On the uniqueness of the solution of an evolution free boundary problem in theory of lubrication

Abstract

We study an evolution free boundary problem issued from hydrodynamic lubrication. The cavitation phenomenon takes place and is described by the Elrod-Adams model. This model is suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The considered model is governed by the parabolic Reynolds equation and the unknows are the pressure of the lubricant contained in the narrow gap between two circular cylinders and its percentage in an elementary volume. The main result of this paper is the uniqueness of the solution for the parabolic free boundary problem. The proof of this result is based on a comparison principle permitting compare two solutions of the problem when their initial values and their values on the boundary can be compared. Previously, we state a continuity result and some monotonicity properties of the solution.