5 $M_{\odot <!--\; ???\; -->}$ and the disk formation time 10⁻³–1 s. Considering possible wind mass loss from the disk, we present both analytic formulae and numerically calculated results for the disk evolution, which is sensitive to the condition that determines the location of the outer disk radius. The systems are shown to be very bright in X-rays in the early phase, but quickly become transient within ≲100 years, with peak luminosities decreasing with time. We suggest that they might account for part of the very faint X-ray transients around the Galactic center region, which generally require a very low mass transfer rate.