  Constant-pressure constant-temperature {\it ab initio} molecular dynamics
simulations at high temperatures have been used to study MgSiO$_3$ liquid, the
major constituent of the Earth's lower mantle to conditions of the Earth's
core-mantle boundary (CMB). We have performed variable-cell {\it ab initio}
molecular dynamic simulations at relevant thermodynamic conditions across one
of the measured melting curves. The calculated equilibrium volumes and
densities are compared with the simulations using an orthorhombic perovskite
configuration under the same conditions. For molten MgSiO$_3$, we have
determined the diffusion coefficients and shear viscosities at different
thermodynamic conditions. Our results provide new constraints on the properties
of molten MgSiO$_3$ at conditions near the core-mantle boundary. The volume
change on fusion is positive throughout the pressure-temperature conditions
examined and ranges from 5% at 88 GPa and 3500 K to 2.9% at 120 GPa and 5000 K.
Nevertheless, neutral or negatively buoyant melts from (Mg,Fe)SiO$_3$
perovskite compositions at deep lower mantle conditions are consistent with
existing experimental constraints on solid-liquid partition coefficients for
Fe. Our simulations indicate that MgSiO$_3$ is liquid at 120 GPa and 4500 K,
consistent with the lower range of experimental melting curves for this
material. Linear extrapolation of our results indicates that the densities of
liquid and solid perovskite MgSiO$_3$ will become equal near 180 GPa.
