  We construct a phase field model including hydrodynamics and elasticity in
one-component systems. It can be used to investigate solid-liquid and
liquid-liquid phase transitions. Upon first-order phase transition, a velocity
field is induced around interfaces in the presence of a density difference
between the two phases even without applied shear flow. As applications, we
present simulation results on two cases of melting, where a solid domain is
placed on a heated wall in one case and is suspended in a warmer liquid under
shear flow in the other. We find that the solid domain moves or rotates as a
whole due to elasticity, releasing latent heat. We also examine the
liquid-liquid phase transition of a highly viscous domain into a less viscous
liquid on a heated wall, where an inhomogeneous velocity field is induced
within a projected part of the domain. In these phase transitions, the
interface temperature is nearly equal to the coexisting temperature $T_{\rm
cx}(p)$ away from the heated wall in the presence of heat flow in the
surrounding liquid.
