Solvability of a one-dimensional problem of fluid flow in poroelastic medium with permeable boundaries

The initial-boundary value problem of a one-dimensional viscous fluid flow in a deformable viscous porous medium with permeable boundaries is considered. The governing equations are the equations of mass conservation for each phase, the equation of momentum conservation for a liquid phase in terms of Darcy's law, the equation of momentum conservation for the whole system, and the rheological equation for porosity. The original system of equations in the Lagrange variables is reduced to a third-order equation for the porosity function. The first part of this paper presents the formulation of the problem, the definition of the classical solution to the considered problem, and the existence and uniqueness theorem for the problem of Holder classes. In the second part of this paper, the local theorem of existence and uniqueness for the problem of Holder classes is proved for an incompressible fluid using the Tikhonov-Schauder fixed-point theorem. The physical principle of the maximum porosity function is determined.