Viscosity overshoot of entangled polymer melts has been observed under shear flow and uniaxial elongational flow, but not under biaxial elongational flow for a long period of time. We confirmed the presence of the viscosity overshoot under biaxial elongational flows observed in a mixed system of ring and linear polymers expressed by coarse-grained molecular dynamics simulations. The overshoot was found to be more pronounced in a weakly entangled melt. Furthermore, the smallest strain rate showing the strain hardening was found to be depending on the linear chain length as εs (N ) ∼ N −1/2 , which is different from the conventional relationship, εs (N ) ∼ N −2 or N −3.4 for melts of linear polymers without or with entanglements, respectively.We have concluded that the cooperative phenomena of ring and linear chains were enhanced under biaxial elongational flow.