A generalization of the Elrod–Adams model of cavitation in lubricated devices is proposed, such that the translation velocity $$V$$ V for the saturation field $$\theta $$ θ can be given any value between $$S/2$$ S / 2 and $$S$$ S , with $$S$$ S being the relative speed of the surfaces. The lack of uniqueness of the classical model when $$V\ne S/2$$ V ≠ S / 2 is explained and a suitable supplementary condition is proposed to fix this issue. The new model is rigorously analyzed, though in the simplified mathematical setting of a one-dimensional problem with a single pressurized region. The main result states the existence of a unique solution globally in time, unless of course the cavitation boundary leaves the domain or disappears. A few preliminary numerical examples are included to illustrate the model.