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In this paper, we investigate some new local Aronson-B\'enilan type gradientestimates for positive solutions of the porous medium equation $$ u_{t}=\Deltau^{m}, $$ under Ricci flow.  As application, the related Harnack inequalities are derived.  Our results generalize known results. These results in the paper can beregard as generalizing the gradient estimates of Lu-Ni-V\'{a}zquez-Villani andHuang-Huang-Li to the Ricci flow.

Local Aronson-Benolan type gradient estimates for the porous medium type equation under Ricci