  We prove that the entanglement entropy of any state evolved under an
arbitrary $1/r^{\alpha}$ long-range-interacting D-dimensional lattice spin
Hamiltonian cannot change faster than a rate proportional to the boundary area
for any $\alpha>D+1$. We also prove that for any $\alpha>2D+2$, the ground
state of such a Hamiltonian satisfies the entanglement area law if it can be
transformed along a gapped adiabatic path into a ground state known to satisfy
the area law. These results significantly generalize their existing
counterparts for short-range interacting systems, and are useful for
identifying dynamical phase transitions and quantum phase transitions in the
presence of long-range interactions.
