Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

0.26 MB·English

by  Joachim Escher

EXISTENCE AND STABILITY OF WEAK SOLUTIONS FOR A DEGENERATE
PARABOLIC SYSTEM MODELLING TWO-PHASE FLOWS IN POROUS
MEDIA
1
1
0 JOACHIMESCHER, PHILIPPELAURENÇOT,ANDBOGDAN–VASILEMATIOC
2

n
Abstract. We prove global existence of nonnegative weak solutions to a degenerate parabolic
a
system which models the interaction of two thin fluid films in a porous medium. Furthermore, we
J
show that these weak solutions converge at an exponential rate towards flat equilibria.
0
2

]
P
1. Introduction
A
. In this paper we consider the following system of degenerate parabolic equations
h
t
a ∂ f = ∂ (f∂ f)+R∂ (f∂ h),
t x x x x
m (t,x) (0, ) (0,L), (1.1)
( ∂th= ∂x(f∂xf)+Rµ∂x[(h f)∂xh]+R∂x(f∂xh), ∈ ∞ ×
[ −

which models two-phase flows in porous media under the assumption that the thickness of the two
1
v fluid layers is small. Indeed, the system (1.1) has been obtained in [4] by passing to the limit of
4 small layer thickness in the Muskat problem studied in [3] (with homogeneous Neumann boundary
6
condition). Similar methods to those presented in [4] have been used in [6] and [8], where it is
9
3 rigorously shown that, in the absence of gravity, appropriate scaled classical solutions of the Stokes’
. and one-phase Hele-Shaw problems with surface tension converge to solutions of thin film equations
1
0 ∂ h+∂ (ha∂3h) = 0,
1 t x x
1
with a = 3 for Stoke’s problem and a = 1 for the Hele-Shaw problem.
:
v Inoursetting f is anonnegative function expressingthe height ofthe interface between thefluids
i
X while h f isthe height of the interface separating the fluid locatedin theupperpart of theporous
≥
r medium from air, cf. Figure 1. We assume that the bottom of the porous medium, which is located
a
at y = 0, is impermeable and that the air is at constant pressure normalised to be zero. The
parameters R and R are given by
µ
ρ+ µ−
R := and R := R,
µ
ρ− ρ+ µ+
−
whereρ−, µ− [resp. ρ+, µ+]denotethedensity and viscosityofthefluidlocatedbelow [resp. above]
in the porous medium. Of course, we have to supplement system (1.1) with initial conditions
f(0)= f , h(0) = h , x (0,L), (1.2)
0 0
∈
2010 Mathematics Subject Classification. 35K65; 35K40; 35D30; 35B35; 35Q35.
Key words and phrases. Degenerate parabolic system, weak solutions, exponential stability, thin film, Liapunov
functional.
Partially supported by theFrench-German procope project 20190SE.
1
2 J.ESCHER,PH.LAURENÇOT,ANDB.–V.MATIOC
y
y = h(x)
g(x)
y = f(x)
.
0 L x
Figure 1. The physical setting
and we impose no-flux boundary conditions at x = 0 and x = L:
∂ f = ∂ h = 0, x = 0,L. (1.3)
x x
Itturnsoutthatthesystemisparabolicifweassumethath > f > 0andR > 0,thatis,thedenser
0 0
fluid lies beneath. Existence and uniqueness of classical solutions to (1.1) have been established in
this parabolic setting in [4]. Furthermore, it is also shown that the steady states of (1.1) are flat
and that they attract at an exponential rate in H2 solutions which are initially close by.
In this paper we are interested in the degenerate case which appears when we allow f = 0 and
0
h = f on some subset of (0,L). Owing to the loss of uniform parabolicity, existence of classical
0 0
solutions can no longer be established by using parabolic theory and we have to work within an
appropriate weak setting. Furthermore, the system is quasilinear and, as a further difficulty, each
equation contains highest order derivatives of both unknowns f and h, i.e. it is strongly coupled.
In order to study the problem (1.1) we shall employ some of the methods used in [2] to investigate
the spreading of insoluble surfactant. However, in our case the situation is more involved since we
have two sources of degeneracy, namely when f and g := h f become zero. It turns out that by
−
choosing (f,g) as unknowns, the system (1.1) is more symmetric:
∂ f = (1+R)∂ (f∂ f)+R∂ (f∂ g),
t x x x x
(t,x) (0, ) (0,L), (1.4)
( ∂tg = Rµ∂x(g∂xf)+Rµ∂x(g∂xg), ∈ ∞ ×
since, up to multiplicative constants, the first equation can be obtained from the second by simply
interchanging f and g. Corresponding to (1.4) we introduce the following energy functionals:
L R
(f,g) := (f lnf f +1)+ (glng g+1) dx
1
E − R −
Z0 (cid:20) µ (cid:21)
and
L
(f,g) := f2+R(f +g)2 dx.
2
E
Z0
It is not difficult to see that both energy functi(cid:2)onals and (cid:3)dissipate along classical solutions of
1 2
E E
(1.4). While in the classical setting the functional plays an important role in the study of the
2
E
stability properties of equilibria [4], in the weak setting we strongly rely on the weaker energy
1
E
which, nevertheless, provides us with suitable estimates for solutions of a regularised problem and
enables us to pass to the limit to obtain weak solutions. Note also that appears quite natural
1
E
A DEGENERATE PARABOLIC SYSTEM MODELLING FLOWS IN POROUS MEDIA 3
in the context of (1.4), while, when considering (1.1), one would not expect to have an energy
functional of this form.
Our main results read as follows:
Theorem 1.1. Assume that R > 0, R > 0. Given f ,g L ((0,L)) with f 0 and g 0 there
µ 0 0 2 0 0
∈ ≥ ≥
exists a global weak solution (f,g) of (1.4) satisfying
(i) f 0, g 0 in (0,T) (0,L),
≥ ≥ ×
(ii) f,g L∞((0,T),L2((0,L))) L2((0,T),H1((0,L))),
∈ ∩
for all T > 0 and
L L T L
f(T)ψdx f ψdx = ((1+R)f∂ f +Rf∂ g)∂ ψdxdt,
0 x x x
− −
Z0 Z0 Z0 Z0
L L T L
g(T)ψdx g ψdx = R (g∂ f +g∂ g)∂ ψdxdt
0 µ x x x
− −
Z0 Z0 Z0 Z0
for all ψ W1 ((0,L)). Moreover, the weak solutions satisfy
∞
∈
(a) f(T) = f , g(T) = g ,
1 0 1 1 0 1
k k k k k k k k
T L 1 R
(b) (f(T),g(T))+ ∂ f 2+ ∂ g 2 dxdt (f ,g ),
1 x x 1 0 0
E 2| | 1+2R| | ≤ E
Z0 Z0 (cid:20) (cid:21)
T L
(c) (f(T),g(T))+ f((1+R)∂ f +R∂ g)2+RR g(∂ f +∂ g)2 dxdt (f ,g )
2 x x µ x x 2 0 0
E ≤ E
Z0 Z0 h i
for almost all T (0, ).
∈ ∞
Remark 1.2. If f = 0 for instance, a solution to (1.4) is (0,g) where g solves the classical porous
0
medium equation ∂ g = R ∂ (g∂ g) in (0, ) (0,L) with homogeneous Neumann boundary
t µ x x
∞ ×
conditions and initial condition g .
0
Additionally to the existence result, we show that the weak solutions constructed in Theorem 1.1
converge at an exponential rate towards the unique flat equilibrium (which is determined by mass
conservation) in the L norm:
2
−
Theorem 1.3 (Exponential stability). Under the assumptions of Theorem 1.1, there exist positive
constants M and ω such that
1 L 2 1 L 2
f(t) f dx + g(t) g dx Me−ωt for a.e. t 0.
0 0
− L − L ≤ ≥
(cid:13) Z0 (cid:13)2 (cid:13) Z0 (cid:13)2
(cid:13) (cid:13) (cid:13) (cid:13)
Remark 1.4.(cid:13)Theorem 1.3 sugg(cid:13)ests t(cid:13)hat degenerate solu(cid:13)tions become classical after evolving over
(cid:13) (cid:13) (cid:13) (cid:13)
a certain finite period of time, and therefore would converge in the H2 norm towards the corre-
−
sponding equilibrium, cf. [4].
The outline of the paper is as follows. In Section 2 we regularise the system (1.4) and prove that
the regularised system has global classical solutions, the global existence being a consequence of
their boundedness away from zero and in H1((0,L)). The purpose of the regularisation is twofold:
on the one hand, the regularised system is expected to be uniformly parabolic and this is achieved
by modifying (1.4) and the initial data such that the comparison principle applied to each equation
4 J.ESCHER,PH.LAURENÇOT,ANDB.–V.MATIOC
separately guarantees that f ε and g ε for some ε > 0. On the other hand, the regularised
≥ ≥
system is expected to be weakly coupled, a property which is satisfied by a suitable mollification of
∂ g in the first equation of (1.4) and ∂ f in the second equation of (1.4). The energy functional
x x 1
E
turns out to provide useful estimates for the regularised system as well. In Section 3 we show that
the classical global solutions of the regularised problem converge, in appropriate norms, towards a
weak solution of (1.4), and that they satisfy similar energy inequalities as the classical solutions of
(1.4) for both energy functionals and . Finally, we give a detailed proof of Theorem 1.3.
1 2
E E
Throughout the paper, we set L := L ((0,L)) and W1 := W1((0,L)) for p [1, ], and
p p p p ∈ ∞
H2α := H2α((0,L)) for α [0,1]. We also denote positive constants that may vary from line to line
∈
and depend only on L, R, R , and (f ,g ) by C or C , i 1. The dependence of such constants
µ 0 0 i
≥
upon additional parameters will be indicated explicitly.
2. The regularised system
In this section we introduce a regularised system which possesses global solutions provided they
are bounded in H1 and also bounded away from zero. In Section 3 we show that these solutions
converge towards weak solutions of (1.4).
We fix two nonnegative functions f and g in L (the initial data of system (1.4)) and first
0 0 2
introduce the space
H2 := f H2((0,L)) : ∂ f(0) = ∂ f(L)= 0 .
B x x
{ ∈ }
We note that for each ε > 0, the elliptic operator (1 ε2∂2) : H2 L is an isomorphism. This
− x B → 2
property is preserved when considering the restriction
(1 ε2∂2): f C2+α([0,L]) : ∂ f(0) = ∂ f(L)= 0 Cα([0,L]), α (0,1).
− x { ∈ x x } → ∈
Given f,g L , we let then
2
∈
F := (1 ε2∂2)−1f, G := (1 ε2∂2)−1g, (2.1)
ε − x ε − x
and consider the following regularised problem
∂ f = (1+R)∂ (f ∂ f )+R∂ ((f ε)∂ G ),
t ε x ε x ε x ε x ε
− (t,x) (0, ) (0,L), (2.2a)
( ∂tgε = Rµ∂x((gε ε)∂xFε)+Rµ∂x(gε∂xgε), ∈ ∞ ×
−
supplemented with homogeneous Neumann boundary conditions
∂ f = ∂ g = 0 x = 0,L, (2.2b)
x ε x ε
and with regularised initial data
f (0) = f := (1 ε2∂2)−1f +ε, g (0) = g := (1 ε2∂2)−1g +ε. (2.2c)
ε 0ε − x 0 ε 0ε − x 0
Note that the regularised initial data (f ,g ) H2 H2 and, invoking the elliptic maximum
0ε 0ε B B
∈ ×
principle, we have
f ε, g ε. (2.3)
0ε 0ε
≥ ≥
Letting F := (1 ε2∂2)−1f and G := (1 ε2∂2)−1g , we obtain by multiplying the relation
0ε − x 0 0ε − x 0
F ε2∂2F = f by F and integrating over (0,L) the following relation
0ε − x 0ε 0 0ε
L
F 2+ε2 ∂ F 2 = f F dx f F ,
k 0εk2 k x 0εk2 0 0ε ≤ k 0k2 k 0εk2
Z0
A DEGENERATE PARABOLIC SYSTEM MODELLING FLOWS IN POROUS MEDIA 5
which gives a uniform L -bound for the regularised initial data:
2
f f +ε√L and g g +ε√L. (2.4)
0ε 2 0 2 0ε 2 0 2
k k ≤ k k k k ≤ k k
Concerning the solvability of problem (2.2), we use quasilinear parabolic theory, as presented in
[1], to prove the following result:
Theorem 2.1. For each ε (0,1) problem (2.2) possesses a unique global nonnegative solution
∈
X := (f ,g ) with
ε ε ε
f ,g C([0, ),H1) C((0, ),H2) C1((0, ),L ).
ε ε B 2
∈ ∞ ∩ ∞ ∩ ∞
Moreover, we have
f ε, g ε for all (t,x) (0, ) (0,L).
ε ε
≥ ≥ ∈ ∞ ×
In order to prove this global result, we establish the following lemma which gives a criterion for
global existence of classical solutions of (2.2):
Lemma 2.2. Given ε (0,1), the problem (2.2) possesses a unique maximal strong solution X =
ε
∈
(f ,g ) on a maximal interval [0,T (ε)) satisfying
ε ε +
f ,g C([0,T (ε)),H1) C((0,T (ε)),H2) C1((0,T (ε)),L ).
ε ε + + B + 2
∈ ∩ ∩
Moreover, if for every T < T (ε) there exists C(ε,T) > 0 such that
+
fε ε/2+C(ε,T)−1, gε ε/2+C(ε,T)−1, and max Xε(t) H1 C(ε,T), (2.5)
≥ ≥ t∈[0,T]k k ≤
then the solution is globally defined, i.e. T (ε) = .
+
∞
Proof. Let ε (0,1) be fixed. To lighten our notation we omit the subscript ε in the remainder
∈
of this proof. Note first that problem (2.2) has a quasilinear structure, in the sense that (2.2) is
equivalent to the system of equations:
∂ X + (X)X = F(X) in (0, ) (0,L),
t
A ∞ ×
X = 0 on (0, ) 0,L , (2.6)

B ∞ ×{ }

 X(0) = X on (0,L),
0
wherethenewvariableisX := (f,g)withX = (f ,g ),andtheoperators andF arerespectively
 0 0 0
B
given by
R(f ε)∂ G
x
X := ∂ X, F(X) := ∂ − .
x x
B Rµ(g ε)∂xF !
−
Letting
(1+R)f 0
a(X) := ,
0 Rµg !
the operator is defined by the relation (X)Y := ∂ (a(X)Y). We shall prove first that (2.6)
x
A A −
has a weak solution defined on a maximal time interval, for which we have a weak criterion for
global existence. We then improve in successive steps the regularity of the solution to show that
it is actually a classical solution, so that this criterion guarantees also global existence of classical
solutions.
6 J.ESCHER,PH.LAURENÇOT,ANDB.–V.MATIOC
Given α [0,1], the complex interpolation space H2α := [L ,H2] is known to satisfy, cf. [1],
B 2 B α
∈
H2α , α 3/4,
H2α = ≤
B
( f H2α : ∂xf = 0 for x = 0,L , α > 3/4.
{ ∈ }
Furthermore, for each β (1/2,2] we define the set
∈
β β
:= f H : ε/2 < f ,
B B
V { ∈ }
which is open in H2. Choose now γ := 1/2 2ξ > 0, where ξ (0,1/18). We infer from (2.2c), that
B
1−ξ 1−ξ − ∈
X . In order to obtain existence of a unique weak solution of (2.6) we verify the
0 B B
∈ V ×V
assumptions of [1, Theorem 13.1]. With the notation of [1, Theorem 13.1] we define
(σ,s,r,τ) := (3/2 3ξ,1+ξ,1 ξ, ξ), 2α := 3/2 3ξ.
− − − −
Since 1 ξ 1/2 > γ, we conclude that 1−ξ Cγ([0,L]), meaning that the elements of a(X)
B
belong t−o Cγ−([0,L]) for X 1−ξ 1−ξ.VSince⊂a(X) is positivebdefinite, we conclude in virtue of
B B
∈ V ×V
γ > 2α 1 that
−
( , ) C1−( 1−ξ 1−ξ, αb(0,L)),
B B
A B ∈ V ×V E
the nobtation αb(0,L) being defined in [1, Sections 4 &8]. Moreover, since (1 ε2∂2)−1 (L ,H2)
E − x ∈ L 2 B
we also have
F C1−( 1−ξ 1−ξ,H−ξ),
∈ V ×V
whereby, in the notation of [1], H−ξ = H−ξ because ξ < 1/2, cf. [1, eq. (7.5)]. Thus, we find all
B
| |
the assumptions of [1, Theorem 13.1] fulfilled, and conclude that, for each X 2, there exists a
0 B
∈ V
3/2−ξ
unique maximal weak H solution X = (f,g) of (2.6), that is
B
−
f,g C([0,T ), 1−ξ) C((0,T ),H3/2−ξ) C1((0,T ),H−1/2−3ξ),
+ B + B + B
∈ V ∩ ∩
and the first equation of (2.6) is satisfied for all t (0,T ) when testing with functions belonging
+
∈
to H1/2+3ξ. Moreover, if X [0,T] is bounded in H1 H1 and bounded away from ∂ 1−ξ for all
B B B B
× V
T > 0, then T = , which yields the desired criterion (2.5).
+
∞ (cid:12)
We show now that this w(cid:12) eak solution has even more regularity, to conclude in the end that
the existence time of the strong solution of (2.6) coincides with that of the weak solution (of
course they are identical on each interval where they are defined). Indeed, given δ > 0, it holds
that X C([δ,T ),H3/2−3ξ) C1([δ,T ),H−1/2−3ξ). Hence, if 0 < 2ρ < 2, we conclude from [1,
+ B + B
∈ ∩
Theorem 7.2] and [7, Proposition 1.1.5] that X is actually Hölder continuous
X Cρ([δ,T ),H3/2−3ξ−2ρ).
+ B
∈
Choosing ρ := ξ and µ := 1 6ξ > 0, we have that H3/2−3ξ−2ρ ֒ Cµ([0,L]), so that the elements
B
− →
of the matrix a(X(t)) belong to Cµ([0,L]) for all t [δ,T ). Defining 2µ := 2 8ξ > 0, we observe
+
∈ −
that µ > 2µ 1 and
−
( (X), ) Cρ([δ,T+), µb((0,L))). b
A B ∈ E
b
Finally, with 2ν := 3/2+ξ, our choice for ξ implies
(X(δ),F(X)) H2ν−2 Cρ([δ,T ),H−ξ) H2ν−2 Cρ([δ,T ),H2ν−2),
B + B B + B
∈ × ⊂ ×
A DEGENERATE PARABOLIC SYSTEM MODELLING FLOWS IN POROUS MEDIA 7
and the assertions of [1, Theorem 11.3] are all fulfilled. Whence, the linear problem
∂ Y + (X)Y = F(X) in (δ,T ) (0,L),
t +
A ×
Y = 0 on (δ,T ) 0,L , (2.7)
 +
B ×{ }

 Y(δ) = X(δ) on (0,L),
possesses a unique strong H2ν solution Y, that is

−
Y C([δ,T ),H2ν−2 H2ν−2) C((δ,T ),H2ν H2ν) C1((δ,T ),H2ν−2 H2ν−2).
+ B B + B B + B B
∈ × ∩ × ∩ ×
3/2−3ξ
In view of [1, Remark 11.1] we conclude that both X and Y are weak H solutions of (2.7),
B
−
whence we infer from [1, Theorem 11.2] that X = Y, and so
f,g C([δ,T ),H−1/2+ξ) C((δ,T ),H3/2+ξ) C1((δ,T ),H−1/2+ξ).
+ B + B + B
∈ ∩ ∩
Interpolating as we did previously and taking into account that δ was arbitrarily chosen, we have
f,g Cθ([δ,T ),C1+θ([0,L])) if we set 4θ ξ. Hence, we find that
+
∈ ≤
(X(δ),(A(X),F(X))) L Cθ([δ,T ), (H2 H2,L L ) (L L )),
2 + B B 2 2 2 2
∈ × H × × × ×
where (H2 H2,L L ) denotes the set of linear operators in L L with domain H2 H2
B B 2 2 2 2 B B
H × × × ×
which are negative infinitesimal generators of analytic semigroups on L L , and, in virtue of [1,
2 2
×
Theorem 10.1],
X C((δ,T ),H2 H2) C1((δ,T ),L L )
+ B B + 2 2
∈ × ∩ ×
for all δ (0,T ). Hence, the strong solution of (2.6), which is obtained by applying [1, Theorem
+
12.1] to t∈hat particular system, exists on [0,T ) and the proof is complete. (cid:3)
+
The remainder of this section is devoted to prove that T (ε) = for the strong solution (f ,g )
+ ε ε
∞
of (2.2) constructed in Lemma 2.2. In view of Lemma 2.2, it suffices to prove that (f ,g ) are a
ε ε
priori bounded in H1 and away from zero. Concerning the latter, we note that we may apply the
parabolic maximum principle to each equation of the regularised system (2.2a) separately. Indeed,
owing to the boundary conditions (2.2b), the constant function (t,x) ε solves the first equation
7→
of (2.2a) and (2.2b) while we have f ε by (2.2c). Consequently, f ε in [0,T (ε)) [0,L] and,
0ε ε +
≥ ≥ ×
using a similar argument for g , we conclude that
ε
f ε, g ε for all (t,x) [0,T (ε)) (0,L). (2.8)
ε ε +
≥ ≥ ∈ ×
Next, owing to (2.2b) and the nonnegativity of f and g , it readily follows from (2.2a) that the
ε ε
L norm of f and g is conserved in time, that is,
1 ε ε
−
f (t) = f = f +εL, g (t) = g = g +εL (2.9)
ε 1 0ε 1 0 1 ε 1 0ε 1 0 1
k k k k k k k k k k k k
for all t [0,T (ε)). The next step is to improve the previous L1 bound to an H1 bound as
+
∈ − −
required by Lemma 2.2. To this end, we shall use the energy for the regularised problem, see
1
E
(2.13) below. As a preliminary step, we collect some properties of the functions (F ,G ) defined in
ε ε
(2.1) in the next lemma.
Lemma 2.3. For all t (0,T (ε))
+
∈
F (t) f (t) , G (t) g (t) , (2.10)
ε 2 ε 2 ε 2 ε 2
k k ≤ k k k k ≤ k k
∂ F (t) ∂ f (t) , ∂ G (t) ∂ g (t) , (2.11)
x ε 2 x ε 2 x ε 2 x ε 2
k k ≤ k k k k ≤ k k
ε ∂2F (t) ∂ f (t) , ε ∂2G (t) ∂ g (t) . (2.12)
k x ε k2 ≤ k x ε k2 k x ε k2 ≤ k x ε k2
8 J.ESCHER,PH.LAURENÇOT,ANDB.–V.MATIOC
Proof. Theproofof (2.10)issimilartothatof (2.4). WenextmultiplytheequationF ε2∂2F = f
ε− x ε ε
by ∂2F , integrate over (0,L) and use the Cauchy-Schwarz inequality to estimate the right-hand
side−anxdεobtain (2.11) and (2.12). (cid:3)
Lemma 2.4. Given T (0,T (ε)), we have that
+
∈
T L 1 R
(f (T),g (T))+ ∂ f 2+ ∂ g 2 dxdt (f (0),g (0)). (2.13)
1 ε ε x ε x ε 1 ε ε
E 2| | 1+2R| | ≤ E
Z0 Z0 (cid:18) (cid:19)
Proof. Using (2.11) and Hölder’s inequality, we get
d L R
(f ,g ) = ∂ (f log(f ))+ ∂ (g log(g )) dx
1 ε ε t ε ε t ε ε
dtE R
Z0 µ
L f ε g ε
= (1+R)∂ f 2+R ε− ∂ f ∂ G +R ε− ∂ g ∂ F +R ∂ g 2 dx
x ε x ε x ε x ε x ε x ε
− | | f g | |
Z0 (cid:18) ε ε (cid:19)
(1+R) ∂ f 2+R ∂ f ∂ G +R ∂ g ∂ F R ∂ g 2
≤− k x εk2 k x εk2k x εk2 k x εk2k x εk2− k x εk2
1 1+2R
∂ f 2 ∂ f 2 2R ∂ f ∂ g +R ∂ g 2
≤− 2k x εk2− 2 k x εk2− k x εk2k x εk2 k x εk2
(cid:18) (cid:19)
1 R
∂ f 2 ∂ g 2.
≤− 2k x εk2− 1+2Rk x εk2
Integrating with respect to time, we obtain the desired assertion. (cid:3)
Since zlnz z + 1 0 for all z [0, ), relation (2.13) gives a uniform estimate in (ε,t)
− ≥ ∈ ∞ ∈
(0,1) (0,T (ε)) of (∂ f ,∂ g ) in L ((0,T),L L ) in dependence only of the initial condition
+ x ε x ε 2 2 2
× ×
(f ,g ). Indeed, on the one hand, since lnz z 1 for all z [0, ), we have
0 0
≤ − ∈ ∞
zlnz z+1 z(z 1) (z 1) = (z 1)2 for all z 0, (2.14)
− ≤ − − − − ≥
so that
L R
(f ,g ) (f 1)2+ (g 1)2 dx C (2.15)
1 0ε 0ε 0ε 0ε 1
E ≤ − R − ≤
Z0 (cid:18) µ (cid:19)
for all ε (0,1) by (2.4). On the other hand, owing to the Poincaré-Wirtinger inequality and (2.9),
∈
we have
1 f (t)
ε 1
f (t) f (t) f (t) + k k C ( ∂ f (t) +1).
ε 2 ε ε 1 x ε 2
k k ≤ − Lk k √L ≤ k k
(cid:13) (cid:13)2
(cid:13) (cid:13)
A similar bound being ava(cid:13)ilable for gε, we in(cid:13)fer from (2.13), (2.15), and the nonnegativity of 1
(cid:13) (cid:13) E
that, for T (0,T (ε)),
+
∈
T T
f (t) 2 + g (t) 2 dt C 1+ ∂ f (t) 2 + ∂ g (t) 2 dt
k ε kH1 k ε kH1 ≤ k x ε k2 k x ε k2
Z0 Z0
(cid:0) (cid:1) C [T +(cid:0) (f ,g )] C (T). (cid:1) (2.16)
1 0ε 0ε 2
≤ E ≤
We next use this estimate to prove that the solution (fε,gε) of (2.2) is bounded in L∞((0,T),H1
×
H1) for all T < T (ε). While the estimates were independent of ε up to now, the next ones have a
+
strong dependence upon ε which explains the need of a regularisation of the original system.
A DEGENERATE PARABOLIC SYSTEM MODELLING FLOWS IN POROUS MEDIA 9
Lemma 2.5. Given (ε,T) (0,1) (0,T (ε)), there exists a constant C(ε,T) > 0 such that the
+
∈ ×
solution (f ,g ) of (2.2) fulfills
ε ε
fε(t) H1 + gε(t) H1 C(ε,T) for all t [0,T]. (2.17)
k k k k ≤ ∈
Proof. We prove first the bound for f . Given z R, let q(z) := z2/2. With this notation, the first
ε
∈
equation of (2.2a) reads
∂ f (1+R)∂2q(f )= R∂ ((f ε)∂ G ).
t ε− x ε x ε− x ε
Multiplying this relation by ∂ q(f ) and integrating over (0,L), we get
t ε
L L L
∂ f ∂ q(f )dx (1+R) ∂2q(f )∂ q(f )dx = R ∂ ((f ε)∂ G )∂ q(f )dx.
t ε t ε − x ε t ε x ε− x ε t ε
Z0 Z0 Z0
Using an integration by parts and Young’s inequality, we come to the following inequality
1+R d 1 R2 L
f ∂ f 2+ ∂ q(f ) 2 f ∂ f 2+ f [∂ ((f ε)∂ G )]2dx.
k ε t εk2 2 dtk x ε k2 ≤ 2k ε t εk2 2 ε x ε− x ε
Z0
p p
Whence, we have shown that
d L
f ∂ f 2+(1+R) ∂ q(f ) 2 R2 f3(∂2G )2+f (∂ f )2(∂ G )2 dx, (2.18)
k ε t εk2 dtk x ε k2 ≤ ε x ε ε x ε x ε
Z0
p (cid:2) (cid:3)
and the second term on the right-hand side of (2.18) may be estimated, in view of (2.8), by
L 1 L 1 L
f (∂ f )2(∂ G )2dx f2(∂ f )2(∂ G )2dx = (∂ q(f ))2(∂ G )2dx
ε x ε x ε ≤ε ε x ε x ε ε x ε x ε
Z0 Z0 Z0
1
∂ G 2 ∂ q(f ) 2.
≤εk x εk∞k x ε k2
Now, sinceG isthesolutionofG ε2∂2G = g withhomogeneous Neumann boundaryconditions
ε ε− x ε ε
at x = 0,L, and g ε, the elliptic maximum principle guarantees that G ε. Hence, ε2∂2G
ε ≥ ε ≥ − x ε ≤
g , and therefore, for x (0,L),
ε
∈
x L
ε2∂ G (x) g dx g and ε2∂ G (x) g dx g ,
x ε ε ε 1 x ε ε ε 1
− ≤ ≤ k k ≤ ≤ k k
Z0 Zx
so that ε2 ∂xGε ∞ gε 1. In view of (2.9), we arrive at
k k ≤ k k
L 1
f (∂ f )2(∂ G )2dx g 2 ∂ q(f ) 2 C(ε) ∂ q(f ) 2. (2.19)
ε x ε x ε ≤ε5k εk1k x ε k2 ≤ k x ε k2
Z0
Concerning the first term on the right-hand side of (2.18), it follows from (2.8) and (2.12) that
L 1 L 4 4
f3(∂2G )2dx f4(∂2G )2dx q(f ) 2 ∂2G 2 q(f ) 2 ∂ g 2.
ε x ε ≤ ε ε x ε ≤ εk ε k∞k x εk2 ≤ ε3k ε k∞k x εk2
Z0 Z0
Inordertoestimate q(fε) ∞,wechoosexε (0,L)suchthatLfε(xε) = fε 1.Bythefundamental
k k ∈ k k
theorem of calculus, we get
x 1
q(f )(x) = q(f )(x )+ ∂ q(f )dx f 2+√L ∂ q(f ) for all x (0,L).
ε ε ε x ε ≤ 2L2k εk1 k x ε k2 ∈
Zxε
10 J.ESCHER,PH.LAURENÇOT,ANDB.–V.MATIOC
Summarising, we obtain in view of (2.18) and (2.19), that
d
∂ q(f ) 2 C(ε)(1+ ∂ g 2)(1+ ∂ q(f ) 2), (2.20)
dtk x ε k2 ≤ k x εk2 k x ε k2
and from (2.16) and (2.20)
t
∂ q(f (t)) 2 (1+ ∂ q(f ) 2)exp C(ε) 1+ ∂ g 2 ds C(ε,T), t [0,T].
k x ε k2 ≤ k x 0ε k2 k x εk2 ≤ ∈
(cid:18) Z0 (cid:19)
(cid:0) (cid:1)
Since f ε, we then have
ε
≥
∂ f (t) 2 C(ε,T) for all t [0,T].
k x ε k2 ≤ ∈
Using (2.9) and the Poincaré-Wirtinger inequality, we finally obtain that
fε(t) H1 C(ε,T) for all t [0,T].
k k ≤ ∈
Moreover, due to the symmetry of (2.2a), g satisfies the same estimate as f , and this completes
ε ε
our argument. (cid:3)
Proof of Theorem 2.1. The proof is adirect consequence of Lemma 2.2, the lower bounds (2.8), and
Lemma 2.5. (cid:3)
We end this section by showing that, though is not dissipated along the trajectories of the
2
E
regularised system (2.2), a functional closely related to is almost dissipated, with non-dissipative
2
E
terms of order ε.
Lemma 2.6. For ε (0,1) and T > 0, we have
∈
T
(f (T),g (T)) + f (1+R)∂ f +R∂ G 2+RR g ∂ (F +g ) 2 dt
2,ε ε ε ε x ε x ε µ ε x ε ε
E | | | |
Z0 h i
T
(f ,g )+εC ̺ (t)dt, (2.21)
2,ε 0ε 0ε 2 ε
≤ E
Z0
with ̺ := ∂ f 2+ ∂ g 2 and
ε k x εk2 k x εk2
L
2 (f ,g ) := (1+R) f 2+R g 2+R (F g +G f ) dx. (2.22)
E2,ε ε ε k εk2 k εk2 ε ε ε ε
Z0
Proof. Wemultiply thefirstequation of (2.2)by (1+R)f +RG andintegrate over (0,L) toobtain
ε ε
L L
∂ f ((1+R)f +RG ) dx = f ((1+R)∂ f +R∂ G )2 dx+I (2.23)
t ε ε ε ε x ε x ε 1,ε
−
Z0 Z0
with
L
I := εR ∂ G ((1+R)∂ f +R∂ G ) dx.
1,ε x ε x ε x ε
Z0
Thanks to Hölder’s inequality and (2.11), we have
I εR(1+R) ∂ G ∂ f + ∂ G 2 εC ∂ f 2+ ∂ g 2 . (2.24)
| 1,ε| ≤ k x εk2k x εk2 k x εk2 ≤ k x εk2 k x εk2
Similarly, multiplying the s(cid:0)econd equation of (2.2) by R(cid:1)(Fε+g(cid:0)ε) and integrating(cid:1)over (0,L) give
L L
R ∂ g (F +g ) dx = RR g (∂ F +∂ g )2 dx+I (2.25)
t ε ε ε µ ε x ε x ε 2,ε
−
Z0 Z0