Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

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Analytical and Numerical Aspects of Partial Differential Equations © de Gruyter 2009

Coupling of a scalar conservation law

with a parabolic problem

Julien Jimenez

Abstract. We deal with the mathematical analysis of the coupling between a purely quasilinear

hyperbolic problem and a parabolic one. First, we use the Schauder–Tikhonov ﬁxed point theorem

and the Holmgren-type duality method to show existence and uniqueness for a class of nonlinear

parabolic problems. In a second part, we recall the method of doubling variables and the vanishing

viscosity method to obtain uniqueness and existence for a class of hyperbolic problems. Then we

combine and adapt the methods stated in the two previous parts to obtain an existence and uniqueness

result for the coupled problem we consider.

Keywords. Parabolic equation, hyperbolic equation, entropy solution, entropy process.

AMS classiﬁcation. 35A05, 35K65, 35L65, 35R05.

1 Introduction

These lecture notes are devoted to the mathematical analysis of the coupling of scalar

conservation laws with parabolic problems. The main motivation for considering such

problems stems from the study of inﬁltration processes in a heterogeneous porous

medium. For instance, in a stratiﬁed subsoil made up of layers with different geo-

logical characteristics, the effects of diffusivity may be negligible in some layers. We

will focus on the case of two layers.

Indeed, let Ω be a bounded domain of R

, n ≥ 1, with a smooth boundary Γ. We

assume that Ω = Ω

∪Ω

, Ω

and Ω

being two disjoint bounded domains of R

, with

Lipschitz boundary denoted by Γ

= ∂Ω

, l ∈ {h, p}. We suppose that the interface

= Γ

∩ Γ

is such that the Lebesgue measure of the set (Γ

∩ (Γ

\ Γ

)) is zero.

The time under consideration is denoted by T > 0. For l ∈ {h, p}, ν

is the outward

unit normal vector deﬁned a.e. on Γ

. We set Q

= (0, T ) × Ω

, Σ

= (0, T ) × Γ

Finally, the interface including the time variable is denoted by Σ

= (0, T ) × Γ

The mathematical model under consideration originates from a combination of con-

servation laws and Darcy’s law and can be described as follows: Find a measurable and

essentially bounded function u on Q := (0, T ) × Ω such that in the sense of distribu-

tions











∂

u + ∇ · (b(x)f(u)) = 0 in Q

∂

u + ∇ · (b(x)f(u)) = ∆φ(u) in Q

u = 0 on (0, T ) × Γ,

u(0, ·) = u

on Ω.

(1.1)

116 Julien Jimenez

Of course, suitable conditions on u across the interface Σ

must be added. These

transmission conditions include the continuity of the ﬂux through the interface, for-

mally written here as

−f(u)b · ν

= (∇φ(u) + f(u)b) · ν

on Σ

Regarding the data of the problem, we assume the following: The initial datum u

belongs to L

∞

(Ω) and there exist m, M ∈ R such that for almost all x ∈ Ω

m ≤ u

(x) ≤ M.

The vector ﬁeld b = (b

, . . . , b

) is an element of W

2,∞

(Ω)

. We denote by M

the Lipschitz constant of b

, i = 1, . . . , n, and set M

i=1

. To simplify the

analysis, we assume

∇ · b(x) = 0 for almost all x ∈ Ω. (1.2)

The continuous function f : R → R is assumed to be Lipschitz continuous on [m, M].

We denote by M

its Lipschitz constant. The right-hand side φ is an increasing contin-

uously differentiable function on R. By normalisation, we suppose φ(0) = 0. We also

suppose that there exists α > 0 such that for all τ ∈ R

(τ) ≥ α. (1.3)

Assumption (1.3) means that the partial differential operator set in Q

is not degener-

ated. This hypothesis is not necessary (we may suppose that this operator is weakly

degenerated) but it avoids technical difﬁculties.

This lecture is organized as follows. In Section 2, we introduce some classical

methods used in the theory of nonlinear parabolic equations. Indeed, thanks to the

Schauder–Tikhonov ﬁxed point theorem and a Holmgren-type duality method, we

prove existence and uniqueness of weak solutions for a class of parabolic problems.

In Section 3, we present the method of doubling variables in order to prove a unique-

ness result for a class of hyperbolic problems. Existence is obtained via the vanishing

viscosity method based upon the notion of entropy process solutions. Then, in Sec-

tion 4, we show an existence and uniqueness result for problem (1.1) by combining

and adapting the methods stated in the two previous sections.

For the reader’s convenience, we collect in the following some notations that will be

used in the sequel. We also give some classical results of functional analysis. However,

we suppose that the deﬁnition and the main properties of Banach, Hilbert, Lebesgue

and Sobolev spaces are known.

The standard norm in H

(Ω) shall be denoted by k · k. The duality pairing between

(Ω) and its dual space H

−1

(Ω) is denoted by h·, ·i. The energetic extension of the

classical differential operator (−∆), supplemented by homogeneous Dirichlet bound-

ary conditions, is an isomorphism from H

(Ω) onto H

−1

(Ω). The L

∞

(Ω)-norm is

denoted by k · k

∞

Let X be a Banach space and T > 0. We denote by L

(0, T ; X), 1 ≤ p < ∞, the

Coupling of a scalar conservation law with a parabolic problem 117

space of functions f : [0, T ] → X such that











f is Bochner measurable,

kfk

(0,T ;X)

kfk

< ∞.

We denote by C([0, T ]; X) the space of continuous functions u : [0, T ] → X with norm

kuk

C([0,T ];X)

:= max

t∈[0,T ]

ku(t)k

The function space

W (0, T ) := {u ∈ L

(0, T ; H

(Ω)); u

∈ L

(0, T ; H

−1

(Ω))},

endowed with the standard norm kuk

W (0,T )

= (kuk

(0,T ;H

(Ω))

+ku

(0,T ;H

−1

(Ω))

)

1/2

and the corresponding inner product, is a Hilbert space, which is continuously embed-

ded in C([0, T ]; L

(Ω)) and compactly embedded in L

(0, T ; L

(Ω))

∼

((0, T )×Ω).

In particular, if u ∈ W (0, T ) then u(0) ∈ L

(Ω) and so u(0) is deﬁned a.e. on Ω. We

will also use the fact that for all u, v ∈ W (0, T ) and almost all t ∈ [0, T ],

Ω

u(t)v(t)dx = hu

(t), v(t)i + hu(t), v

(t)i. (1.4)

We refer, e.g., to [5, Chap. XVIII] (see also [6] for an English translation of [5]) for

more details on the function space W (0, T ). If u ∈ W (0, T ) then u is identiﬁed with a

function ˜u deﬁned on [0, T ]×Ω by setting, for all t ∈ [0, T ], x ∈ Ω, ˜u(t, x) = [u(t)](x).

The time derivative u

can be identiﬁed with the partial derivative ∂

˜u. In the sequel,

we will skip the “tilde”, and ∂

u will denote the derivative u

as well as the partial

derivative ∂

˜u.

A sequence (ρ

)

j∈N

is called a sequence of molliﬁers in R

if, for all j ∈ N, ρ

nonnegative, ρ

∈ C

∞

; R), supp ρ

⊂ B(0,

) and

(x)dx = 1.

Let x ∈ R

. Then x is said to be a Lebesgue point of f ∈ L

loc

) if

lim

r→0

meas(B(x, r))

B(x,r)

|f(x) − f(y)|dy = 0,

where B(x, r) = {y ∈ R

; kx − yk < r}. We refer to [7] for the properties of a

Lebesgue point and only mention that if f is continuous at x then x is a Lebesgue

point of f. For f ∈ L

loc

), almost every point is a Lebesgue point. Therefore, if

v ∈ L

(O) (O is an open bounded subset of R

) then

lim

j→∞

O×O

(v(y) − v(x))ρ

(y − x)dxdy = 0.

The following lemma is due to Mignot and Bamberger (see [9, p. 31]); a general-

ization can be found in [18].

118 Julien Jimenez

Lemma 1.1. Let β : R → R be a continuous and increasing function such that

lim sup

|λ|→∞

|β(λ)|

|λ|

< ∞.

Then, for any function u ∈ L

(0, T ; L

(Ω)) with ∂

u ∈ L

(0, T ; H

−1

(Ω)) and β(u) ∈

(0, T ; H

(Ω)), the following integration-by-parts formula holds in L

(0, T ):

h∂

u, β(u)i =

Ω

u(·,x)

β(r)dr

dx.

For the following well-known Gronwall lemma, we refer, e.g., to [5, p. 672].

Lemma 1.2 (Gronwall). Let T > 0, φ ∈ L

∞

(0, T ), φ ≥ 0 a.e. on (0, T ), µ ∈ L

(0, T ),

µ ≥ 0 a.e. on (0, T ). Suppose there exists κ > 0 such that, for almost all s ∈ (0, T ),

φ(s) ≤ κ +

µ(t)φ(t)dt.

Then, for almost all s ∈ (0, T ),

φ(s) ≤ κ exp



µ(t)dt



The following auxiliary result can be found in [16].

Proposition 1.3. Let w ∈ H

(Ω) and let g : R → R be a Lipschitz continuous function.

Then g ◦ w ∈ H

(Ω) and for all i = 1, . . . , n

∂

g(w) = g

(w)∂

w a.e. on Ω.

We also make use of a variant of the well-known Schauder–Tikhonov ﬁxed point

theorem that can be found in [9, p. 30].

Theorem 1.4. Let X be a reﬂexive and separable Banach space and K ⊂ X, K 6=

∅, be a closed, bounded and convex set. Let T : K → K be a “weakly-weakly”

sequentially continuous mapping, i.e., for any sequence (x

)

n∈N

⊂ K that converges

weakly towards some x ∈ X, the sequence (T (x

))

n∈N

converges weakly towards

T (x). Then T has at least one ﬁxed point in K.

Finally, we employ the following result that goes back to Eymard, Gallouët and

Herbin, see [8].

Theorem 1.5. Let O be an open bounded set of R

and (u

)

j∈N

a sequence of measur-

able essentially bounded functions on O. If there exists A > 0 such that

∞

(O)

≤ A for all j ∈ N

then there exist a subsequence (u

ϕ(j)

)

j∈N

and a function π ∈ L

∞

((0, 1) × O) such that

for any continuous and bounded function f : O × (−A, A) → R

lim

j→∞

f(x, u

ϕ(j)

(x))ξdx =

f(x, π(α, x))ξdxdα ∀ ξ ∈ L

(O).

Coupling of a scalar conservation law with a parabolic problem 119

Let sgn : R → R be given by

sgn(x) =











−1 if x < 0,

0 if x = 0,

1 if x > 0,

and denote by sgn

(η > 0) its Lipschitz continuous approximation given by

sgn

(x) =











−1 if x ≤ −η,

if − η < x < η,

1 if x ≥ η.

Furthermore, for all x ∈ R, η > 0, we set

sgn

(x) := max(sgn

(x), 0) and sgn

−

(x) := max(−sgn

(x), 0).

2 The parabolic problem

In this part, we collect some tools that will be useful in the sequel in order to obtain

existence and uniqueness results for a class of nonlinear parabolic equations. One may

assume that we treat the special case Ω

= ∅ so that Ω

= Ω. We consider the initial-

boundary value problem











∂

u + ∇ · (b(x)f(u)) = ∆φ(u) in Q,

u = 0 on (0, T ) × Γ,

u(0, ·) = u

on Ω.

(2.1)

The assumptions on the data are the same as in Section 1. In particular, u

∈

∞

(Ω) with m ≤ u

(x) ≤ M for almost all x ∈ Ω. The vector ﬁeld b ∈ W

2,∞

(Ω)

satisﬁes (1.2). The function f is Lipschitz continuous on [m, M] and φ is an increasing

continuously differentiable function on R that fulﬁlls (1.3) with φ(0) = 0.

First, we provide a deﬁnition of a weak solution to (2.1).

Deﬁnition 2.1. A function u ∈ W (0, T ) with

m ≤ u(t, x) ≤ M for almost all (t, x) ∈ Q (2.2)

is said to be a weak solution to (2.1) if for almost all t ∈ (0, T ) and all v ∈ H

(Ω) the

equation

h∂

u(t), vi +

Ω

(∇φ(u(t)) − b(x)f(u(t))) · ∇vdx = 0 (2.3)

is fulﬁlled and if

u(0) = u

in L

(Ω). (2.4)

Remark 2.2. As is mentioned in Section 1, we have identiﬁed in Deﬁnition 2.1 the

function u ∈ W (0, T ) with the function ˜u deﬁned on Q = (0, T ) ×Ω such that ˜u(t, x) =

[u(t)](x) for almost all t ∈ (0, T ), x ∈ Ω.

120 Julien Jimenez

2.1 Existence: the ﬁxed point method

This part is devoted to the proof of the following theorem:

Theorem 2.3. There exists at least one weak solution to (2.1).

The main idea is to use the Schauder–Tikhonov ﬁxed point theorem. To do so, we

need a result about linear parabolic problems due to J. L. Lions that can be found in [5,

Chap. XVIII, p. 629] and also in [19, Chap. IV, p. 397].

Theorem 2.4 (Lions). For any t ∈ [0, T ], let a(t; ·, ·) be a bilinear form on H

(Ω) ×

(Ω) such that t 7→ a(t; u, v) is measurable on [0, T ] for all u, v ∈ H

(Ω). The

bilinear form is assumed to be uniformly bounded and strongly positive, i.e., there are

constants β, γ > 0 such that for all u, v ∈ H

(Ω), t ∈ [0, T ],

|a(t; u, v)| ≤ βkukkvk and a(t; u, u) ≥ γkuk

Then, for any u

∈ L

(Ω) and g ∈ L

(0, T ; H

−1

(Ω)), there exists a unique solution

u ∈ W (0, T ) such that for almost all t ∈ (0, T ) and all v ∈ H

(Ω)

(t), vi + a(t; u(t), v) = hg(t), vi

and u(0) = u

in L

(Ω).

Furthermore there exists a constant C > 0 depending on β and γ such that

kuk

W (0,T )

≤ C(ku

(Ω)

+ kgk

(0,T ;H

−1

(Ω))

). (2.5)

Proof of Theorem 2.3. The proof can be divided into three steps. In a ﬁrst step, we

introduce a modiﬁed problem with bounded coefﬁcients (see (2.6)) that is equivalent

to (2.2)–(2.4). In a second step, we consider a linearized problem related to (2.6). With

the help of Theorem 2.4, we prove that this linear problem has a unique solution. This

allows us to deﬁne a mapping T on W (0, T ) whose possible ﬁxed points are solutions

to (2.6). Finally, in the third step, we show that Theorem 1.4 can be applied to the

mapping T .

Step 1: Truncation process. We introduce the following auxiliary problem: Find u

in W (0, T ) such that a.e. on (0, T ) and for any v ∈ H

(Ω),











h∂

u(t), vi +

Ω

(φ

∗

(t))∇u(t) − b(x)f(u

∗

(t))) · ∇vdx = 0,

u(0) = u

in L

(Ω),

(2.6)

where, for almost all (t, x) ∈ Q,

∗

(t, x) =











m if u(t, x) < m,

u(t, x) if m ≤ u(t, x) ≤ M,

M if u(t, x) > M.

Coupling of a scalar conservation law with a parabolic problem 121

Let us prove that (2.2)–(2.4) and (2.6) are equivalent. It is clear that if u satisﬁes

(2.2)–(2.4) then u fulﬁlls (2.6).

Conversely, suppose that u is a solution to (2.6). Let us show that u satisﬁes (2.2).

We consider in (2.6) the test function v

= sgn

(u − M ) (thanks to Proposition 1.3,

∈ W (0, T )) and we integrate over (0, s) for any s of (0, T ]. This yields

h∂

u, v

idt +

Ω

(φ

∗

)∇u − b(x)f(u

∗

)) · ∇v

dx = 0. (2.7)

For the evolution term, we use Lemma 1.1 and the fact that u

≤ M a.e. on Ω to obtain

h∂

u, v

idt =

h∂

(u − M), v

idt =

Ω

u(s,x)−M

sgn

(r)dr

dx.

To treat the convection term, we refer to the deﬁnition of sgn

and u

∗

. We ﬁnd

Ω

b(x)f(u

∗

)∇v

dxdt =

{x∈Ω; 0<u(t,x)−M <η}

b(x)f(u

∗

)∇v

dxdt

Ω

b(x)f(M)∇v

dxdt.

By the Gauss–Green formula and (1.2), we have

Ω

b(x)f(M)∇v

dxdt = −

Ω

f(M)∇ · b(x)v

dxdt = 0.

We use Proposition 1.3 and the deﬁnition of sgn

for the diffusion term to deduce

Ω

∗

)∇u · ∇v

dxdt =

{x∈Ω; 0<u(t,x)−M <η}

∗

)∇u · ∇udxdt ≥ 0.

Then, from (2.7), for all s ∈ (0, T ], η > 0,

Ω

u(s,x)−M

sgn

(r)dr

dx ≤ 0.

Thus, for all s ∈ (0, T ], we ﬁnd

lim

η→0

Ω

u(s,x)−M

sgn

(r)dr

dx =

Ω

(u(s, x) − M)

dx ≤ 0,

where (u(s, x) − M )

= sgn

(u(s, x) − M)((u(s, x) − M )) for almost all x ∈ Ω. It

follows that u ≤ M a.e. on Q.

To prove that u ≥ m a.e. on Q, the reasoning is similar: We just consider the

test function sgn

−

(u − m) in (2.6). Then u satisﬁes (2.2), (2.4) and, consequently,

122 Julien Jimenez

(2.3) (since u

∗

= u a.e. on Q). Thus, the existence result for (2.1) is equivalent to an

existence result to (2.6).

Step 2: Study of a linear problem. For any given w ∈ W (0, T ), we consider now

the linear problem: Find U

∈ W (0, T ) such that, a.e. on (0, T ) and for all v ∈ H

(Ω)











h∂

, vi +

Ω

(φ

∗

)∇U

− f(w

∗

)b(x)) · ∇vdx = 0,

(0) = u

in L

(Ω).

(2.8)

In order to prove the existence of a unique solution U

∈ W (0, T ) to problem (2.8),

we set for all t ∈ [0, T ] and all u, v ∈ H

(Ω)

a(t; u, v) :=

Ω

∗

(t, x))∇u(x) · ∇v(x)dx

and

hg(t), vi :=

Ω

f(w

∗

(t, x))b(x) · ∇v(x)dx.

We have g ∈ L

(0, T ; H

−1

(Ω)) (by the Cauchy–Schwarz inequality) and, for almost

all t ∈ (0, T ),

|a(t; u, v)| ≤ kφ

∞

kukkvk ≤ kφ

∞,[m,M]

kukkvk and a(t; u, u) ≥ αkuk

where kφ

∞,[m,M]

:= max

τ∈[m,M]

|φ

(τ)|.

Hence, Theorem 2.4 ensures that there exists a unique solution U

∈ W (0, T ) to

(2.8). Moreover, it also follows from Theorem 2.4 that there exists C > 0 such that

W (0,T )

≤ C. (2.9)

Since, for any w ∈ W (0, T ), the problem (2.8) has a unique solution U

∈ W (0, T ),

we may deﬁne the operator

T : W (0, T ) → W (0, T ), w 7→ U

The aim is now to prove that T admits at least one ﬁxed point. Indeed, if there

exists u ∈ W (0, T ) such that T (u) = u then u will be a weak solution to (2.6).

Step 3: Use of the Schauder–Tikhonov theorem. We set

K := {v ∈ W (0, T ); v(0) = u

and kvk

W (0,T )

≤ C},

where C is the a priori bound given by (2.9). It is clear that K is non-empty, bounded,

convex and closed. By (2.9), we also have T (K) ⊂ K. Therefore, in order to apply

the Schauder–Tikhonov ﬁxed point theorem (Theorem 1.4), it remains to prove that

T is “weakly-weakly” sequentially continuous. Let (w

)

n∈N

⊂ K be a sequence that

converges weakly towards w ∈ K. We have to prove that (T (w

))

n∈N

converges

weakly towards T (w).

Coupling of a scalar conservation law with a parabolic problem 123

By deﬁnition of T , the function U

= T (w

) satisﬁes, a.e. in (0, T ) and for all

v ∈ H

(Ω),







h∂

, vi +

Ω

(φ

∗

)∇U

− f(w

∗

)b(x)) · ∇vdx = 0,

(0) = u

in L

(Ω).

(2.10)

In order to take the limit with respect to n in (2.10), we need to specify some

convergence results for the sequences (w

)

n∈N

and (U

)

n∈N

. Since W (0, T )

,→

(0, T ; L

(Ω))

∼

(Q), there exists a subsequence, still denoted by (w

)

n∈N

, that

converges strongly in L

(Q) to w.

From (2.9), we have kU

W (0,T )

≤ C. Hence, there exists a subsequence, still

denoted by (U

)

n∈N

, such that U

* U in W (0, T ), i.e.,

∂

* ∂

U in L

(0, T ; H

−1

(Ω)) and U

* U in L

(0, T ; H

(Ω)). (2.11)

Since W (0, T )

,→ L

(Q), the strong convergence

→ U in L

(Q)

follows. Moreover, since W(0, T ) ,→ C([0, T ], L

(Ω)), we have

(0) * U(0) in L

(Ω).

As U

(0) = u

for all n ∈ N, we deduce that U(0) = u

in L

(Ω).

Now it is possible to take the limit with respect to n in (2.10). Since f is Lipschitz

continuous on [m, M], b is bounded and w

→ w in L

(Q), we also have

bf(w

∗

) → bf(w

∗

) in L

(Q)

Since m ≤ w

∗

≤ M and φ

is continuous, the subsequence can be chosen such

that φ

∗

)∇U

* φ

∗

)∇U in L

(Q)

. Taking the limit in (2.10) now shows that

U ∈ W (0, T ) fulﬁlls a.e. in (0, T ) and for all v ∈ H

(Ω)

h∂

U, vidt +

Ω

(φ

∗

)∇U − b(x)f(w

∗

))∇vdx = 0

as well as U(0) = u

. Hence, U = T (w).

Up to now, we have shown that there exists a subsequence of (T (w

))

n∈N

that con-

verges towards T (w). To see that the whole sequence (T (w

))

n∈N

converges towards

T (w), we remark that the limit of any convergent subsequence (T (w

))

n∈N

satisﬁes

(2.8). Since (2.8) has a unique solution, we deduce by contradiction that the whole

sequence converges towards T (w).

Thus the assumptions of the Schauder–Tikhonov theorem are satisﬁed, and we may

conclude that there exists u ∈ W (0, T ) such that u = T (u). Eventually, this ﬁxed

point u is a weak solution to (2.6) and thus to (2.1).

124 Julien Jimenez

2.2 Uniqueness: the Holmgren-type duality method

We are now interested in the uniqueness of solutions to (2.1), which shall be proved

by contradiction. Let u and ˆu be two weak solutions to (2.1). The idea is to use the

−1

(Ω)-norm in order to estimate (u − ˆu). Since (−∆) : H

(Ω) → H

−1

(Ω) is an

isomorphism, the H

−1

(Ω)-norm of u − ˆu is equivalent to k(−∆)

−1

(u − ˆu)k.

For any t ∈ [0, T ], let z(t) ∈ H

(Ω) be the weak solution to







−∆z(t) = u(t) in Ω,

z(t) = 0 on ∂Ω,

and analogously ˆz(t) ∈ H

(Ω) be the weak solution of the foregoing Poisson problem

with right-hand side ˆu(t), i.e., for all v ∈ H

(Ω),

Ω

∇z(t) · ∇vdx =

Ω

u(t)vdx (resp.

Ω

∇ ˆz(t) · ∇vdx =

Ω

ˆu(t)vdx). (2.12)

Remember that u, ˆu ∈ C([0, T ]; L

(Ω)).

Since ∂

u ∈ L

(0, T ; H

−1

(Ω)), we are able to deﬁne ∂

z(t) (resp. ∂

ˆz(t)) for al-

most all t ∈ (0, T ) as elements of H

(Ω) characterized by

Ω

∇(∂

z(t)) · ∇vdx = h∂

u, vi (resp.

Ω

∇(∂

ˆz(t)) · ∇vdx = h∂

ˆu, vi) (2.13)

for all v ∈ H

(Ω). Indeed, the isomorphism (−∆)

−1

: H

−1

(Ω) → H

(Ω) can be ex-

tended to a linear and continuous mapping of L

(0, T ; H

−1

(Ω)) onto L

(0, T ; H

(Ω)).

We subtract the equations (2.3) for the solutions u and ˆu and take v = (z − ˆz)(t).

We then integrate from 0 to s, s ∈ (0, T ], and obtain

h∂

(u− ˆu), z− ˆzidt+

Ω

(∇(φ(u)−φ( ˆu))−(f(u)−f( ˆu))b(x))·∇(z− ˆz)dxdt = 0.

(2.14)

From (2.13), it follows with an integration by parts that

h∂

(u − ˆu), z − ˆzidt =

Ω

∇(∂

(z − ˆz)) · ∇(z − ˆz)dxdt

Ω

|∇(z − ˆz)|

(s)dx −

Ω

|∇(z − ˆz)|

(0)dx.

With (2.12) for t = 0 and since u(0) = ˆu(0) = u

, we ﬁnd

Ω

|∇(z − ˆz)|

(0)dx =

Ω

(u(0) − ˆu(0))

dx = 0.

We choose v = φ(u(t)) − φ( ˆu(t)) in (2.12) written for z(t) and ˆz(t), integrate over

(0, s) (with Q

= (0, s) × Ω) and get, using (1.3),

Ω

∇(φ(u) − φ( ˆu)) · ∇(z − ˆz)dxdt =

Ω

(u − ˆu)(φ(u) − φ(ˆu))dxdt

≥ αku − ˆuk

)