  We present two-dimensional hydrodynamic simulations of self-gravitating
protostellar disks subject to axisymmetric infall from envelopes and
irradiation from the central star, to explore disk fragmentation due to
gravitational instability (GI), and the fragmented clump evolution. We assume
that the disk is built gradually and smoothly by the infall, resulting in good
numerical convergence. We confirm that for disks around solar-mass stars,
infall at high rates at radii beyond ~50 AU leads to disk fragmentation. At
lower infall rates <1e-5 Msun/yr, however, irradiation suppresses
fragmentation. We find that, once formed, the fragments or clumps migrate
inward on typical type-I time scales of ~2e3 yr initially, but later migration
deviates from the type-I time scale when the clump becomes more massive than
the local disk mass, and/or when they starts to open gaps. As they migrate, the
clumps accrete from the disk at a rate 1e-3 to 1e-1 MJupiter/yr, consistent
with analytic estimates that assume a 1-2 Hill radii cross section. Most clumps
can grow to their isolation masses >0.1 Msun quickly. The eventual fates of
these clumps, however, diverges depending on the migration speed: 3 out of 13
clumps become massive enough (brown dwarf mass) to open gaps in the disk and
essentially stop migrating; 4 are tidally destroyed during inward migration; 6
migrate across the inner simulated disk boundary. A simple analytic model for
clump evolution is derived to explain these different fates. Overall, our
results indicate that fast migration, accretion, and tidal destruction of the
clumps pose challenges to the scenario of giant planet formation by GI in situ,
but may provide a formation mechanism for close binary systems.
