On the solvability of a variational inequality in the filtration theory

In this paper, we proved the generalized solvability of a problem describing the process of unsteady saturated-unsaturated fluid filtration in a porous medium with the condition of unilateral permeability to parts of the boundary. It should be noted that the variational inequality that arises in this case is a variational inequality of a variable type: in the saturated filtration zone – elliptical and parabolic – otherwise. In the generalized formulation of the problem under consideration, a classical transition based on the Kirchhoff transform to an equivalent variational problem that is more convenient for research was used. In this paper, we considered the most interesting case, from the point of applications, when the Kirchhoff transform maps the real axis into a semi-axis bounded below: $[-\gamma,+\infty).$ It is assumed that the value of the Kirchhoff transform at a point $-\gamma$ is zero. Along with the original problem with restriction, we considered the so-called “predefined problem” without restrictions $u(x,t) \geq -\gamma$, the solution of which on the set $(-\infty,-\gamma)$ is assumed to be zero. Definitions of generalized solutions to these problems in the form of variational inequalities were given. The proof of the existence theorem for a generalized solution of the “predefined problem” was carried out using the methods of half-sampling and penalty. In conclusion, it was proved that the solution to the “predetermined problem” is the solution to the original one.