On local Solvability of the Problem of Fluid Motion in a Deformable Porous Medium in the Class of Continuous Functions
Abstract
The process of viscous fluid filtration in a deformable porous medium with mainly viscous properties is described by a system of equations that includes the equation of mass conservation for the liquid phase and the poroelastic skeleton, the law of momentum conservation in the form of Darcy law that considers the motion of the solid skeleton, the law of momentum conservation for the system as a whole, and the equation for effective pressure and porosity in the form of Maxwell’s rheological law. When liquid density and the poroelastic skeleton are taken to be constant, the system is closed and it reduces to one nonlinear equation of the porosity function in the Lagrange variables for the one-dimensional case. This paper is devoted to the mathematical justification of the proposed model. Two theorems on the local solvability of the problem of fluid filtration in a poroelastic medium are proved. Paragraph 1 provides an overview of the work. In paragraph 2, we state the problem and formulate the main results of the paper. In paragraph 3, we establish a local existence theorem for a smooth solution of an initial-boundary value problem in the class of continuous functions (theorem 1). In paragraph 4, we establish the local solvability of the problem in Holder classes (theorem 2). Theorem 1 is proved on the basis of the Hilbert theorem for the boundary value problem for an ordinary differential equation of the second order, and Theorem 2 is proved on the basis of the Tikhonov-Schauder theorem on a fixed point. The main point is the proof of the physical maximum principle for porosity.
DOI 10.14258/izvasu(2017)4-25
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