Application is made of Pao's theory for viscoelastic flow together with measurements of the relaxation spectrum to analyze the flow behavior of polymer melts. One result for linear polyethylene is that over a limited range of shear rate, the shear stress becomes a many‐valued function of shear rate. This behavior arises as a natural consequence of the viscoelastic nature of polymers and appears to unify many previous experimental observations of the behavior of polymer melts at high shear rates. The theory predicts an instability in the flow curve of linear polyethylene. It predicts that this discontinuity occurs at different values depending on whether the shear stress is increasing or decreasing. The recoverable strain is predicted to pass through a maximum and the velocity of a material particle in the unstable region can have a number of values at different instants in time. One interpretation of this would be that the flow can occur as a “stick‐slip” process. The effect of hydrostatic pressure on the flow curve is also discussed. All of the results of this analysis have been observed experimentally by others.