An unconditional existence result for the quasi-variational elastohydrodynamic free boundary value problem

In this paper a two-dimensional quasi-variational inequality arising in elastohydrodynamic lubrication is studied for non-constant viscosity. So far, existence results for such piezo-viscous problems require an L∞ property for an auxiliary problem. For the usual pressure-viscosity relations, this property needs small data assumptions which are not observed in experimental conditions. In the present work, such small data assumptions are proved unnecessary for existence results. Besides well-established monotonicity behavior for the viscosity-pressure relation, the only condition used here is on the asymptotic behavior for this law as the pressure tends to infinity. If the procedure used here, namely the introduction of a reduced pressure by Grubin transform followed by a regularization procedure, appears somewhat classical, the way in which an upper bound is obtained is completely new.