On the existence of a solution of one variational inequality of the nonlinear filtration theory

Abstract

© 2019 IOP Publishing Ltd. A theorem on the existence of a generalized solution of one variational inequality describing the process of nonlinear nonstationary filtration of a liquid in a porous medium with the condition of one-way permeability on a part of the boundary is proved. The case is considered, in which the Kirchhoff transformation used in the determination of a generalized solution maps the real axis to the semi-axis bounded from below. In investigating the solvability of the resulting variational inequality with a lower-bound constraint on the solution, an auxiliary problem with no constraints is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem studied in the paper. The solvability of the auxiliary problem is established by means of using the semidiscretization method with a penalty and the Galerkin method.