The propagation law of the cross spectrum density is employed to derive the analytical expression of the elements of the cross spectrum density matrix in the observation plane for partially coherent vortex beam after propagation under the condition of paraxial approximation. Based on the derived result, the intensity distribution in the observation plane is analyzed. It is shown that different from the completely coherent vortex beam, the partially coherent votex beam has an intensity of the center-point in the observation plane, which gradually becomes prominent after propagation, and the intensity distribution in the observation plane tends to the distribution of Gaussian-like type with the increase of propagation length. The evolution of intensity distribution depends on the topological charge and correlation length of the source beam. On the condition that other parameters of the source beam are invariable, the beam will evolve fast if the topological charge is small and the correlation length is short. Finally, for the first-order partially coherent vortex beam, the detail of the evolution of the beam shape is investigated by studying the extremum of the intensity in the observation plane. And the theoretical proof is presented for the rule of the evolution of the beam.