Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 63, pp. 1–26.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

GLOBAL WEAK SOLUTIONS TO DEGENERATE COUPLED

TRANSPORT PROCESSES IN PARTIALLY SATURATED

DEFORMABLE ELASTIC-INELASTIC POROUS MEDIA

MICHAL BENEŠ

Abstract. In this work we prove the existence of global weak solutions to a
degenerate and strongly coupled parabolic system arising from the transport

processes through partially saturated deformable porous materials. The hygro-
thermal model is coupled with quasi-static evolution equations modeling elastic

and inelastic mechanical deformations. Physically relevant Newton boundary

conditions are considered for water pressure and temperature of the porous
system. The traction boundary condition is imposed on the deformable solid

skeleton of the porous material. Degeneration occurs in both elliptic and

parabolic part of the balance equation for mass of water. The coupling between
water pressure, temperature, stress tensor and internal variables occurs in

transport coefficients, constitutive functions and the decomposition of the total

strain tensor into elastic and plastic parts due to mechanical effect and strain
tensor due to thermal expansion.

1. Introduction

Let Ω be a bounded domain in R2 with Lipschitz boundary ∂Ω. Let T ∈ (0,∞)
be fixed throughout the paper, I := [0, T ], ΩT := Ω × I and ∂ΩT := ∂Ω × I. We
consider the system

∂tθ(p) +∇ · qp = 0 in ΩT , (1.1)

∂t [θ(p)ϑ+ ρϑ] +∇ · qϑ +∇ · [ϑqp] = σd : ∂tε
p` in ΩT , (1.2)

−∇ · [σ − Iχ(p)] = f in ΩT , (1.3)

∂tε
p` = B(ϑ,σ,α) in ΩT , (1.4)

∂tα = C(ϑ,σ,α) in ΩT (1.5)

with the boundary conditions

qp · n = γp(x)(p− p∞) on ∂ΩT , (1.6)

qϑ · n = γϑ(x)(ϑ− ϑ∞) on ∂ΩT , (1.7)

σ · n = t̆ on ∂ΩT (1.8)

2010 Mathematics Subject Classification. 35A01, 35D30, 35B65, 35B45, 35B50, 35K15, 35K40.
Key words and phrases. Second-order parabolic systems, global solution; porous media;

smoothness and regularity; coupled transport processes; elastic-inelastic solids;

internal variables; traction problem; constitutive equations; coercivity; convexity.
c©2020 Texas State University.

Submitted October 8, 2019. Published June 19, 2020.

1



2 M. BENEŠ EJDE-2020/63

and the initial conditions

p(·, 0) = p0, ϑ(·, 0) = ϑ0, εp`(·, 0) = εp`0 , α(·, 0) = α0 in Ω. (1.9)

System (1.1)–(1.9) arises from the coupled moisture movement and heat transport
through partially saturated deformable porous media [6, 28, 31]. The model includes
classical plasticity, elasto-plasticity, inelastic behaviour, hygro-thermal effects, creep
as well as relaxation. In (1.1)–(1.9), p : ΩT → R, ϑ : ΩT → R, σ : ΩT → R4,
εp` : ΩT → R4 and α : ΩT → Rd, d ∈ N, are the unknown functions. In particular,
p corresponds to the water pressure, ϑ represents the temperature, σ denotes the
symmetric stress tensor, εp` is the tensor of plastic deformation and α stands for
the vector of internal state variables (taking into account e.g. the work-hardening
of the porous material). By qp we denote the flow velocity of liquid water and qϑ is

the heat flux through the porous material. p0 : Ω → R, ϑ0 : Ω → R, εp`0 : Ω → R4

and α0 : Ω→ Rd are given functions describing the initial state of the system. By n
we denote the unit outward normal vector with respect to Ω along ∂Ω. In (1.6) and
(1.7), γϑ : ∂Ω→ R represents the heat transfer coefficient function, γp : ∂Ω→ R is
a coefficient function associated with a measure of the permeability of the boundary
to the moisture flow, ϑ∞ is the temperature of the external environment and p∞ is
a fictitious water pressure related to the ambient conditions (the relative humidity,
gas pressure and temperature). In (1.3), f : ΩT → R2 stands for given body forces
and, in (1.8), t̆ : ∂ΩT → R2 represents surface tractions. Further, θ : R → R,
χ : R → R, B : R × R4 × Rd → R4 and C : R × R4 × Rd → Rd are given
functions of primary unknowns. Namely, θ represents the water content, which is
the volume of water per volume of porous medium [31, Chapter 3.7.6], B and C
are given constitutive functions describing the elasto-plastic behavior of the solid
material, see e.g. [22, 23]. ρ is a real positive constant associated with the density
of the solid skeleton. For notational simplicity, we normalized the density of water,
the heat capacity of water and the heat capacity of solid microstructure to 1. On
the right-hand side of (1.2), σd represents the deviatoric part of the stress tensor
σ, σd = σ − 1

3 tr(σ)I, where I denotes the identity matrix. By ε we denote the

symmetric small strain tensor composed of the plastic part εp`, the elastic part εe`

and the thermal dilatation strain εϑ. The deformation of the domain Ω is described
by small displacement theory and the strain tensor ε is defined by

εij(u) =
1

2

( ∂ui
∂xj

+
∂uj
∂xi

)
, i, j = 1, 2, (1.10)

where the vector u : ΩT → R2, u = (u1, u2), describes the displacement of the
material.

In the model, we suppose the following constitutive equations

qp = − (κ(σ)kR(p)/ν(ϑ)) (∇p− eg) , (1.11)

qϑ = −λ(p, ϑ)∇ϑ, (1.12)

εij = εe`ij + εp`ij + εϑij =
1

2

( ∂ui
∂xj

+
∂uj
∂xi

)
, (1.13)

εe`ij =
∂P

∂σij
(σ,α), (1.14)

εϑij = βδij(ϑ− ϑref ). (1.15)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 3

Here, κ : R4 → R is the intrinsic permeability, kR : R → R represents the relative
hydraulic conductivity, λ : R2 → R is the thermal conductivity function, ν : R→ R
is the temperature dependent kinematic viscosity of the fluid and eg := (0, 1) stands
for the normalized gravity vector. Note that the relation (1.14) is a general form of
Hooke’s law with a given function P . In (1.15), δij is the Kronecker delta, β is the
thermal expansion coefficient and ϑref stands for the given reference temperature.

Under isothermal conditions, the quasi-static problem (1.3)–(1.5) and (1.14) with
boundary and initial conditions (1.8) and (1.9)3,4 has been first introduced and the-
oretically studied in [22], [23] and [27]. From the mathematical point of view, the
existence and uniqueness theorems have been proven in [29] and [28, Chapter 9]. At
the same time, the existence and uniqueness results for the displacement boundary-
value problem were given in [19]. Existence and uniqueness results and the con-
tinuous dependence of the solution with respect to the input data for the problem
with mixed boundary conditions have been proven in [32]. Uncoupled thermo-visco-
plastic processes, where the absolute temperature has been introduced in the model
as the internal variable, have been theoretically studied in [33, 34]. More recently,
existence results for the coupled thermo-mechanical models can be found e.g. in
[4, 5, 17, 18].

Further, assuming hygro-thermal processes and ignoring mechanical phenomena,
the existence of the weak solution to the problem (1.1)–(1.2) with homogeneous
Dirichlet boundary conditions or mixed homogeneous Dirichlet-Neumann boundary
conditions is given in [8] and [10], respectively.

Our aim is to prove the existence of the solution to the fully coupled hygro-
thermo-mechanical model (1.1)–(1.9). From theoretical point of view, the difficulty
lies in the coupling of equations and nontrivial structure of the system with non-
symmetrical parabolic part (see also [35]). Moreover, one easily verifies that (1.1)
is a degenerate equation where the degeneracy occurs in both elliptic and parabolic
terms (θ′(p)→ 0 and kR(p)→ 0 as p→ −∞).

The rest of this paper is organized as follows. In Section 2, we introduce basic
notation and suitable function spaces, specify our assumptions on the data in the
problem and present auxiliary results which will be used throughout the paper. In
Section 3, we formulate the problem in the variational sense and state the main re-
sult of the paper, the existence of the global weak solution to (1.1)–(1.9). The main
result is proved in Section 4 by constructing approximates and limiting procedure.

2. Preliminaries

2.1. Notation and function spaces. Vectors, vector functions and matrices are
denoted by boldface letters. Throughout the paper, we will always use positive
constants c, c1, c2, . . . , which are not specified and which may differ from line to
line. We suppose q, q′ ∈ [1,∞), q′ denotes the conjugate exponent to q, q > 1,
1/q + 1/q′ = 1. Lq(Ω) denotes the usual Lebesgue space equipped with the norm
‖ · ‖Lq(Ω) and W k,q(Ω), k ≥ 0 (k need not to be an integer, see [24]), denotes the
usual Sobolev-Slobodecki space with the norm ‖ · ‖Wk,q(Ω). By X ′ we denote the
space of all continuous linear forms on Banach space X.

By Lq(I;X) we denote the usual Bochner space (see e.g. [1]). Further, we define
C(I;X), the space of functions u : I → X continuous in I, equipped with the
norm ‖u‖C(I;X) = maxt∈I ‖u(t)‖X , where ‖ · ‖X denotes the norm in the space X.



4 M. BENEŠ EJDE-2020/63

Finally, let S be the Hilbert space of all symmetric tensor functions such that

S = {τ = (τij); τij ∈ L2(Ω), τij = τji, i, j = 1, 2}
with the inner product

(τ ,σ)S =

∫
Ω

τijσij dx. (2.1)

Here and subsequently, summation convention is used, i.e. summation is performed
over repeated indices.

2.2. Structure and data properties. We now introduce our assumptions on
functions and coefficients in (1.1)–(1.9).

(i) θ ∈ C1(R) is a positive and strictly monotone function such that

0 < θ(ξ) ≤ Cθ < +∞, 0 < θ′(ξ) ≤ CL < +∞ ∀ξ ∈ R (Cθ, CL = const); (2.2)

(ii) χ ∈ C1(R) is Lipschitz continuous and κ, kR, ν, λ are continuous functions
satisfying

0 < κ1 ≤ κ(ξ) ≤ κ2 < +∞ (κ1, κ2 = const) ∀ξ ∈ R4, (2.3)

0 < kR(ξ) < k2 < +∞ (k2 = const) ∀ξ ∈ R, (2.4)

0 < ν1 ≤ ν(ξ) ≤ ν2 < +∞ (ν1, ν2 = const) ∀ξ ∈ R, (2.5)

0 < λ1 ≤ λ(ξ1, ξ2) ≤ λ2 < +∞ (λ1, λ2 = const) ∀ξ1, ξ2 ∈ R; (2.6)

(iii) the constitutive functions P , Bij and Ci are assumed to be smooth enough,
such that Bij = Bji and

∂P

∂τij
(0, ξ) = 0, (2.7)∣∣∣ ∂2P

∂τij∂ξk
(τ , ξ)

∣∣∣+
∣∣∣ ∂2P

∂τij∂τkl
(τ , ξ)

∣∣∣+
∣∣∣ ∂2P

∂ξj∂ξk
(τ , ξ)

∣∣∣ ≤ CP (CP = const), (2.8)

∂2P

∂τij∂τkl
(τ , ξ)ζijζkl ≥ c1ζijζij , c1 > 0, (2.9)∣∣∣∂Bij

∂τkl
(τ , ξ)

∣∣∣+
∣∣∣∂Bij
∂ξk

(τ , ξ)
∣∣∣+
∣∣∣Bij(τ , ξ)

∣∣∣ ≤ CB (CB = const), (2.10)∣∣∣ ∂Ci
∂τkl

(τ , ξ)
∣∣∣+
∣∣∣∂Ci
∂ξj

(τ , ξ)
∣∣∣ ≤ Cc (Cc = const) (2.11)

for all τ ∈ R4, ξ ∈ Rd and ζ ∈ R4;
(iv) the given body forces f in C(I; [L2(Ω)]2) and surface tractions t̆ in

C(I; [L2(∂Ω)]2) satisfy the compatibility condition∫
Ω

f(t) · v dx+

∫
∂Ω

t̆(t) · v dS = 0 (2.12)

for every t from I and v ∈ R, where

R =
{
v ∈ [W 1,2(Ω)]2; v = (a1 − bx2, a2 + bx1), a1, a2, b ∈ R

}
; (2.13)

(v) the functions from (1.6), (1.7) and (1.9) have the following properties:

γp ∈ L∞(∂Ω), γϑ ∈ L∞(∂Ω), (2.14)

such that

0 < γ1 < γp(·) < γ2 on ∂ΩT (γ1, γ2 = const), (2.15)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 5

0 < γ3 < γϑ(·) < γ4 on ∂ΩT (γ3, γ4 = const), (2.16)

further

p0 ∈ L∞(Ω) ∩ L∞(∂Ω), ϑ0 ∈ L2(Ω), εp`0 ∈ L2(Ω)4, α0 ∈ [L2(Ω)]d, (2.17)

p∞ ∈ C(I;L∞(∂Ω)), ϑ∞ ∈ C(I;L∞(∂Ω)), (2.18)

such that

p1 < p∞(·) < p2 on ∂ΩT (p1, p2 = const), (2.19)

ϑ1 < ϑ∞(·) < ϑ2 on ∂ΩT (ϑ1, ϑ2 = const). (2.20)

2.3. Auxiliary results. Here we present some auxiliary results which will be fre-
quently used throughout the paper.

Remark 2.1 ([2, Section 1.1]). Let us note that (i) implies that there is a (strictly)
convex C1-function Φ : R → R, such that b(z) − b(0) = Φ′(z) ∀z ∈ R. Introduce
the Legendre transform

B(z) := (b(z)− b(0))z − Φ(z) + Φ(0) =

∫ z

0

(b(z)− b(s)) ds.

It is not difficult to verify that (see [2])

B(z) =

∫ 1

0

(b(z)− b(sz))z ds ≥ 0 ∀z ∈ R, (2.21)

B(s)−B(r) ≥ (b(s)− b(r))r ∀r, s ∈ R. (2.22)

The following theorem is proven in [24, Theorem 6.4.2], see also [26, Section 2.4].

Theorem 2.2. Let Ω ⊂ R2 and q ≥ 1. Then there exists exactly one continuous
linear mapping T : W 1,2(Ω) → Lq(∂Ω), such that T(u) = u on ∂Ω for all u ∈
C∞(Ω).

As usual, we will denote the trace of u ∈ W 1,2(Ω) on ∂Ω again by u. We often
use the inequality (see the proof of [26, Theorem 1.2] or [13, eq. (3.32)])∫

∂Ω

|u|2 dS ≤ η
∫

Ω

|∇u|2 dx+ c(η)

∫
Ω

|u|2 dx (2.23)

for all u ∈W 1,2(Ω) and all sufficiently small η > 0.
The following useful assertion is proved in [13, Lemma 2 and 3]: let 0 < m <∞

and

{wk}∞k=1 ⊂ L2(I;W 1,2(Ω)) ∩ L∞(I;Lm+1(Ω)),

ess sup0≤t≤T

∫
Ω

|wk(t)|m+1 dx+

∫ T

0

‖wk(t)‖2W 1,2(Ω) dt < c, k = 1, 2, . . .

Moreover, let wk → w a.e. on ΩT . Then

wk → w in Lq+1(ΩT ), 0 ≤ q < m+ 2,

wk → w in Ls+1(∂ΩT ), 0 < s < (2 + min{s,m}+ 1)/2.
(2.24)

The next lemma (Korn’s inequality) follows from [28, Chapter 10.2.2]. First,
define

VR =
{

v ∈ [W 1,2(Ω)]2 :

3∑
i=1

q2
i (v) = 0

}
, (2.25)



6 M. BENEŠ EJDE-2020/63

qi(v) =

∫
Γ

vi dS (i = 1, 2), q3(v) =

∫
Γ

(x1v2 − x2v1) dS,

where Γ is an arbitrary part of the Lipschitz boundary Γ̃ of a domain Ω̃ ⊂ Ω such

that Γ is open in Γ̃ and its measure is positive (Γ ⊂ ∂Ω is also allowed).

Lemma 2.3 (Korn’s inequality). The inequality∫
Ω

εij(v)εij(v) dx ≥ c‖v‖2[W 1,2(Ω)]2 (2.26)

holds for all v ∈ VR. The constant c in the inequality (2.26) depends only on the
domain Ω.

Lemma 2.4. VR is the orthogonal complement of R with respect to [W 1,2(Ω)]2 in
the sense of the inner product

(u,v) =

∫
Ω

εij(u)εij(v) dx+

3∑
i=1

qi(u)qi(v).

The proof of the above lmma is analogous to the proof of [28, Chapter 7.3,
Lemma 3.2]. Let ω : [W 1,2(Ω)]2 → [L2(Ω)]4 be the mapping defined by

ωij(u) =
1

2

( ∂ui
∂xj

+
∂uj
∂xi

)
(2.27)

for all u ∈ [W 1,2(Ω)]2. Further, let K = ω([W 1,2(Ω)]2). We have the following
lemma, see [28, Chapter 9, Lemma 2.1].

Lemma 2.5. K is a closed subspace in S.

Finally, let H be the orthogonal complement of the subspace K in S, i.e. let

S = K ⊕H. (2.28)

3. Main result

The aim of this paper is to prove the existence of a weak solution to problem
(1.1)–(1.9). Recall that the primary unknowns in the model are p, ϑ, σ, εp` and
α. It is worth noting that the displacement field u is determined from (1.10), of
course, except for a rigid-body translation and rotation.

We first formulate our problem in a variational sense.

Definition 3.1. By a weak solution of (1.1)–(1.9) we mean functions p, ϑ, σ, εp`

and α such that

p ∈ L2(I;W 1,2(Ω)), ϑ ∈ L2(I;W 1,2(Ω)), σ ∈ L2(I; [L2(Ω)]4),

εp` ∈ C(I; [L2(Ω)]4), α ∈ C(I; [L2(Ω)]d),

which satisfy the variational equations

−
∫

ΩT

θ(p)∂tϕdxdt+

∫
ΩT

[(κ(σ)kR(p)/ν(ϑ)) (∇p− eg)] · ∇ϕdxdt

+

∫
∂ΩT

γp(x)pϕdS dt

=

∫
Ω

θ(p0)ϕ(x, 0) dx+

∫
∂ΩT

γp(x)p∞ϕdS dt

(3.1)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 7

for any test function ϕ ∈ C∞(ΩT ), ϕ(x, T ) = 0 and for all x ∈ Ω;

−
∫

ΩT

[θ(p)ϑ+ ρϑ]∂tψ dx dt+

∫
ΩT

λ(p, ϑ)∇ϑ · ∇ψ dxdt

+

∫
∂ΩT

γϑ(x)ϑψ dS dt

+

∫
ΩT

ϑ[(κ(σ)kR(p)/ν(ϑ)) (∇p− eg)] · ∇ψ dx dt

+

∫
∂ΩT

ϑγp(x)(p− p∞)ψ dS dt

=

∫
ΩT

σd : B(ϑ,σ,α)ψ dxdt

+

∫
Ω

[θ(p0)ϑ0 + ρϑ0]ψ(x, 0) dx+

∫
∂ΩT

γϑ(x)ϑ∞ψ dS dt

(3.2)

for any test function ψ ∈ C∞(ΩT ), ψ(x, T ) = 0 for all x ∈ Ω;∫
ΩT

σ : ε(v) dxdt+

∫
ΩT

∇χ(p) · v dx dt =

∫
ΩT

f · v dxdt+

∫
∂ΩT

t̆ · v dS dt (3.3)

for any v ∈ C∞(ΩT )2 and∫
Ω

(
εe`(t) + εp`(t) + εϑ(t)

)
: τ dx = 0 (3.4)

for all τ ∈ H and a.a. t ∈ I and, finally,

εp`(t) = εp`0 +

∫ t

0

B (ϑ(s),σ(s),α(s)) ds, (3.5)

α(t) = α0 +

∫ t

0

C(ϑ(s),σ(s),α(s)) ds. (3.6)

The main result of this paper reads as follows.

Theorem 3.2 (Main result). Let the assumptions (i)–(v) be satisfied. Then there
exists at least one weak solution of the system (1.1)–(1.9).

4. Proof of the main result

To prove the main result of the paper we use the method of semi-discretization
in time by constructing temporal approximations and limiting procedure. We first
approximate our problem by semi-discretizing the equations in time by the semi-
implicit scheme and decompose the problem into hygral, thermal and mechanical
parts. The decoupled steady problems are easier to solve combining the theory of
pseudomonotone and potential operators and the Lax-Milgram lemma. We next
construct piecewise constant time interpolants and derive suitable a priori estimates
and employ the sequential weak-compactness arguments. Since we deal with the
nonlinear problem, we need strong convergence. In this work, we apply the Aubin-
Lions lemma [12] and use the technique introduced by Alt and Luckhaus in [2].
Finally, we pass to the limit in the discrete weak formulation to obtain the solution
of the original problem (1.1)–(1.9).



8 M. BENEŠ EJDE-2020/63

4.1. Discretized problem. Let h > 0 be a time step and suppose that T/h is an
integer. We will work with a sequence {hN}N∈N such that limN→+∞ hN → 0. Let

us fix r ∈ N and, for simplicity, assume that hN = T/(2N−1r). In what follows we
often omit the index N and write simply h instead of hN . Further, let us define

(p∞)nN :=
1

h

∫ nh

(n−1)h

p∞(·, s) ds,

(ϑ∞)nN :=
1

h

∫ nh

(n−1)h

ϑ∞(·, s) ds,

fnN :=
1

h

∫ nh

(n−1)h

f(·, s) ds,

t̆nN :=
1

h

∫ nh

(n−1)h

t̆(·, s) ds,

p0
N := p0, ϑ0

N := ϑ0,

(εp`)0
N := εp`0 , α0

N := α0,

n = 1, . . . , 2N−1r.
We approximate our evolution problem by a semi-implicit discrete scheme. Then

we define, in each time step n = 1, . . . , 2N−1r, the set of functions pnN , ϑnN , σnN ,
(εp`)nN and αnN as a solution of the following recurrence steady problem: for the

given functions pn−1
N ∈ L∞(Ω), ϑn−1

N ∈ L2(Ω), (εp`)n−1
N ∈ [L2(Ω)]4 and αn−1

N ∈
[L2(Ω)]d, n = 1, . . . , 2N−1r, find pnN ∈ W 1,s(Ω) with some s > 2, ϑnN ∈ W 1,2(Ω),
σnN ∈ [L2(Ω)]4, (εp`)nN ∈ [L2(Ω)]4 and αnN ∈ [L2(Ω)]d, such that

1

h

∫
Ω

(
θ(pnN )− θ(pn−1

N )
)
ϕdx

+

∫
Ω

(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇ϕdx+

∫
∂Ω

γp(x)pnNϕdS

=

∫
∂Ω

γp(x)(p∞)nNϕdS

(4.1)

for any ϕ ∈W 1,2(Ω);

1

h

∫
Ω

(
θ(pnN )ϑnN − θ(pn−1

N )ϑn−1
N

)
ψ dx

+
ρ

h

∫
Ω

(
ϑnN − ϑn−1

N

)
ψ dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )∇ϑnN · ∇ψ dx

+

∫
Ω

ϑnN
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇ψ dx

+

∫
∂Ω

γϑ(x)ϑnNψ dS +

∫
∂Ω

ϑnNγp(x)(pnN − (p∞)nN )ψ dS

=

∫
Ω

(σd)
n
N : B

(
ϑn−1
N ,σnN ,α

n
N

)
ψ dx+

∫
∂Ω

γϑ(x)(ϑ∞)nNψ dS

(4.2)

for any ψ ∈W 1,2(Ω), where (σd)
n
N := σnN − 1

3 tr(σ
n
N )I;∫

Ω

σnN : ε(v) dx+

∫
Ω

∇χ(pnN ) · v dx =

∫
Ω

fnN · v dx+

∫
∂Ω

t̆nN · v dS (4.3)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 9

for any v ∈ [W 1,2(Ω)]2 and∫
Ω

( ∂P
∂σij

(σnN ,α
n
N ) + (εp`ij )nN + βδij(ϑ

n−1
N − ϑref )

)
τij dx = 0 (4.4)

for all τ ∈ H and, finally,

(εp`)nN = (εp`)n−1
N + hB

(
ϑn−1
N ,σnN ,α

n
N

)
, (4.5)

αnN = αn−1
N + hC

(
ϑn−1
N ,σnN ,α

n
N

)
. (4.6)

The next result proves the xistence of the solution to (4.1)–(4.6).

Theorem 4.1. Let pn−1
N ∈ L∞(Ω), ϑn−1

N ∈ L2(Ω), (εp`)n−1
N ∈ [L2(Ω)]4 and αn−1

N ∈
[L2(Ω)]d be given and the assumptions (i)–(v) be satisfied. Then for every h ∈
(0, h0) with h0 small enough there exists a solution of (4.1)–(4.6).

Proof. The existence of pnN ∈ W 1,2(Ω), the solution to the problem (4.1), follows
from [36, Chapter 2.4]. Moreover, according to [14, Theorem 3, Chapter 4.1] we
also have pnN ∈ W 1,s(Ω) with some s > 2. The W 1,s-regularity of pnN will be used
later.

With pnN ∈W 1,2(Ω) in hand, we now consider the problem to find the elements
σnN ∈ [L2(Ω)]4, (εp`)nN ∈ [L2(Ω)]4 and αnN ∈ [L2(Ω)]d satisfying (4.3)–(4.6). Let us
defineM, the class of statically admissible stress tensors. In particular, we require
σ ∈ S such that∫

Ω

σ : ε(v) dx =

∫
Ω

fnN · v dx+

∫
∂Ω

t̆nN · v dS −
∫

Ω

∇χ(pnN ) · v dx (4.7)

for all v ∈ [W 1,2(Ω)]2. Let us fix arbitrary σ ∈ M. For all h, such that hCc < 1,
based on the Banach contraction principle, we have a unique solution α ∈ [L2(Ω)]d

of the equation
α = αn−1

N + hC
(
ϑn−1
N ,σ,α

)
. (4.8)

For arbitrary σ1,σ2 ∈M we define α1,α2 ∈ [L2(Ω)]d satisfying

αi = αn−1
N + hC

(
ϑn−1
N ,σi,αi

)
, i = 1, 2. (4.9)

Using (2.11) we have

‖α1 −α2‖[L2(Ω)]d ≤ h‖C(ϑn−1
N ,σ1,α1)−C(ϑn−1

N ,σ2,α2)‖[L2(Ω)]d

≤ hCc
(
‖σ1 − σ2‖M + ‖α1 −α2‖[L2(Ω)]d

)
and hence

‖α1 −α2‖[L2(Ω)]d ≤
hCc

1− hCc
‖σ1 − σ2‖M. (4.10)

Define εp`i ∈ [L2(Ω)]4, i = 1, 2, by

εp`i = (εp`)n−1
N + hB

(
ϑn−1
N ,σi,αi

)
, i = 1, 2. (4.11)

Using (2.10) we can write

‖εp`1 − ε
p`
2 ‖S ≤ h‖B(ϑn−1

N ,σ1,α1)−B(ϑn−1
N ,σ2,α2)‖S

≤ hCB
(
‖σ1 − σ2‖M + ‖α1 −α2‖[L2(Ω)]d

)
.

(4.12)

Choose σ1 ∈M and define α1 ∈ [L2(Ω)]d and εp`1 ∈ S by

α1 = αn−1
N + hC

(
ϑn−1
N ,σ1,α1

)
, (4.13)

εp`1 = (εp`)n−1
N + hB

(
ϑn−1
N ,σ1,α1

)
. (4.14)



10 M. BENEŠ EJDE-2020/63

We now look for σ̃1 ∈M and define the mapping Z by σ̃1 = Z(σ1) such that∫
Ω

[ ∂P
∂σij

(σ̃1,α1) + (εp`ij )1 + βδij(ϑ
n−1
N − ϑref )

]
τij dx = 0 (4.15)

for all τ ∈ H. We find σ̃1 ∈M in the form

σ̃1 = ω + σ0, (4.16)

where ω ∈ H and σ0 is an arbitrary fixed element of M. The left-hand side in
(4.15) is a Gateaux differential of the functional

Φ(ω) =

∫
Ω

[
P (ω + σ0,α1) + (εp`ij )1ωij + βδij(ϑ

n−1
N − ϑref )ωij

]
dx (4.17)

defined on H. As we now see, the problem is to find all critical points of Φ. From
(2.8) and (2.9) we have

|P (ω + σ0,α)| ≤ c (1 + |ω|+ |α|)2
, (4.18)

further

P (ω + σ0,α) ≥ c1|ω|2 − c2|α|2 − c3, (4.19)∫
Ω

( ∂P
∂σij

(ω + σ0,α1)− ∂P

∂σij
(ω̂ + σ0,α1)

)
(ωij − ω̂ij) dx ≥ c‖ω − ω̂‖2S . (4.20)

From (4.19) we further have

Ψ(ω)→∞ as ‖ω‖S →∞. (4.21)

We can now use the theory in [36, Chapter 4.1, Theorem 4.2], see also [28, Chap-
ter 7.2, Theorem 2.1 and Theorem 2.2], to conclude that there exists ω ∈ H, a
point of minimum of Φ in H. The uniqueness of such a point follows from (4.20).
Note that σ̃1 ∈M in (4.16) is independent of the choice of σ0 ∈M.

For arbitrary σ1,σ2 ∈M define σ̃1 = Z(σ1) and σ̃2 = Z(σ2), respectively. We
have∫

Ω

[ ∂P
∂σij

(σ̃1,α1) + (εp`ij )1 + βδij(ϑ
n−1
N − ϑref )

]
((σ̃ij)1 − (σ̃ij)2) dx = 0, (4.22)∫

Ω

[ ∂P
∂σij

(σ̃2,α2) + (εp`ij )2 + βδij(ϑ
n−1
N − ϑref )

]
((σ̃ij)1 − (σ̃ij)2) dx = 0. (4.23)

Subtracting (4.23) from (4.22) and using (2.8) and (4.20) we obtain

‖σ̃1 − σ̃2‖M ≤ c
(
‖εp`1 − ε

p`
2 ‖S + ‖α1 −α2‖[L2(Ω)]d

)
. (4.24)

From this, (4.10) and (4.12), we deduce

‖Z(σ1)−Z(σ2)‖M := ‖σ̃1 − σ̃2‖M ≤ ch‖σ1 − σ2‖M. (4.25)

Hence, there exists h0 > 0 small enough, such that for all h, 0 < h ≤ h0, Z realizes
a contraction. Thus there exists a unique fixed point Z(σ) = σ inM. We now set

σnN := σ and compute αnN := α1 and (εp`)nN := εp`1 from (4.13) and (4.14) (where
we set σ1 := σnN ), respectively.

Finally, we obtain the function ϑnN ∈ W 1,2(Ω) by solving (4.2) using the Lax-
Milgram lemma. We also employ the W 1,s-regularity of pnN (with some s > 2) and

the embedding W 1,s(Ω) ↪→ C(Ω) (recall that Ω is a bounded domain in R2 with



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 11

Lipschitz boundary). Let pnN ∈ W 1,s(Ω) with some s > 2 be the solution of (4.1).
For φ, ψ ∈W 1,2(Ω) we define

a(φ, ψ) :=
1

h

∫
Ω

[θ(pnN ) + ρ]φψ dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )∇φ · ∇ψ dx

+

∫
Ω

φ
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇ψ dx

+

∫
∂Ω

[γϑ(x) + γp(x)(pnN − (p∞)nN )]φψ dS.

The map a : W 1,2(Ω)×W 1,2(Ω)→ R is clearly bilinear.
Using the Hölder inequality we have

a(φ, ψ)

≤ 1

h
‖θ(pnN ) + ρ‖L∞(Ω) ‖φ‖L2(Ω) ‖ψ‖L2(Ω)

+ c1 ‖φ‖W 1,2(Ω) ‖ψ‖W 1,2(Ω)

+ c2
∥∥κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
∥∥
L∞(Ω)

‖∇pnN‖[Ls(Ω)]2 ‖φ‖L2s/(s−2)(Ω) ‖ψ‖W 1,2(Ω)

+ c3
∥∥κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
∥∥
L∞(Ω)

‖φ‖L2(Ω) ‖ψ‖W 1,2(Ω)

+ c4 ‖γϑ(x) + γp(x)(pnN − (p∞)nN )‖L∞(∂Ω) ‖φ‖L2(∂Ω) ‖ψ‖L2(∂Ω)

≤ c ‖φ‖W 1,2(Ω) ‖ψ‖W 1,2(Ω)

for all φ, ψ ∈W 1,2(Ω). Hence, a is continuous.
Moreover, for sufficiently small h, it is also coercive, as it satisfies

a(φ, φ) :=
1

h

∫
Ω

ρ|φ|2 dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )|∇φ|2 dx+

∫
∂Ω

γϑ(x)|φ|2 dS

+

∫
Ω

φ
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇φdx

+
1

h

∫
Ω

θ(pnN )|φ|2 dx+

∫
∂Ω

γp(x)(pnN − (p∞)nN )|φ|2 dS

=
1

h

∫
Ω

ρ|φ|2 dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )|∇φ|2 dx+

∫
∂Ω

γϑ(x)|φ|2 dS

+
1

2h

∫
Ω

θ(pnN )|φ|2 dx+
1

2h

∫
Ω

θ(pn−1
N )|φ|2 dx

+
1

2

∫
∂Ω

γp(x)(pnN − (p∞)nN )|φ|2 dS

≥ 1

h

∫
Ω

ρ|φ|2 dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )|∇φ|2 dx+

∫
∂Ω

γϑ(x)|φ|2 dS

+
1

2h

∫
Ω

θ(pnN )|φ|2 dx+
1

2h

∫
Ω

θ(pn−1
N )|φ|2 dx

− 1

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

(
δ‖φ‖2W 1,2(Ω) + C(δ)‖φ‖2L2(Ω)

)
≥
(ρ
h
− C(δ)

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

)
‖φ‖2L2(Ω)

+
(
c1 −

δ

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

)
‖φ‖2W 1,2(Ω),

(4.26)



12 M. BENEŠ EJDE-2020/63

where we have used (2.23) in the estimate

1

2

∫
∂Ω

γp(x)(pnN − (p∞)nN )|φ|2 dS

≤ 1

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

(
δ‖φ‖2W 1,2(Ω) + C(δ)‖φ‖2L2(Ω)

)
and the identity (choosing ϕ = 1

2φ
2 in (4.1))∫

Ω

φ
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇φ dx

= − 1

2h

∫
Ω

(
θ(pnN )− θ(pn−1

N )
)
φ2 dx− 1

2

∫
∂Ω

γp(x) ((p∞)nN − pnN )φ2 dS.

(4.27)

Now, it is easy to see in (4.26), that one can choose δ and then h small enough such
that (ρ

h
− C(δ)

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

)
> 0, (4.28)(

c1 −
δ

2
‖γp(x)(pnN − (p∞)nN )‖L∞(∂Ω)

)
> 0. (4.29)

Hence, there exists h0 > 0 (small enough) such that for all positive h ≤ h0, the
bilinear form a is continuous and coercive. Applying the Lax-Milgram lemma we
conclude that there exists ϑnN ∈W 1,2(Ω) the solution to the problem (4.2). �

Remark 4.2. The solution of the first boundary value problem (4.3)–(4.6) also
gives the displacement vector unN ∈ VR which can be found from the compatible
strain tensor

(εij)
n
N :=

∂P

∂σij
(σnN ,α

n
N ) + (εp`ij )nN + βδij(ϑ

n−1
N − ϑref ). (4.30)

The displacement field is determined except for a rigid-body displacement (transla-
tion and rotation) of the material (according to Lemma 2.4, every v ∈ [W 1,2(Ω)]2

can be written as a sum v = v̂ + vR, v̂ ∈ VR and vR ∈ R).

4.2. A priori estimates. In this part we prove some uniform estimates (with
respect to N). In the following estimates, many different constants will appear.
Recall that, for simplicity of notation, we denote by c generic constants which may
change their numerical value from one formula to another but do not depend on N
and the functions under consideration.

In view of (1.11) and (2.4), degeneration occurs in the nonlinear transport co-
efficient kR which is not assumed to be bounded below by some positive constant.
This difficulty is avoided by deriving the uniform bound of pnN in Ω and on the
boundary ∂Ω. Let ` ∈ R be an arbitrary real fixed number and set (ξ− `)− = ξ− `
for ξ < ` and (ξ − `)− = 0 for ξ ≥ `, ξ ∈ R. Let us first observe that

Φ`(ξ1)− Φ`(ξ2) ≤ [θ(ξ1)− θ(ξ2)](ξ1 − `)− (4.31)

for all ξ1, ξ2 ∈ R, where Φ`(ξ) =
∫ ξ
`
θ′(s)(s − `)− ds. It is easy to check that

Φ`(ξ) ≥ 0 for any ξ ∈ R. Now let k be sufficiently small, such that

k < p0 − x2 a.e. in Ω. (4.32)

k < p1 − x2 a.e. on ∂Ω. (4.33)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 13

We choose ϕ = (pnN − x2 − k)− as a test function in (4.1) to obtain

1

h

∫
Ω

(
θ(pnN )− θ(pn−1

N )
)

(pnN − x2 − k)− dx

+

∫
Ω

(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)
∇(pnN − x2 − k) · ∇(pnN − x2 − k)− dx

+

∫
∂Ω

γp(x) (pnN − (p∞)nN ) (pnN − x2 − k)− dS = 0.

(4.34)

In view of (4.32) and (4.33) we see that the second and third integrals are nonneg-
ative. Hence, using (4.31), we obtain∫

Ω

[Φk+x2
(pnN )− Φk+x2

(pn−1
N )] dx ≤ 0. (4.35)

Thus, Φk+x2(pn−1
N ) = 0 implies Φk+x2(pnN ) = 0. By induction and taking into

account (4.32), there exists k ∈ R such that

k ≤ k + x2 ≤ pnN (4.36)

a.e. in Ω, n = 1, . . . , 2N−1r.
Note that since pnN ∈ W 1,s

ΓD
(Ω) ⊂ C(Ω) (recall that s > 2 and Ω is the two-

dimensional bounded domain with Lipschitz boundary) we also have k ≤ pnN (x)

for all x ∈ Ω, n = 1, . . . , 2N−1r. The upper bound can be shown analogously.
Consequently,

k ≤ pnN (x) ≤ k for all x ∈ Ω, n = 1, . . . , 2N−1r (k, kconst). (4.37)

Let us stress that the constants k and k are independent of N . Hence, by (2.3),
(2.4) and (2.5), there exist constants K1 and K2 such that

0 < K1 ≤
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)
≤ K2 a.e. in Ω, n = 1, . . . , 2N−1r. (4.38)

We now test (4.1) with ϕ = pnN and use (2.22) to obtain

1

h

∫
Ω

(
B(pnN )−B(pn−1

N )
)

dx

+

∫
Ω

(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)
|∇pnN |2 dx+

∫
∂Ω

γp(x)|pnN |2 dS

≤
∫

Ω

(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)
eg · ∇pnN dx+

∫
∂Ω

γp(x)(p∞)nNp
n
N dS.

(4.39)

Applying Young’s inequality to the right-hand side and summing for n = 1, 2, . . . , k
and using (4.38) we obtain∫

Ω

B(pkN ) dx+ c1h

k∑
n=1

∫
Ω

|∇pnN |2 dx+ c2h

k∑
n=1

∫
∂Ω

|pnN |2 dS

≤
∫

Ω

B(p0
N ) dx+ c3kh+ c4h

k∑
n=1

∫
∂Ω

|(p∞)nN |2 dS, k = 1, 2, . . . , 2N−1r,

and whence ∫
Ω

B(pkN ) dx+ h

k∑
n=1

‖pnN‖2W 1,2(Ω) + h

k∑
n=1

‖pnN‖2L2(∂Ω) ≤ c. (4.40)



14 M. BENEŠ EJDE-2020/63

Similarly, using ψ = 2ϑnN as a test function in (4.2) and re-arranging the parabolic
part we arrive at

1

h

∫
Ω

(
θ(pnN )[ϑnN ]2 − θ(pn−1

N )[ϑn−1
N ]2

)
dx+

ρ

h

∫
Ω

(
[ϑnN ]2 − [ϑn−1

N ]2
)

dx

+
1

h

∫
Ω

(
θ(pnN )− θ(pn−1

N )
)

[ϑnN ]2 dx+

∫
Ω

λ(pn−1
N , ϑn−1

N )|∇ϑnN |2 dx

+

∫
Ω

2ϑnN
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇ϑnN dx

+ 2

∫
∂Ω

γϑ(x)|ϑnN |2 dS + 2

∫
∂Ω

|ϑnN |2γp(x)(pnN − (p∞)nN ) dS

≤ 2

∫
Ω

(σd)
n
N : B

(
ϑn−1
N ,σnN ,α

n
N

)
ϑnN dx+ 2

∫
∂Ω

γϑ(x)(ϑ∞)nNϑ
n
N dS.

(4.41)

One is allowed to use ϕ = [ϑnN ]2 as a test function in (4.1) to obtain

1

h

∫
Ω

(
θ(pnN )− θ(pn−1

N )
)

[ϑnN ]2 dx

+

∫
Ω

2ϑnN
(
κ(σn−1

N )kR(pn−1
N )/ν(ϑn−1

N )
)

(∇pnN − eg) · ∇ϑnN dx

+

∫
∂Ω

|ϑnN |2γp(x)(pnN − (p∞)nN ) dS

= 0.

(4.42)

Subtracting (4.42) from (4.41) and, again, re-arranging the parabolic part we deduce∫
Ω

(θ(pnN ) + ρ) [ϑnN ]2 dx−
∫

Ω

(
θ(pn−1

N ) + ρ
)

[ϑn−1
N ]2 dx

+ h

∫
Ω

λ(pn−1
N , ϑn−1

N )|∇ϑnN |2 dx+ 2h

∫
∂Ω

γϑ(x)|ϑnN |2 dS

≤ 2h

∫
∂Ω

γϑ(x)(ϑ∞)nNϑ
n
N dS − h

∫
∂Ω

|ϑnN |2γp(x)(pnN − (p∞)nN ) dS

+ 2h

∫
Ω

(σd)
n
N : B

(
ϑn−1
N ,σnN ,α

n
N

)
ϑnN dx.

Using (2.15), (2.16), (2.18), (2.19), (2.20) and (4.37) and applying the discrete
Gronwall lemma we obtain

max
n=1,...,k

∫
Ω

|ϑnN |2 dx+ h

k∑
n=1

∫
Ω

|∇ϑnN |2 dx+ h

k∑
n=1

∫
∂Ω

|ϑnN |2 dS

≤ c1 + c2h

k∑
n=1

∫
Ω

|(σd)nN : B
(
ϑn−1
N ,σnN ,α

n
N

)
|2 dx, k = 1, . . . , 2N−1r.

(4.43)

We now proceed with the estimates for σnN , (εp`)nN and αnN . First, substituting
v = unN , unN ∈ VR, into (4.3) we obtain∫

Ω

σnN : ε(unN ) dx =

∫
Ω

fnN · unN dx+

∫
∂Ω

t̆nN · unN dS −
∫

Ω

∇χ(pnN ) · unN dx. (4.44)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 15

Employing the Green formula∫
Ω

∇χ(pnN ) · unN dx =

∫
∂Ω

χ(pnN )n · unN dx−
∫

Ω

Iχ(pnN ) : ε(unN ) dx (4.45)

for the last term on the right-hand side in (4.44) and using (1.13)–(1.15), (2.7),
(2.8) and (2.9) we arrive at

c‖σnN‖2[L2(Ω)]4 ≤
∫

Ω

(σij)
n
N :

∂P

∂σij
(σnN ,α

n
N ) dx

=

∫
Ω

fnN · unN dx+

∫
∂Ω

t̆nN · unN dS −
∫
∂Ω

χ(pnN )n · unN dS

+

∫
Ω

Iχ(pnN ) : ε(unN ) dx−
∫

Ω

σnN :
[
(εp`)nN + βI(ϑnN − ϑref )

]
dx.

Using the Young inequality with parameter δ we obtain

c‖σnN‖2[L2(Ω)]4 ≤C(δ)
(
‖fnN‖2[L2(Ω)]2 + ‖t̆nN‖2[L2(∂Ω)]2

)
+ δ‖unN‖2[L2(Ω)]2

+ C(δ)‖χ(pnN )‖2[L2(∂Ω)] + δ‖unN‖2[L2(∂Ω)]2

+ C(δ)‖Iχ(pnN )‖2[L2(Ω)]4 + δ‖ε(unN )‖2[L2(Ω)]4 + δ‖σnN‖2[L2(Ω)]4

+ C(δ)
(
‖(εp`)nN‖2[L2(Ω)]4 + ‖βI(ϑnN − ϑref )‖2[L2(Ω)]4

)
.

By (4.37) we can further simplify the latter inequality as

‖σnN‖[L2(Ω)]4 ≤ δ‖unN‖[L2(Ω)]2 + δ‖unN‖[L2(∂Ω)]2 + δ‖ε(unN )‖[L2(Ω)]4

+ C(δ)
(

1 + ‖(εp`)nN‖[L2(Ω)]4 + ‖ϑnN‖L2(Ω)

)
.

(4.46)

Using Theorem 2.2 and Lemma 2.26 we can write

‖unN‖[L2(Ω)]2 ≤ ‖unN‖[W 1,2(Ω)]2 ≤ c‖ε(unN )‖[L2(Ω)]4 , (4.47)

‖unN‖[L2(∂Ω)]2 ≤ ‖unN‖[W 1,2(Ω)]2 ≤ c‖ε(unN )‖[L2(Ω)]4 . (4.48)

Clearly, from (1.13) we have

‖ε(unN )‖[L2(Ω)]4

≤ ‖(εe`)nN‖[L2(Ω)]4 + ‖(εp`)nN‖[L2(Ω)]4 + ‖βI(ϑnN − ϑref )‖[L2(Ω)]4 .
(4.49)

From (1.14) and using (2.8) we obtain

‖(εe`)nN‖[L2(Ω)]4 ≤ c
(
1 + ‖σnN‖[L2(Ω)]4 + ‖αnN‖[L2(Ω)]d

)
. (4.50)

From (4.5) and (4.6) one sees immediately that

(εp`)nN = (εp`)n−1
N + hB(ϑn−1

N ,σnN ,α
n
N )

= (εp`)0
N + h

n∑
j=1

B(ϑj−1
N ,σjN ,α

j
N )

(4.51)

and
αnN = αn−1

N + hC
(
ϑn−1
N ,σnN ,α

n
N

)
= α0

N + h

n∑
j=1

C(ϑj−1
N ,σjN ,α

j
N ).

(4.52)



16 M. BENEŠ EJDE-2020/63

Using (4.47)–(4.52) in (4.46) we deduce

‖σnN‖[L2(Ω)]4

≤ (C(δ) + δc) + δc‖σnN‖[L2(Ω)]4 + (C(δ) + δc) ‖ϑnN‖L2(Ω)

+ δc
(
‖α0

N‖[L2(Ω)]d + h

n∑
j=1

‖C(ϑj−1
N ,σjN ,α

j
N )‖[L2(Ω)]d

)
+ (C(δ) + δc)

(
‖(εp`)0

N‖[L2(Ω)]4 + h

n∑
j=1

‖B(ϑj−1
N ,σjN ,α

j
N )‖[L2(Ω)]4

)
(4.53)

and, choosing δ sufficiently small and using (2.10), we arrive at

‖σnN‖[L2(Ω)]4 ≤ c1 + c2‖ϑnN‖L2(Ω) + c3hδ

n∑
j=1

‖C(ϑj−1
N ,σjN ,α

j
N )‖[L2(Ω)]d . (4.54)

Note that from (4.43), by (2.10), we have

‖ϑnN‖L2(Ω) ≤ c1 + c2h

n∑
j=1

‖σjN‖[L2(Ω)]4 . (4.55)

Using (4.55) and (2.11) in (4.54) we obtain

‖σnN‖[L2(Ω)]4 ≤ c1 + c2h

n∑
j=1

(
‖ϑj−1

N ‖L2(Ω) + ‖σjN‖[L2(Ω)]4 + ‖αjN‖[L2(Ω)]d

)
. (4.56)

Similarly, from (4.52) we also have

‖αnN‖[L2(Ω)]d ≤ c1 + c2h

n∑
j=1

(
‖ϑj−1

N ‖L2(Ω) + ‖σjN‖[L2(Ω)]4 + ‖αjN‖[L2(Ω)]d

)
. (4.57)

Adding (4.55)–(4.57) and applying the discrete Gronwall inequality [36, Chap-
ter 1.6] (provided h is sufficiently small) we arrive at

‖αnN‖[L2(Ω)]d + ‖σnN‖[L2(Ω)]4 + ‖ϑnN‖L2(Ω) ≤ c, n = 1, . . . , 2N−1r. (4.58)

Finally, from (4.51) and (2.10) we also have

‖(εp`)nN‖[L2(Ω)]4 ≤ c, n = 1, . . . , 2N−1r. (4.59)

4.3. Construction of temporal interpolants and passage to the limit. Us-
ing the sequences {pnN} , {ϑnN} , {σnN} ,

{
(εp`)nN

}
and {αnN}, we define the piecewise

constant interpolants ζN (t) = ζnN for t ∈ ((n − 1)h, nh] and, in addition if neces-

sary, we extend ζN for t ≤ 0 by ζN (t) = ζ0 for t ∈ [−h, 0]. For any function ζ

we often use the simplified notation ζ := ζ(t) and ∂−ht ζ(t) := ζ(t)−ζ(t−h)
h . Then,

by (4.1)–(4.3), pN ∈ L∞(I;W 1,s(Ω)) with some s > 2, ϑN ∈ L∞(I;W 1,2(Ω)),

σN ∈ L∞(I; [L2(Ω)]4), εp`N ∈ L∞(I; [L2(Ω)]4) and αN ∈ L∞(I; [L2(Ω)]d) satisfy



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 17

the equations∫
Ω

∂−ht θ(pN (t))ϕdx

+

∫
Ω

(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
(∇pN (t)− eg) · ∇ϕdx

+

∫
∂Ω

γp(x)pN (t)ϕdS

=

∫
∂Ω

γp(x)(p∞)N (t)ϕdS

(4.60)

for all ϕ ∈W 1,2(Ω) and t ∈ (0, T ];∫
Ω

∂−ht [θ(pN (t))ϑN (t)]ψ dx+ ρ

∫
Ω

∂−ht ϑN (t)ψ dx

+

∫
Ω

λ(pN (t− h), ϑN (t− h))∇ϑN (t) · ∇ψ dx

+

∫
Ω

ϑN (t)
(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
(∇pN (t)− eg) · ∇ψ dx

+

∫
∂Ω

γϑ(x)ϑN (t)ψ dS +

∫
∂Ω

ϑN (t)γp(x)(pN (t)− (p∞)N (t))ψ dS

=

∫
∂Ω

γϑ(x)(ϑ∞)N (t)ψ dS

(4.61)
for all ψ ∈W 1,2(Ω) and t ∈ (0, T ];∫

Ω

σN (t) : ε(v) dx+

∫
Ω

∇χ(pN (t)) · v dx

=

∫
Ω

fN (t) · v dx+

∫
∂Ω

t̆N (t) · v dS

(4.62)

for all v ∈ [W 1,2(Ω)]2 and t ∈ (0, T ] and

εp`N (t) = εp`N (t− h) + hB
(
ϑN (t− h),σN (t),αN (t)

)
, (4.63)

αN (t) = αN (t− h) + hC
(
ϑN (t− h),σN (t),αN (t)

)
. (4.64)

From (4.37), (4.40), (4.43), (4.58) and (4.59) we see that

sup
0≤t≤T

∫
Ω

B(p̄N (t)) dx+

∫ T

0

‖p̄(t)‖2W 1,2(Ω) dt+

∫ T

0

‖p̄N (t)‖2L2(∂Ω) dt ≤ c, (4.65)

‖p̄N‖L∞(ΩT ) ≤ c, (4.66)

‖p̄N‖L∞(∂ΩT ) ≤ c, (4.67)

sup
0≤t≤T

∫
Ω

|ϑ̄N (t)|2 dx+

∫ T

0

‖ϑ̄N (t)‖2W 1,2(Ω) dt+

∫ T

0

‖ϑ̄N (t)‖2L2(∂Ω) dt ≤ c, (4.68)

sup
0≤t≤T

‖σN (t)‖[L2(Ω)]4 ≤ c, (4.69)

sup
0≤t≤T

‖(εp`)N (t)‖[L2(Ω)]4 ≤ c, (4.70)

sup
0≤t≤T

‖αN (t)‖[L2(Ω)]d ≤ c. (4.71)



18 M. BENEŠ EJDE-2020/63

It follows from (4.65) and (4.68)–(4.71) that the sequences {pN}, {ϑN}, {σN},
{εp`N} and {αN} are bounded in the spaces L2(I;W 1,2(Ω)), L2(I;W 1,2(Ω)),
L∞(I; [L2(Ω)]4), L∞(I; [L2(Ω)]4) and L∞(I; [L2(Ω)]d), respectively, so that sub-

sequences {pNk
}, {ϑNk

}, {σNk
}, {εp`Nk

} and {αNk
} can be chosen such that

pNk
⇀ p weakly in L2(I;W 1,2(Ω)),

ϑNk
⇀ ϑ weakly in L2(I;W 1,2(Ω)),

σNk
⇀ σ weakly star in L∞(I; [L∞(Ω)]4),

εp`Nk
⇀ εp` weakly star in L∞(I; [L2(Ω)]4),

αNk
⇀ α weakly star in L∞(I; [L2(Ω)]d).

Since the problem is nonlinear, we need strong convergence to identify the limits
with the weak solution of problem (1.1)–(1.9).

To simplify the notation, let us write, for a moment, pN , ϑN , σN , εp`N , αN ,

instead of pNk
, ϑNk

, σNk
, εp`Nk

, αNk
, respectively. To show that pN converges to

p almost everywhere on ΩT we follow [2]. Let k ∈ N and use

ϕ(t) = ∂kht pN (s)

for jh ≤ t ≤ (j + k)h with (j − 1)h ≤ s ≤ jh and 1 ≤ j ≤ T
h − k, as a test function

in (4.60). For the parabolic term, we can write∫ (j+k)h

jh

∫
Ω

∂−ht θ(pN (t)) ∂kht pN (t) dxdt

=
1

kh2

∫ jh

(j−1)h

∫
Ω

(θ(pN (t+ kh))− θ(pN (t))) (pN (t+ kh)− pN (t)) dxdt.

Hence, summing over j = 1, . . . ,m− k we obtain the estimate

m−k∑
j=1

∫ (j+k)h

jh

∫
Ω

∂−ht θ(pN (t)) ∂kht pN (t) dxdt

≥ 1

kh2

∫ T−kh

0

∫
Ω

(θ(pN (t+ kh))− θ(pN (t))) (pN (t+ kh)− pN (t)) dxdt.

(4.72)

Similarly, for the elliptic term, using (4.65), we have

m−k∑
j=1

∫ (j+k)h

jh

∫
Ω

(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
×∇pN · ∇∂kht pN dx dt

=

k∑
`=1

m−k∑
j=1

∫ (j+`)h

(j+`−1)h

∫
Ω

(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
×∇pN · ∇∂kht pN dx dt

=

k∑
`=1

∫ T−kh+`h

`h

∫
Ω

(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
×∇pN (t) · ∇∂kht pN (t− `h) dxdt



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 19

≤ c1
h

∫
ΩT

|
(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
×∇pN |2 dxdt+

c2
h

∫
ΩT

|∇pN |2 dx dt

≤ C

h

and by a similar computation and using (4.67) we arrive at

m−k∑
j=1

∫ (j+k)h

jh

∫
∂Ω

γp(x)pN∂
kh
t pN dS dt ≤ C

h
, (4.73)

m−k∑
j=1

∫ (j+k)h

jh

∫
∂Ω

γp(x)(p∞)N∂
kh
t pN dS dt ≤ C

h
. (4.74)

Combining (4.72)–(4.74) we obtain∫ T−kh

0

(θ(pN (s+ kh))− θ(pN (s))) (pN (s+ kh)− pN (s)) ds ≤ Ckh. (4.75)

Using the compactness argument one can show in the same way as in [2, Lemma 1.9]
and [13, Eqs. (2.10)–(2.12)] that

θ(pN )→ θ(p) in L1(ΩT ) and almost everywhere on ΩT . (4.76)

Since θ is strictly monotone, it follows from (4.76) that (see [21, Proposition 3.35])

pN → p almost everywhere on ΩT (4.77)

and taking into account the estimates (4.66) and (4.67) we conclude that

pN → p in Lq+1(ΩT ), 0 ≤ q < +∞,
pN → p in Ls+1(∂ΩT ), 0 ≤ s < +∞.

(4.78)

In what follows, we study the convergence of ϑN . From (4.61) we have∫
ΩT

∂−ht [(θ(pN (t)) + ρ)ϑN (t)]ψ dx dt

= −
∫

ΩT

λ(pN (t− h), ϑN (t− h))∇ϑN (t) · ∇ψ dxdt

−
∫

ΩT

ϑN (t)
(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
× (∇pN (t)− eg) · ∇ψ dx dt

−
∫
∂ΩT

γϑ(x)ϑN (t)ψ dS dt

−
∫
∂ΩT

ϑN (t)γp(x)(pN (t)− (p∞)N (t))ψ dS dt

+

∫
∂ΩT

γϑ(x)(ϑ∞)N (t)ψ dS dt

(4.79)

for all ψ ∈ W 1,2(Ω). First, by means of an interpolation argument, see [1, Theo-
rem 5.8], we deduce

L2(I;W 1,2(Ω)) ∩ L∞(I;L2(Ω)) ↪→ L4(ΩT ). (4.80)



20 M. BENEŠ EJDE-2020/63

Now, let ψ ∈ L4(I;W 1,4(Ω)). For the convective term in (4.79), by Hölder’s in-
equality and using (4.80), we have∣∣∣ ∫

ΩT

ϑN (t)
(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
(∇pN (t)− eg) · ∇ψ dxdt

∣∣∣
≤ c‖ϑN (t)

(
κ(σN (t− h))kR(pN (t− h))/ν(ϑN (t− h))

)
× (∇pN (t)− eg) ‖[L4/3(ΩT )]2‖ψ‖L4(I;W 1,4(Ω))

≤ c‖ϑN‖L4(ΩT )

(
‖∇pN‖[L2(ΩT )]2 + 1

)
‖ψ‖L4(I;W 1,4(Ω)).

(4.81)
The remaining terms on the right-hand side of (4.79) can be handled in a more
straightforward and simpler way. Using (4.65)–(4.68) we conclude that∣∣∣ ∫

ΩT

∂−ht [(θ(pN ) + ρ)ϑN ]ψ dxdt
∣∣∣ ≤ c‖ψ‖L4(I;W 1,4(Ω))

for all ψ ∈ L4(I;W 1,4(Ω)) and hence∥∥∂−ht [(θ(pN ) + ρ)ϑN ]
∥∥
L4/3(I;W 1,4(Ω)′)

≤ c.

Further, from (4.65)–(4.68) we also have

‖ (θ(pN ) + ρ)ϑN‖L4/3(I;W 1,4/3(Ω)) ≤ c.
Since

W 1,4/3(Ω) ↪→↪→W 1−β,4/3(Ω) ↪→W 1,4(Ω)′,

where β is a small positive real number, the Aubin-Lions lemma [12] yields the
existence of g ∈ L4/3(I;W 1−β,4/3(Ω)) such that (along a selected subsequence)

(θ(pN ) + ρ)ϑN → g strongly in L4/3(I;W 1−β,4/3(Ω)) (4.82)

and almost everywhere on ΩT . In view of (4.76) and (4.82) we can write

ϑN =
(θ(pN ) + ρ)ϑN
θ(pN ) + ρ

→ g

θ(p) + ρ
= ϑ almost everywhere on ΩT . (4.83)

Now, using (4.68), (4.83) and (2.24) we also have

ϑN → ϑ in Lq+1(ΩT ), 0 ≤ q < 3,

ϑN → ϑ in Ls+1(∂ΩT ), 0 < s < 2.
(4.84)

We now prove that the sequence {αN (t)} converges to the function α(t) strongly
in [L2(Ω)]d, uniformly with respect to t ∈ I. To this aim, let us define

CN (t) =

{
C(ϑn−1

N ,σnN ,α
n
N ) for t ∈ ((n− 1)hN , nhN ], n = 1, . . . , 2N−1r,

C(ϑ0
N ,σ

1
N ,α

1
N ) for t = 0

(4.85)
and

ΨN (t) =

∫ t

0

CN (s)ds. (4.86)

Indeed, for t = 0, we have

‖αN (0)−ΨN (0)‖[L2(Ω)]d ≤ chN . (4.87)

Now, let t ∈ (0, T ] be arbitrary, say t ∈ ((n− 1)hN , nhN ]. We have

‖αN (t)−ΨN (t)‖[L2(Ω)]d ≤
∥∥αN (t)−

∫ t

0

CN (s)ds
∥∥

[L2(Ω)]d
≤ chN . (4.88)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 21

Clearly, we can write

‖αN (t)−αM (t)‖[L2(Ω)]d

≤ ‖αN (t)−ΨN (t)‖[L2(Ω)]d

+ ‖αM (t)−ΨM (t)‖[L2(Ω)]d + ‖ΨM (t)−ΨN (t)‖[L2(Ω)]d

(4.89)

and, according to (4.87) and (4.88), we obtain

‖αN (t)−αM (t)‖[L2(Ω)]d

≤ chN + chM +
∥∥∫ t

0

CM (s)ds−
∫ t

0

CN (s)ds
∥∥

[L2(Ω)]d
.

(4.90)

For the last term we have∥∥∫ t

0

CM (s)ds−
∫ t

0

CN (s)ds
∥∥

[L2(Ω)]d

≤
∫ t

0

‖CM (s)−CN (s)‖[L2(Ω)]d ds

≤ c1
∫ t

0

(∥∥ϑN (s− hN )− ϑM (s− hM )
∥∥
L2(Ω)

+ ‖σN (s)− σM (s)‖[L2(Ω)]4

)
ds+ c2

∫ t

0

‖αN (s)−αM (s)‖[L2(Ω)]d ds

(4.91)

and thus (4.90) becomes

‖αN (t)−αM (t)‖[L2(Ω)]d

≤ c1
∫ t

0

‖αN (s)−αM (s)‖[L2(Ω)]d ds+ chN + chM

+ c1

∫ t

0

(∥∥ϑN (s− hN )− ϑM (s− hM )
∥∥
L2(Ω)

+ ‖σN (s)− σM (s)‖[L2(Ω)]4

)
ds.

(4.92)

A process similar to that used above implies that the same estimate can be obtained

for εp`N , i.e. ∥∥εp`N (t)− εp`M (t)
∥∥

[L2(Ω)]4

≤ c1
∫ t

0

‖αN (s)−αM (s)‖[L2(Ω)]d ds+ chN + chM

+ c1

∫ t

0

(∥∥ϑN (s− hN )− ϑM (s− hM )
∥∥
L2(Ω)

+ ‖σN (s)− σM (s)‖[L2(Ω)]4

)
ds.

(4.93)

What remains is to estimate the last terms on the right-hand sides of (4.92) and
(4.93). From (4.62) we have∫

Ω

[σN (t)− σM (t)] : ε(v) dx+

∫
Ω

∇ · I[χ(pN (t))− χ(pM (t))] · v dx

=

∫
Ω

[fN (t)− fM (t)] · v dx+

∫
∂Ω

[t̆N (t)− t̆M (t)] · v dS

(4.94)



22 M. BENEŠ EJDE-2020/63

for any v ∈ [W 1,2(Ω)]2. Setting v = uN (t)−uM (t) and applying the Green formula
to the second term we obtain∫

Ω

[σN (t)− σM (t)] : [ε(uN (t))− ε(uM (t))] dx

+

∫
∂Ω

[χ(pN (t))− χ(pM (t))]n · [uN (t)− uM (t)] dS

−
∫

Ω

I[χ(pN (t))− χ(pM (t))] : [ε(uN (t))− ε(uM (t))] dx

=

∫
Ω

[fN (t)− fM (t)] · [uN (t)− uM (t)] dx

+

∫
∂Ω

[t̆N (t)− t̆M (t)] · [uN (t)− uM (t)] dS.

(4.95)

In view of (4.30) and (1.14), (4.95) becomes

∫
Ω

[σN (t)− σM (t)] :
[∂P
∂σ

(σN (t),αN (t))− ∂P

∂σ
(σM (t),αM (t))

]
dx

+

∫
Ω

[σN (t)− σM (t)] : [εp`N (t)− εp`M (t)] dx

+

∫
Ω

[σN (t)− σM (t)] : βI(ϑN (t)− ϑM (t)) dx

+

∫
∂Ω

[χ(pN (t))− χ(pM (t))]n · [uN (t)− uM (t)] dS

−
∫

Ω

I[χ(pN (t))− χ(pM (t))] : [ε(uN (t))− ε(uM (t))] dx

=

∫
Ω

[fN (t)− fM (t)] · [uN (t)− uM (t)] dx

+

∫
∂Ω

[t̆N (t)− t̆M (t)] · [uN (t)− uM (t)] dS.

(4.96)

Using (4.20), (2.26) and Theorem 2.2 we deduce

‖σN (t)− σM (t)‖2[L2(Ω)]4

≤ c1 ‖σN (t)− σM (t)‖[L2(Ω)]4 ‖αN (t)−αM (t)‖[L2(Ω)]d

+ c2 ‖σN (t)− σM (t)‖[L2(Ω)]4

∥∥∥εp`N (t)− εp`M (t)
∥∥∥

[L2(Ω)]4

+ c3 ‖σN (t)− σM (t)‖[L2(Ω)]4

∥∥ϑN (t)− ϑM (t)
∥∥
L2(Ω)

+ c4 ‖pN (t)− pM (t)‖L2(∂Ω) ‖ε(uN (t))− ε(uM (t))‖[L2(Ω)]4

+ c5 ‖pN (t)− pM (t)‖L2(Ω) ‖ε(uN (t))− ε(uM (t))‖[L2(Ω)]4

+ c6
∥∥fN (t)− fM (t)

∥∥
[L2(Ω)]2

‖ε(uN (t))− ε(uM (t))‖[L2(Ω)]4

+ c7

∥∥∥t̆N (t)− t̆M (t)
∥∥∥

[L2(∂Ω)]2
‖ε(uN (t))− ε(uM (t))‖[L2(Ω)]4 .

(4.97)



EJDE-2020/63 DEGENERATE COUPLED FLOWS IN DEFORMABLE POROUS MEDIA 23

Indeed, from (1.13)–(1.15) we have

‖ε(uN (t))− ε(uM (t))‖[L2(Ω)]4

≤
∥∥∂P
∂σ

(σN (t),αN (t))− ∂P

∂σ
(σM (t),αM (t))

∥∥
[L2(Ω)]4

+
∥∥∥εp`N (t)− εp`M (t)

∥∥∥
[L2(Ω)]4

+ c
∥∥ϑN (t)− ϑM (t)

∥∥
L2(Ω)

≤ c1 ‖σN (t)− σM (t)‖[L2(Ω)]4 + |αN (t)−αM (t)‖[L2(Ω)]d

+
∥∥εp`N (t)− εp`M (t)

∥∥
[L2(Ω)]4

+ c2
∥∥ϑN (t)− ϑM (t)

∥∥
L2(Ω)

.

(4.98)

Finally, using (4.98) in (4.97) and applying the Young inequality with parameter
δ > 0 we arrive at

‖σN (t)− σM (t)‖[L2(Ω)]4

≤ c1(δ) ‖αN (t)−αM (t)‖[L2(Ω)]d

+ c2(δ)
∥∥∥εp`N (t)− εp`M (t)

∥∥∥
[L2(Ω)]4

+ c3(δ)
∥∥ϑN (t)− ϑM (t)

∥∥
L2(Ω)

+ c4(δ) ‖pN (t)− pM (t)‖L2(∂Ω) + c5(δ) ‖pN (t)− pM (t)‖L2(Ω)

+ c6(δ)
∥∥fN (t)− fM (t)

∥∥
[L2(Ω)]2

+ c7(δ)
∥∥∥t̆N (t)− t̆M (t)

∥∥∥
[L2(∂Ω)]2

.

(4.99)

Substituting (4.99) into (4.92) and (4.93) we obtain

‖αN (t)−αM (t)‖[L2(Ω)]d

≤ c1
∫ t

0

‖αN (s)−αM (s)‖[L2(Ω)]d ds

+ c2

∫ t

0

∥∥∥εp`N (s)− εp`M (s)
∥∥∥

[L2(Ω)]4
ds+ c3hN + c3hM

+ c4

∫ t

0

∥∥ϑN (s− hN )− ϑM (s− hM )
∥∥
L2(Ω)

ds

+ c5

∫ t

0

∥∥ϑN (s)− ϑM (s)
∥∥
L2(Ω)

ds

+ c6

∫ t

0

‖pN (s)− pM (s)‖L2(∂Ω) ds+ c7

∫ t

0

‖pN (s)− pM (s)‖L2(Ω) ds

+ c8

∫ t

0

∥∥fN (s)− fM (s)
∥∥

[L2(Ω)]2
ds

+ c9

∫ t

0

∥∥∥t̆N (s)− t̆M (s)
∥∥∥

[L2(∂Ω)]2
ds

(4.100)



24 M. BENEŠ EJDE-2020/63

and∥∥εp`N (t)− εp`M (t)
∥∥

[L2(Ω)]4

≤ c1
∫ t

0

‖αN (s)−αM (s)‖[L2(Ω)]d ds

+ c2

∫ t

0

∥∥∥εp`N (s)− εp`M (s)
∥∥∥

[L2(Ω)]4
ds+ c3hN + c3hM

+ c4

∫ t

0

∥∥ϑN (s− hN )− ϑM (s− hM )
∥∥
L2(Ω)

ds

+ c5

∫ t

0

∥∥ϑN (s)− ϑM (s)
∥∥
L2(Ω)

ds

+ c6

∫ t

0

‖pN (s)− pM (s)‖L2(∂Ω) ds+ c7

∫ t

0

‖pN (s)− pM (s)‖L2(Ω) ds

+ c8

∫ t

0

∥∥fN (s)− fM (s)
∥∥

[L2(Ω)]2
ds

+ c9

∫ t

0

∥∥∥t̆N (s)− t̆M (s)
∥∥∥

[L2(∂Ω)]2
ds,

(4.101)

respectively. Adding (4.100) with (4.101), applying the Gronwall lemma and using
(4.78) and (4.84), we conclude that

‖αN (t)−αM (t)‖[L2(Ω)]d → 0 as M, N →∞,

‖εp`N (t)− εp`M (t)‖[L2(Ω)]4 → 0 as M, N →∞

independently of t ∈ I. This means that {αN (t)} and
{
εp`N (t)

}
are Cauchy

sequences and thus αN (t) → α(t) in [L2(Ω)]d and εp`N (t) → εp`(t) in [L2(Ω)]4

uniformly with respect to t ∈ I.
By similar arguments, from (4.99) and in view of (4.78) and (4.84) we have

σN → σ in L2(I; [L2(Ω)]4)

and almost everywhere on ΩT .
The above established convergences are sufficient for taking the limit N →∞ in

(4.60)–(4.64) (along a selected subsequence) to get the weak solution of the system
(1.1)–(1.9) in the sense of Definition 3.1. This completes the proof of the main
result stated in Theorem 3.2.

Acknowledgments. The author has been financially supported by the European
Regional Development Fund (project No. CZ.02.1.01/0.0/0.0/16−019/0000778)
within activities of the Center of Advanced Applied Sciences (CAAS).

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Michal Beneš

Department of Mathematics, Faculty of Civil Engineering, Czech Technical University
in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic

Email address: michal.benes@cvut.cz


	1. Introduction
	2. Preliminaries
	2.1. Notation and function spaces
	2.2. Structure and data properties
	2.3. Auxiliary results

	3. Main result
	4. Proof of the main result
	4.1. Discretized problem
	4.2. A priori estimates
	4.3. Construction of temporal interpolants and passage to the limit
	Acknowledgments

	References