Global weak solutions to degenerate coupled diffusion-convection-dispersion processes and heat transport in porous media

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by  Michal Beneš

GLOBAL WEAK SOLUTIONS TO DEGENERATE COUPLED
DIFFUSION-CONVECTION-DISPERSION PROCESSES AND
HEAT TRANSPORT IN POROUS MEDIA
7
1
0 MICHALBENESˇ ANDLUKA´SˇKRUPICˇKA
2

n
Abstract. In this contribution we prove the existence of weak solutions to
a
degenerate parabolic systems arising from the coupled moisture movement,
J
transport of dissolved species and heat transfer through partially saturated

2 porous materials. Physically motivated mixed Dirichlet-Neumann boundary
conditions and initial conditions are considered. Existence of a global weak

] solutionoftheproblemisprovedbymeansofsemidiscretizationintimeandby
P
passingtothelimitfromdiscreteapproximations. Degenerationoccursinthe
A
nonlinear transport coefficients which are not assumed to be bounded below
. andabovebypositiveconstants. Degeneraciesinalltransportcoefficientsare
h overcomebyprovingsuitablea-prioriL∞-estimatesfortheapproximationsof
t
a primaryunknowns ofthesystem.
m
[

1 1. Introduction
v
7 LetΩbeaboundeddomaininR2,Ω∈C0,1 andletΓ andΓ be opendisjoint
D N
9 subsetsof∂Ω (notnecessarilyconnected)suchthatΓ 6=∅andthe ∂Ω\(Γ ∪Γ )
D D N
3
is a finite set. Let T ∈ (0,∞) be fixed throughout the paper, I = (0,T) and
0
Q =Ω×I denotes the space-time cylinder, Γ =Γ ×I and Γ =Γ ×I.
0 T DT D NT N
. We shall study the following initial boundary value problem in QT
1
0 (1.1) ∂tb(u)=∇·[a(θ)∇u],
7
(1.2) ∂ [b(u)w]=∇·[b(u)D (u)∇w]+∇·[wa(θ)∇u],
1 t w
: (1.3) ∂ [b(u)θ+̺θ]=∇·[λ(θ,u)∇θ]+∇·[θa(θ)∇u],
v t
i with the mixed-type boundary conditions
X
r (1.4) u=0, w =0, θ =0 onΓDT,
a
(1.5) ∇u·n=0, ∇w·n=0, ∇θ·n=0 onΓ
NT
and the initial conditions
(1.6) u(·,0)=u , w(·,0)=w , θ(·,0)=θ inΩ.
0 0 0
Thesystem(1.1)–(1.6)arisesfromthecoupledmoisturemovement,transportof
dissolvedspeciesandheattransferthroughtheporoussystem[4,20]. Forsimplicity,
thegravitytermsandexternalsourcesarenotincludedsincethey donotaffectthe
analysis. Forspecificapplicationswereferthereadertoe.g. [19]. Hereu:Q →R,
T
w : Q → R and θ : Q → R are the unknown functions. In particular, u
T T
1991MathematicsSubjectClassification. 35A05,35D05,35B65,35B45,35B50,35K15,35K40.
Key words and phrases. Initial-boundary value problems for second-order parabolic systems;
global existence, smoothness and regularity of solutions; coupled transport processes in porous
media.
1
2 MICHALBENESˇANDLUKA´SˇKRUPICˇKA
correspondstotheKirchhofftransformationofthematricpotential[2],wrepresents
concentration of dissolved species and θ represents the temperature of the porous
system. Further, a : R → R, D : R → R, b : R → R, λ : R2 → R, u : Ω → R,
w 0
w : Ω → R, and θ : Ω → R are given functions, ̺ is a real positive constant and
0 0
n is the outward unit normal vector. In this paper we study the existence of the
weak solution to the system (1.1)–(1.6).
Nowadays, description of heat, moisture or soluble/non-soluble contaminant
transportinconcrete,soilorrockporousmatrixisfrequentlybasedontimedepen-
dent models. Coupled transport processes (diffusion processes, heat conduction,
moister flow, contaminant transport or coupled flows through porous media) are
typicallyassociatedwithsystemsofstronglynonlineardegenerateparabolicpartial
differential equations of type (written in terms of operators A, Ψ, F)
(1.7) ∂ Ψ(u)−∇·A(u,∇u)=F(u),
t
where u stands for the unknown vector of state variables. There is no complete
theory for such general problems. However, some particular results assuming spe-
cialstructureofoperatorsAandΨandgrowthconditionsonF canbefoundinthe
literature. Most theoretical results on parabolic systems exclude the case of non-
symmetrical parabolic parts [2, 8, 13]. Giaquinta and Modica in [10] proved the
local-in-time solvability of quasilinear diagonal parabolic systems with nonlinear
boundary conditions (without assuming any growth condition), see also [22]. The
existence ofweak solutions to more generalnon-diagonalsystems like (1.7) subject
to mixed boundary conditions has been provenin [2]. The authors proved an exis-
tenceresultassumingtheoperatorΨtobeonly(weak)monotoneandsubgradient.
Thisresulthasbeenextendedin[8],wheretheauthorspresentedthelocalexistence
ofthe weak solutions for the systemwith nonlinear Neumann boundary conditions
and under more general growth conditions on nonlinearities in u. These results,
however, are not applicable if Ψ does not take the subgradient structure, which is
typical of coupled transport models in porous media. Thus, the analysis needs to
exploit the specific structure of such problems.
The existence of a local-in-time strong solution for moisture and heat transfer
in multi-layer porous structures modelling by the doubly nonlinear parabolic sys-
tem is proven in [5]. In [21], the author proved the existence of the solution to
the purely diffusive hygro-thermal model allowing non-symmetrical operators Ψ,
but requiring non-realistic symmetry in the elliptic part. In [7, 12], the authors
studied the existence, uniqueness and regularity of coupled quasilinear equations
modeling evolution of fluid species influenced by thermal, electrical and diffusive
forces. In [15, 16, 17], the authors studied a model of specific structure of a heat
and mass transfer arising from textile industry and proved the global existence for
one-dimensional problems in [15, 16] and three-dimensional problems in [17].
Inthepresentpaperweextendourpreviousexistenceresultforcoupledheatand
mass flows in porous media [6] to more general problem (including the convection-
dispersionequation)modelingcoupledmoisture,soluteandheattransportinporous
media. This leads to a fully nonlinear degenerate parabolic system with natural
(critical) growths and degeneracies in all transport coefficients.
The rest of this paper is organized as follows. In Section 2, we introduce basic
notation and suitable function spaces and specify our assumptions on data and
coefficient functions in the problem. In Section 3, we formulate the problem in the
DEGENERATE COUPLED FLOWS IN POROUS MEDIA 3
variationalsenseandstatethemainresult,theglobal-in-timeexistenceoftheweak
solution. The main result is proved by an approximation procedure in Section 4.
First we formulate the semi-discrete scheme and prove the existence of its solu-
tion. Thecruciala-prioriestimatesanduniformboundnessoftimeinterpolantsare
provedin part 4.2. Finally, we conclude that the solutions of semi-discrete scheme
converge and the limit is the solution of the original problem (Subsection 4.3).
Remark 1.1. The present analysis can be straightforwardly extended to a setting
with nonhomogeneous boundary conditions (see [6] for details). Here we work
with homogeneous boundary conditions, ignoring the gravity terms and excluding
external sources to simplify the presentation and avoid unnecessary technicalities
in the existence result.
2. Preliminaries
2.1. Notations and some properties of Sobolev spaces. Vectors and vector
functions are denoted by boldface letters. Throughout the paper, we will always
usepositiveconstantsC,c,c ,c ,...,whicharenotspecifiedandwhichmaydiffer
1 2
fromline to line. Throughoutthis paper we suppose s,q,s′ ∈[1,∞], s′ denotes the
conjugate exponent to s > 1, 1/s+1/s′ = 1. Ls(Ω) denotes the usual Lebesgue
space equipped with the norm k·k and Wk,s(Ω), k ≥ 0 (k need not to be
Ls(Ω)
an integer, see [14]), denotes the usual Sobolev-Slobodecki space with the norm
k·k . We define W1,2(Ω) := v ∈W1,2(Ω); v =0 . By E∗ we denote
the sWpak,cse(Ωo)f allcontinuous,ΓDlinear formns on Banachsp(cid:12)aΓcDe E aond by h·,·i we denote
(cid:12)
the duality between E and E∗. By Ls(I;E) we denote the Bochner space (see [1]).
Therefore, Ls(I;E)∗ =Ls′(I;E∗).
2.2. Structure and data properties. We start by introducing our assumptions
on functions in (1.1)–(1.6).
(i) b is a positive continuous strictly monotone function such that
0<b(ξ)≤b <+∞ ∀ξ ∈R (b =const),
2 2
(b(ξ )−b(ξ ))(ξ −ξ )>0 ∀ξ ,ξ ∈R, ξ 6=ξ .
1 2 1 2 1 2 1 2
(ii) a, D ∈C(R) and λ ∈C(R2) such that
w
0<a(ξ), 0<D (ξ) ∀ξ ∈R,
w
0<λ(ξ,ζ) ∀ξ,ζ ∈R.
(iii) (Initial data) Assume
u , w , θ ∈L∞(Ω),
0 0 0
such that
(2.1) −∞<u <u <u <0 a.e. in Ω (u ,u =const).
1 0 2 1 2
2.3. Auxiliary results.
Remark 2.1 ([2], Section 1.1). Let us note that (i) implies that there is a (strictly)
convex C1-function Φ : R → R, Φ(0) = 0, Φ′(0) = 0, such that b(z)−b(0) =
Φ′(z)∀z ∈R. Introduce the Legendre transform
1 z
B(z):= (b(z)−b(sz))zds= (b(z)−b(s))ds.
Z Z
0 0
4 MICHALBENESˇANDLUKA´SˇKRUPICˇKA
Let us present some properties of B [2]:
1
B(z):= (b(z)−b(sz))zds≥0 ∀z ∈R,
Z
0
B(s)−B(r)≥(b(s)−b(r))r ∀r,s∈R,
b(z)z−Φ(z)+Φ(0)=B(z)≤b(z)z ∀z ∈R.
3. The main result
Theaimofthispaperistoprovetheexistenceofaweaksolutiontotheproblem
(1.1)–(1.6). First we formulate our problem in a variational sense.
Definition 3.1. A weak solution of (1.1)–(1.6) is a triplet [u,w,θ] such that
u∈L2(I;W1,2(Ω)), w ∈L2(I;W1,2(Ω))∩L∞(Q ), θ ∈L2(I;W1,2(Ω))∩L∞(Q ),
ΓD ΓD T ΓD T
which satisfies
(3.1) − b(u)∂ φdxdt+ a(θ)∇u·∇φdxdt= b(u )φ(x,0)dx
t 0
Z Z Z
QT QT Ω
for any φ∈L2(I;W1,2(Ω))∩W1,1(I;L∞(Ω)) with φ(·,T)=0;
ΓD
(3.2) − b(u)w∂ ηdxdt+ b(u)D (u)∇w·∇ηdxdt
t w
Z Z
QT QT
+ wa(θ)∇u·∇ηdxdt= b(u )w η(x,0)dx
0 0
Z Z
QT Ω
for any η ∈L2(I;W1,2(Ω))∩W1,1(I;L∞(Ω)) with η(·,T)=0;
ΓD
(3.3) − [b(u)θ+̺θ]∂ ψdxdt+ λ(θ,u)∇θ·∇ψdxdt
t
Z Z
QT QT
+ θa(θ)∇u·∇ψdxdt= [b(u )θ +̺θ ]ψ(x,0)dx
0 0 0
Z Z
QT Ω
for any ψ ∈L2(I;W1,2(Ω))∩W1,1(I;L∞(Ω)) with ψ(·,T)=0.
ΓD
The main result of this paper reads as follows.
Theorem 3.2 (Main result). Let the assumptions (i)–(iii) be satisfied. Then there
exists at least one weak solution of the system (1.1)–(1.6).
To prove the main result of the paper we use the method of semidiscretization
in time by constructing temporal approximations and limiting procedure. The
proof can be divided into three steps. In the first step we approximate our prob-
lem by means of a semi-implicit time discretization scheme (which preserve the
pseudo-monotone structure of the discrete problem) and prove the existence and
W1,s(Ω)-regularity (with some s > 2) of temporal approximations. In the second
step we construct piecewise constant time interpolants and derive suitable a-priori
estimates. The key point is to establish L∞-estimates to overcome degeneracies in
transport coefficients. Finally, in the third step we pass to the limit from discrete
approximations.
DEGENERATE COUPLED FLOWS IN POROUS MEDIA 5
4. Proof of the main result
4.1. Approximations. Let us fix p∈N and set τ :=T/p (a time step). Further,
let us consider u0 := u , w0 := w , θ0 := θ a.e. on Ω. We approximate our
p 0 p 0 p 0
evolution problem by a semi-implicit time discretization scheme. Then we define,
in each time step n = 1,...,p, a triplet [un,wn,θn] as a solution of the following
p p p
recurrence steady problem.
For a given triplet [un−1,wn−1,θn−1], n = 1,...,p, un−1 ∈ L∞(Ω), wn−1 ∈
p p p p p
L∞(Ω), θn−1 ∈ L∞(Ω), find [un,wn,θn], such that un ∈ W1,s(Ω), wn ∈ W1,s(Ω),
p p p p p ΓD p ΓD
θn ∈W1,s(Ω) with some s>2 and
p ΓD
b(un)−b(un−1)
(4.1) p p φdx+ a(θn−1)∇un·∇φdx=0
Z τ Z p p
Ω Ω
for any φ∈W1,2(Ω);
ΓD
b(un)wn−b(un−1)wn−1
p p p p
(4.2) ηdx
Z τ
Ω
+ b(un−1)D (un−1)∇wn·∇ηdx+ wna(θn−1)∇un·∇ηdx=0
Z p w p p Z p p p
Ω Ω
for any η ∈W1,2(Ω);
ΓD
b(un)θn−b(un−1)θn−1 θn−θn−1
p p p p p p
(4.3) ψdx+ ̺ ψdx
Z τ Z τ
Ω Ω
+ λ(θn−1,un−1)∇θn·∇ψdΩ+ θna(θn−1)∇un·∇ψdΩ=0
Z p p p Z p p p
Ω Ω
for any ψ ∈W1,2(Ω).
ΓD
Theorem 4.1 (Existence of the solution to (4.1)–(4.3)). Let un−1 ∈ L∞(Ω),
p
wn−1 ∈ L∞(Ω), θn−1 ∈ L∞(Ω) be given and the assumptions (i)–(iii) be satis-
p p
fied. Then there exists [un,wn,θn], such that un ∈ W1,s(Ω), wn ∈ W1,s(Ω) and
p p p p ΓD p ΓD
θn ∈W1,s(Ω) with some s>2 satisfying (4.1)–(4.3).
p ΓD
Proof. The proof rests on the W1,p-regularity of elliptic problems presented in
[9, 11] and the embedding W1,s(Ω) ⊂ L∞(Ω) if s > 2 (recall that Ω is a bounded
ΓD
domain in R2).
The existence of un ∈ W1,s(Ω) with some s > 2 and θn ∈ W1,2(Ω), solutions
p ΓD p ΓD
to problems (4.1) and (4.3), respectively, is proven in [6]. The existence of wn ∈
p
W1,2(Ω), the solution to (4.2), can be handled in the same way.
ΓD
Now, with wn ∈W1,2(Ω) in hand, rewrite the equation(4.2) in the form(trans-
p ΓD
ferring the lower-order terms to the right hand side)
b(un−1)D (un−1)∇wn·∇ηdx
Z p w p p
Ω
b(un)wn−b(un−1)wn−1
=− p p p p ηdx− wna(θn−1)∇un·∇ηdx.
Z τ Z p p p
Ω Ω
6 MICHALBENESˇANDLUKA´SˇKRUPICˇKA
Since un−1 ∈ L∞(Ω), un ∈ W1,s(Ω) with some s > 2, wn−1 ∈ L∞(Ω), θn−1 ∈
L∞(Ω),pboth integrals opn the rΓiDght hand side make sensepfor any η ∈ W1,pr′(Ω),
ΓD
r′ = r/(r − 1) with some r > 2. Now we are able to apply [9, Theorem 4] to
obtain wn ∈ W1,s(Ω) with some s >2. Analysis similar to the above implies that
p ΓD
θn ∈W1,s(Ω) with some s>2. (cid:3)
p ΓD
4.2. A-priori estimates. In this part we prove some uniform estimates (with
respect to p) for the time interpolants of the solution. In the following estimates,
many different constants will appear. For simplicity of notation, C represents
generic constants which may change their numerical value from one formula to
another but do not depend on p and the functions under consideration.
4.2.1. Construction of temporal interpolants. With the sequences un,wn,θn con-
p p p
structed in Section 4.1, we define the piecewise constant interpolants φ¯ (t) = φn
p p
for t ∈ ((n − 1)τ,nτ] and, in addition, we extend φ¯ for t ≤ 0 by φ¯ (t) = φ
p p 0
for t ∈ (−τ,0]. For a function ϕ we often use the simplified notation ϕ := ϕ(t),
ϕ (t) := ϕ(t−τ), ∂−τϕ(t) := ϕ(t)−ϕ(t−τ), ∂τϕ(t) := ϕ(t+τ)−ϕ(t). Then, follow-
τ t τ t τ
ing (4.1)–(4.3), the piecewise constant time interpolants u¯ ∈ L∞(I;W1,s(Ω)),
p ΓD
w¯ ∈ L∞(I;W1,s(Ω)) and θ¯ ∈ L∞(I;W1,s(Ω)) (with some s > 2) satisfy the
p ΓD p ΓD
equations
(4.4) ∂−τb(u¯ (t))φdx+ a(θ¯ (t−τ))∇u¯ (t)·∇φdx=0
t p p p
Z Z
Ω Ω
for any φ∈W1,2(Ω),
ΓD
(4.5) ∂−τ[b(u¯ (t))w¯ (t)]ηdx+ b(u¯ (t−τ))D (u¯ (t−τ))∇w¯ (t)·∇ηdx
t p p p w p p
Z Z
Ω Ω
+ w¯ (t)a(θ¯ (t−τ))∇u¯ (t)·∇ηdx=0
p p p
Z
Ω
for any η ∈W1,2(Ω) and
ΓD
(4.6) ∂−τ b(u¯ (t))θ¯ (t)+̺θ¯ (t) ψdx
t p p p
Z
Ω (cid:2) (cid:3)
+ λ(θ¯ (t−τ),u¯ (t−τ))∇θ¯ (t)·∇ψdx
p p p
Z
Ω
+ θ¯ (t)a(θ¯ (t−τ))∇u¯ (t)·∇ψdx=0
p p p
Z
Ω
for any ψ ∈W1,2(Ω).
ΓD
4.2.2. L∞-bound for u¯ , w¯ and θ¯ . Firstwe provethe L∞-estimate foru¯ . Let us
p p p p
set
b(u¯ )−b(u ), u¯ <u ,
(4.7) φ:=[b(u¯ )−b(u )] = p 1 p 1
p 1 − (cid:26) 0, u¯p ≥u1,
DEGENERATE COUPLED FLOWS IN POROUS MEDIA 7
as a test function in (4.4). Note that φ vanishes on Γ . It is a simple matter to
D
deduce
1
(4.8) [b(u¯ (t))−b(u )]2dx
2Z p 1 −
Ω
+ a(θ¯ (s−τ))b′(u¯ (s))|∇u¯ (s)|2χ dxds=0
Z p p p {u¯p<u1}
Qt
for almost every t ∈I. Hence we conclude that the set {x∈Ω; u¯ (x,t)<u } has
p 1
a measure zero for almost every t∈I.
Now setting
b(u¯ )−b(u ), u¯ >u ,
(4.9) φ=[b(u¯ )−b(u )] = p 2 p 2
p 2 + (cid:26) 0, u¯p ≤u2,
we obtain, using similar arguments,
1
[b(u¯ )−b(u )]2dx=0 for almost every t∈I.
2Z p 2 +
Ω
Hence the set {x∈Ω; u¯ (x,t)>u } has a measure zero for almost every t ∈ I.
p 2
Finally, combining the previous arguments, we deduce
(4.10) ku¯pkL∞(QT) ≤C,
where C does not depend on p.
Now we prove a similar estimate for w¯ . Let ℓ be an odd integer. Using φ =
p
[ℓ/(ℓ+1)](w¯ )ℓ+1 as a test function in (4.4) and η =(w¯ )ℓ in (4.5) and combining
p p
both equations we obtain
(4.11)
1 1 1 1
b(u¯ (s))[w¯ (s)]ℓ+1dx− b(u¯ (s−τ))[w¯ (s−τ)]ℓ+1dx
p p p p
τ ℓ+1Z τ ℓ+1Z
Ω Ω
1 1 1 ℓ
+ b(u¯ (s−τ))[w¯ (s−τ)]ℓ+1dx+ b(u¯ (s−τ))[w¯ (s)]ℓ+1dx
p p p p
τ ℓ+1Z τ ℓ+1Z
Ω Ω
1
− b(u¯ (s−τ))w¯ (s−τ)[w¯ (s)]ℓdx
p p p
τ Z
Ω
+ ℓ[w¯ (s)]ℓ−1b(u¯ (s−τ))D (u¯ (s−τ))∇w¯ (s)·∇w¯ (s)dx
p p w p p p
Z
Ω
=0.
Applying the Young’s inequality we can write for the term in the third line
(4.12)
1 1 1
b(u¯ (s−τ))w¯ (s−τ)[w¯ (s)]ℓdx≤ b(u¯ (s−τ))[w¯ (s−τ)]ℓ+1dx
p p p p p
τ Z τ ℓ+1Z
Ω Ω
1 ℓ
+ b(u¯ (s−τ))[w¯ (s)]ℓ+1dx.
p p
τ ℓ+1Z
Ω
8 MICHALBENESˇANDLUKA´SˇKRUPICˇKA
Combining (4.11) and (4.12) we deduce
1 1 1 1
(4.13) b(u¯ (s))[w¯ (s)]ℓ+1dx− b(u¯ (s−τ))[w¯ (s−τ)]ℓ+1dx
p p p p
τ ℓ+1Z τ ℓ+1Z
Ω Ω
+ ℓ[w¯ (s)]ℓ−1b(u¯ (s−τ))D (u¯ (s−τ))∇w¯ (s)·∇w¯ (s)dx
p p w p p p
Z
Ω
≤0.
Now, integrating (4.13) over s from 0 to t we get
(4.14) (w¯ (t))ℓ+1b(u¯ (t))dx
p p
Z
Ω
+ (ℓ+1)ℓ[w¯ (s)]ℓ−1b(u¯ (s−τ))D (u¯ (s−τ))|∇w¯ (s)|2dxds
p p w p p
Z
Ωt
≤ (w )ℓ+1b(u )dx.
0 0
Z
Ω
Note that the second integralin (4.14) is nonnegative (ℓ is supposed to be the odd
integer). Moreover,from (4.14) and (4.10) it follows that
(4.15) kw¯pkL∞(0,T;Lℓ+1(Ω)) ≤C,
where the constant C is independent of ℓ and p. Now, let ℓ → +∞ in (4.15), we
get
(4.16) kw¯pkL∞(QT) ≤C.
In the same manner we arrive at the estimate for θ¯ , i.e.
p
(4.17) kθ¯pkL∞(QT) ≤C.
4.2.3. Energy estimates for u¯ , w¯ and θ¯ . We test (4.4) with φ =u¯ (t) and inte-
p p p p
grate (4.4) over t from 0 to s. For the parabolic term we can write
s 1 s
(4.18) ∂−τb(u¯ (t))u¯ (t)dxdt≥ B(u¯ (t))−B(u )dxdt.
Z Z t p p τ Z Z p 0
0 Ω s−τ Ω
Further, using (4.10) and (4.18), applying the usual estimates for the elliptic part
(see also [2]), we obtain the a-prioriestimate
T
(4.19) sup B(u¯ (t))dx+ |∇u¯ (t)|2dxdt≤C.
p p
0≤t≤TZΩ Z0 ZΩ
Now it follows that there exists a function u ∈ L2(I;W1,2(Ω)) such that, along a
ΓD
selectedsubsequence(lettingp→∞),wehaveu¯ (t)⇀uweaklyinL2(I;W1,2(Ω)).
p ΓD
Now we prove similar result for w¯ (t). Using η(t)=2w¯ (t) as a test function in
p p
(4.5) we get
∂−τb(u¯ (t))2w¯ (t)2dx+ ∂−τw¯ (t)2w¯ (t)b(u¯ (t−τ))dx
Z t p p Z t p p p
Ω Ω
+2 b(u¯ (t−τ))D (u¯ (t−τ))∇w¯ (t)·∇w¯ (t)dx
p w p p p
Z
Ω
(4.20) + a(θ¯ (t−τ))∇u¯ (t)·2w¯ (t)∇w¯ (t)dx=0.
p p p p
Z
Ω
DEGENERATE COUPLED FLOWS IN POROUS MEDIA 9
One is allowed to use φ(t)=w¯ (t)2 as a test function in (4.4) to obtain
p
(4.21) [∂−τb(u¯ (t))]w¯ (t)2dx+ a(θ¯ (t−τ))∇u¯ (t)·∇w¯ (t)2dx=0.
t p p p p p
Z Z
Ω Ω
Combining (4.20) and (4.21) we deduce
1
(4.22) ∂−τ w¯ (t)2b(u¯ (t)) dx+ [w¯ (t)−w¯ (t−τ)]2b(u¯ (t−τ))dx
Z t p p Z τ p p p
Ω (cid:2) (cid:3) Ω
+2 b(u¯ (t−τ))D (u¯ (t−τ))∇w¯ (t)·∇w¯ (t)dx=0.
p w p p p
Z
Ω
In view of (4.10) we have
(4.23) b(u¯ (t)), b(u¯ (t−τ)), D (u¯ (t−τ))>C in Ω×(−τ,T).
p p w p
Recall that C does not depend on p. Now, integrating (4.22) with respect to time
t we obtain
T
sup |w¯ (t)|2dΩ+ kw¯ (t)k2 dΩ≤C.
0≤t≤TZΩ p Z0 p WΓ1D,2(Ω)
From this we can write
(4.24) kw¯ k ≤C.
p L2(I;W1,2(Ω))
ΓD
Similarly, we use ψ(t)=2θ¯ (t) as a test function in (4.5) to obtain
p
(4.25) ∂−τb(u¯ (t))2θ¯ (t)2dx+ ∂−τθ¯ (t)2θ¯ (t)b(u¯ (t−τ))dx
t p p t p p p
Z Z
Ω Ω
+2 λ(θ¯ (t−τ),u¯ (t−τ))∇θ¯ (t)·∇θ¯ (t)dx
p p p p
Z
Ω
+ a(θ¯ (t−τ))∇u¯ (t)·2θ¯ (t)∇θ¯ (t)dx=0.
p p p p
Z
Ω
Using φ(t)=θ¯ (t)2 as a test function in (4.4) we get
p
(4.26) ∂−τb(u¯ (t))θ¯ (t)2dx+ a(θ¯ (t−τ))∇u¯ (t)·∇θ¯ (t)2dx=0.
t p p p p p
Z Z
Ω Ω
Combining (4.25) and (4.26) we deduce
(4.27) ∂−τ θ¯ (t) 2b(u¯ (t)) dx+ 1 θ¯ (t)−θ¯ (t−τ) 2b(u¯ (t−τ))dx
ZΩ t h(cid:0) p (cid:1) p i ZΩ τ (cid:2) p p (cid:3) p
+2 λ(θ¯ (t−τ),u¯ (t−τ))∇θ¯ (t)·∇θ¯ (t)dx=0.
p p p p
Z
Ω
Integrating (4.27) with respect to time t we obtain the a-priori estimate (using
(4.10) and (4.17))
T
(4.28) sup |θ¯ (t)|2dx+ kθ¯ (t)k2 dt≤C.
0≤t≤TZΩ p Z0 p WΓ1D,2(Ω)
From this we have
(4.29) kθ¯ k ≤C.
p L2(I;W1,2(Ω))
ΓD
10 MICHALBENESˇANDLUKA´SˇKRUPICˇKA
4.2.4. Furtherestimates. Inordertoshowthatu¯ convergestoualmosteverywhere
p
on Q we follow [2]. Let k ∈N and use
T
φ(t)=∂kτu¯ (s)
t p
for jτ ≤t≤(j+k)τ with (j−1)τ ≤s≤jτ and 1≤j ≤ T −k, as a test function
τ
in (4.4). For the parabolic term, we can write
(j+k)τ
∂−τb(u¯ (t))∂kτu¯ (t)dxdt
Z Z t p t p
jτ Ω
1 jτ
= (b(u¯ (t+kτ))−b(u¯ (t)))(u¯ (t+kτ)−u¯ (t)) dxdt.
kτ2 Z Z p p p p
(j−1)τ Ω
Hence, summing over j =1,...,p−k we get the estimate
p−k (j+k)τ
(4.30) ∂−τb(u¯ (t))∂kτu¯ (t)dxdt
t p t p
Z Z
Xj=1 jτ Ω
1 T−kτ
≥ (b(u¯ (t+kτ))−b(u¯ (t)))(u¯ (t+kτ)−u¯ (t)) dxdt.
kτ2 Z Z p p p p
0 Ω
Similarly,fortheellipticterm,afteralittlelengthybutstraightforwardcomputation
we obtain
p−k (j+k)τ
(4.31) a(θ¯ (t−τ))∇u¯ ·∇∂kτu¯ dxdt
Z Z p p t p
Xj=1 jτ Ω
k p−k (j+ℓ)τ
= a(θ¯ (t−τ))∇u¯ ·∇∂kτu¯ dxdt
Z Z p p t p
Xℓ=1Xj=1 (j+ℓ−1)τ Ω(cid:0) (cid:1)
k T−kτ+ℓτ
= a(θ¯ (t−τ))∇u¯ (t)·∇∂kτu¯ (t−ℓτ)dxdt
p p t p
Z Z
Xℓ=1 ℓτ Ω
c c
≤ 1 |a(θ¯ (t−τ))∇u¯ |2dxdt+ 2 |∇u¯ |2dxdt
p p p
τ Z τ Z
QT QT
C
≤ .
τ
Combining (4.30)–(4.31) and using (4.19) we obtain
T−kτ
(4.32) (b(u¯ (s+kτ))−b(u¯ (s)))(u¯ (s+kτ)−u¯ (s))ds≤Ckτ.
p p p p
Z
0
Usingthecompactnessargumentonecanshowinthesamewayasin[2,Lemma1.9]
and [8, Eqs. (2.10)–(2.12)]
(4.33) b(u¯ )→b(u) in L1(Q )
p T
and almost everywhere on Q . Since b is strictly monotone, it follows from (4.33)
T
that [13, Proposition 3.35]
(4.34) u¯ →u almost everywhere on Q .
p T
Further, in much the same way as in (4.32), we arrive at
T−kτ
(4.35) |b(u¯ (s+kτ))w¯ (s+kτ)−b(u¯ (s))w¯ (s)|2ds≤Ckτ.
p p p p
Z
0