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Solvability of the system of equations of one-dimensional movement of a viscous liquid in a deformable viscous porous medium

The initial-boundary value problem for the system of one-dimensional motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Hölder classes.


Introduction
The problem of fluid filtration in a deformable porous medium is relevant for solving important applied problems [1][2][3]. In particular, it is the problems of hemodynamics, as well as the movement of other physiological fluids in the muscle tissue [4]. It is necessary to take into account the compressibility of the fluid and the skeleton when solving problems of the motion of gases or carbonated liquids in porous media. The purpose of this study is to take into account the compressibility of a porous skeleton and justify the filtration model of a viscous compressible fluid in a deformable porous medium. The numerical studies of various initial boundary-value problems for systems of equations describing process of fluid filtration through deformable viscoelastic medium were carried out in [2,5]. The local solvability of the problem of filtration of compressible fluid in poroelastic medium was established in [6]. Structurally similar systems of equations was considered in [7,9]. In these studies, based on a number of simplifying assumptions, the original system were reduced to one higher order equation. The local solvability of the Cauchy problem in Sobolev spaces was established in [7]. Travelling wave solutions have been studied in [8,9].

Problem statement
The system of equations describing the one-dimensional unsteady motion of a compressible fluid in a viscous porous medium in the domain (x, t) ∈ Q T = Ω × (0, T ), Ω = (0, 1), is as follows [5,10]: The problem is written in the Eulerian coordinates (x, t). The real density of the solid particles ρ s are assumed constant, and clapeyron dependence is taken for fluid p f = Rρ f , R = const > 0. At the boundary of the region Ω, the velocities of the phases v s , v f are set, and at the initial moment of time -the density ρ 0 f (x) and porosity φ 0 (x). The following conditions are considered for the system (1)- (4): After passing to Lagrange variables by velocity of solid phase we convert the system of equation and get following system of equations for porosity and density of fluid [6]: where For system (6)-(7) conditions (5) could be written as where
Since the function ψ = G(φ) is strictly monotone, at φ ∈ (0, 1), that the inverse function is exist: Here where K 1 is an arbitrary positive constant, while the positive constant K 2 will be given later. We note that on the set V following inequalities hold: Let us construct an operator Λ mapping V in V . Suppose thatω,ρ ∈ V . Using (10), we define the function ω by the equality From the representation (11) it follows that smoothness ω is determined by the smoothness of functionsρ,ω, ρ 0 , p 0 and g. Therefor there exists a value t 1 = t 1 (m 0 , M 0 , m 1 , M 1 ), such that for all t 0 ≤ t 1 the following inequality holds 0 < m 0 /2 ≤ φ(x, t) ≤ (M 0 + 1)/2, (x, t) ∈ Q t 0 . In particular, we have an estimate We also have the estimate for function ω(x, t): G(m 0 /2) ≤ ω(x, t) + G(φ 0 ) ≤ G((M 0 + 1)/2). Using (9),ρ andω(x, t), we find the function ρ(x, t) as a solution of the problem (here and elsewhere, we assume that the initial and boundary conditions are matched): The equation for ρ(x, t) is uniformly parabolic. In view of the properties ofω(x, t) and ρ 0 (x) problem (12) has a classical solution [13]. In addition, we have the following estimate: Under the additional condition smallness for the value of the time interval the following statement holds. Lemma 1. There exists such a t 2 , that when t 0 ≤ min(t 1 , t 2 ), the classical solution of problem (12) satisfies the following inequality in Q t 0 : 0 <m 1 ≤ ρ(x, t) + ρ 0 (x) ≤M 1 < ∞.
Proof. Further, setting U (x, t) = ρ(x, t) + ρ 0 (x), we can express problem (13) in the form Passing to the limit as ε → 0, we find that z (0) = 0, i.e. U ≥ 0. We can express problem (13) in the form: Following [13], we move from function U (x, t) to a new function w(x, t) related to it by equality w(x, t) = e −λt ϕ(x)U (x, t), where and number λ will be indicated later. Function w because of (14) is the solution of the equation the function w reaches a positive maximum at t = 0, such Therefore, we obtain upper bound for U : U ≤ e λt M 1 (1 + g 0 /((1 − M 0 )R)). Then there exists a valuet 2 = ln 2 1/λ , that for all t ≤t 2 we have the estimate for ρ from above from Lemma 1. To obtain a lower estimate we represent equation (14) in the form (z(x, t) = 1/U (x, t)) Then in the same way we obtain the estimate Then there exists a valuet 3 = ln 2 1/λ 1 , that for all t ≤t 3 we have the estimate for ρ from above from Lemma 1. Choosing t ≤ t 2 = min{t 2 ,t 3 }, we proof Lemma 1.
Following [6], we obtain the necessary estimates. Thus, the operator Λ maps the set V into itself for sufficiently small values of t 0 . Using the obtained estimates, we can easily show the continuity of the operator Λ in the norm of the space C 2+β,1+β/2 (Q t 0 ). By the Tikhonov-Schauder theorem, there exists a fixed point (ρ, ω) ∈ V of the operator Λ. Uniqueness is established in the standard way [6]. Theorem 1 is proved.

The case of an incompressible media
If the density of the liquid phase is constant (ρ f = const) the system of equations (6)- (7) can be reduced to one equation for the porosity of φ in Lagrange variables: This equation is supplemented by the following initial-boundary conditions: Definition 2. The solution of problem (15)-(16) is the function (φ t , φ xxt ) ∈ C α,β (Q T ), such that 0 < φ < 1. This functions satisfy the equation (15) and the initial and boundary conditions (16) and regarded as continuous functions in Q T .
Proof. For the function G we have for s > 0 where c 1 , c 2 , c 3 , c 4 are arbitrary constants, It is enough to check the non-negativity of functionsL, L. Lemma 3 is proved. Proof. Proof fully follows by [14]. Further we use the theory of elliptic equations for the function ∂G/∂t set forth in [11]. Theorem 3 is proved.