Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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lim

n→+∞

lim

µ→+∞

lim

ε→0



Ω

A(x, t)Du

.Du

dx ds dt



= 0,

(55)

and

lim

µ→+∞

lim

ε→0

Ω

dx ds dt = 0 for any n ≥ 1. (56)

Proof of (52)

. In view of the deﬁnition (50) of W

, lemma 3.1 applies with

h = S

for ﬁxed n ≥ K. As a consequence (52) holds true.

Proof of (53). For ﬁxed n ≥ 1, we have

)Φ

)DW

= S

)Φ

n+1

))DW

(57)

a.e. in Q, and where suppS

⊂ [−(n + 1), n + 1]. Since S

is smooth

and bounded, (19) and (36) lead to S

)Φ

n+1

)) converges to

(u)Φ(T

n+1

(u)) a.e. in Q and in L

∞

(Q) weak ?, as ε tends to 0. For

ﬁxed µ > 0, we have

* (T

(u) −T

(u)

) weakly in L

(0, T ; H

(Ω)) (58)

and a.e. in Q and in L

∞

(Q) weak ?, as ε tends to 0. As a consequence of

(57) and (58) we deduce that

lim

ε→0

Ω

)Φ

)DW

dx ds dt (59)

Ω

(u)Φ(u)D

(u) − T

(u)

dx ds dt

for any µ > 0.

Appealing now to (47) and passing to the limit as µ → +∞ in (59)

allows to conclude that (53) holds true.

Proof of (54)

. For ﬁxed n ≥ 1, and by the same arguments that those that

lead to (53), we have

)Φ

)Du

= S

)Φ

n+1

))DT

n+1

a.e. in Q.

From (19) and (36), it follows that for any µ > 0

lim

ε→0

Ω

)Φ

)Du

dx ds dt

Ω

)Φ

n+1

))DT

n+1

dx ds dt

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161

with the help of (58) passing to the limit, as µ tends to +∞, in the above

equality leads to (53).

Proof of (55). For any n ≥ 1 ﬁxed, we have suppS

⊂ [−(n + 1), −n] ∪

[n, n + 1]. As a consequence



Ω

)A(x, t)Du

.Du

dx ds dt



≤ T C

{n≤|u

|≤n+1}

A(x, t)Du

.Du

dx dt,

for any n ≥ 1, and any µ > 0, where C is a constant independent of n, µ.

With the help of (40) passing to the limit, as ε tends to zero, µ tends to

+∞ and n tends to +∞ and to establish (55).

Proof of (56)

. For ﬁxed n ≥ 1, and in view (20), (36) and (58), Lebesgue’s

convergence theorem implies that for any µ > 0 and any n ≥ 1

lim

ε→0

Ω

(x, t, T

n+1

))S

dx ds dt

Ω

f(x, t, T

n+1

(u))S

(u)



(u) − T

(u)



dx ds dt.

Now for ﬁxed n ≥ 1, using (47) permits to pass to the limit as µ tends

to +∞ in the above equality to obtain (56).

We now turn back to the proof of lemma 3.2, due to (51), (52), (53),

(54), (55) and (56), we are in a position to pass to the lim-sup when ε tends

to zero, then to the limit-sup when µ tends to +∞ and then to the limit as

n tends to +∞ in (51). We obtain using the deﬁnition of W

that for any

K ≥ 0

lim

n→+∞

lim

µ→+∞

lim

ε→0

Ω

)A(x, t)

×Du



) − T

(u)



dx ds dt ≤ 0.

Since S

)DT

) = DT

) for K ≤ n.

The above inequality implies that for K ≤ n

lim

ε→0

Ω

A(x, t)Du

.DT

) dx ds dt (60)

≤ lim

n→+∞

lim

µ→+∞

lim

ε→0

Ω

)A(x, t)DT

n+1

).DT

(u)

dx ds dt.

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162

The right hand side of (61) is computed as follows. Due to (37) it follows

that for ﬁxed n ≥ 1

)DT

n+1

) * S

(u)DT

n+1

(u) weakly in (L

(Q))

when ε tends to 0. The strong convergence of T

(u)

to T

(u) in

(0, T ; H

(Ω)) as µ tends to +∞, then allows to conclude that

lim

µ→+∞

lim

ε→0

Ω

)A(x, t)DT

n+1

).DT

(u)

dx ds dt (61)

Ω

(u)A(x, t)DT

n+1

(u).DT

(u) dx ds dt

Ω

A(x, t)DT

(u).DT

(u) dx ds dt

as soon as K ≤ n, since S

(r) = 1 for |r| ≤ n.

Recalling (60) and (61) allows to conclude (49) holds true and the proof

of lemma 3.2 is complete.

? Step 6 : The strong convergence of truncates. In this step we

prove the following monotonicity estimate :

Lemma 3.3. The subsequence of u

deﬁned in step 3 satisﬁes for any

K ≥ 0

lim

ε→0

Ω

A(x, t)[DT

)−DT

(u)].[DT

)−DT

(u)]dx dt ds = 0.

(62)

And

) −→ T

(u) strongly in L

(0, T, H

(Ω)) (63)

as ε goes to zero.

Proof of Lemma 3.3. Let K ≥ 0 be ﬁxed. we have

Ω

A(x, t)

) − DT

(u)

) − DT

(u)

dx dt ds

(64)

Ω

A(x, t)DT

).DT

) dx dt ds

−

Ω

A(x, t)DT

(u)DT

) dx dt ds

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163

−

Ω

A(x, t)

) − DT

(u)

.DT

(u) dx dt ds.

Using (49) of lemma 3.2, we obtain

lim

ε→0

≤

Ω

A(x, t)DT

).DT

) dx dt ds (65)

≤

Ω

A(x, t)DT

(u).DT

(u) dx dt ds.

As a consequence of (37) we have for all K > 0

lim

ε→0

Ω

A(x, t)

) − DT

(u)

.DT

(u) dx dt ds = 0 (66)

(65) and (66) allow to pass to the lim-sup as ε tends to zero in (64) and to

obtain (62) and (63) of lemma 3.3.

? Step 7. In this step, u is shown to satisﬁes (14) and (15). Let S be

a function in W

2,∞

(IR) such that S

has a compact support. Let K be a

positive real number such that suppS

⊂ [−K, K]. Pointwise multiplication

of the approximate equation (22) by S

) leads to

∂b

(x, u

)

∂t

− div



)A(x, t)Du



+ S

)A(x, t)Du

.Du

(67)

− div



)Φ

)



)Φ

)Du

(x, t, u

) = 0 in D

(Q).

where b

(x, r) =

∂b

(x, s)

∂s

(s) ds. In what follows we pass to the limit

as ε tends to 0 in each term of (67).

? Since S is bounded, and b

(x, u

) converges to b

(x, u) a.e. in Q and in

∞

(Q) weak ?. Then

∂b

(x,u

)

∂t

converges to

∂b

(x,u)

∂t

in D

(Q) as ε tends to

? Since suppS

⊂ [−K, K], we have forS

)A(x, t)Du

)A(x, t)DT

) a.e. in Q. The pointwise convergence of u

to u as ε

tends to 0, the bounded character of S

and (37) imply that

)A(x, t)DT

) * S

(u)A(x, t)DT

(u) weakly in (L

(Q))

as ε tends to 0. And the term S

(u)A(x, t)DT

(u) = S

(u)A(x, t)Du a.e.

in Q.

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164

? Since suppS

⊂ [−K, K], we have S

)A(x, t)Du

.Du

A(x, t)DS

).DT

) a.e. in Q. Due to (29) and (36), DS

) con-

verges to DS

(u) weakly in (L

(Q))

as ε tends to 0, and (63) of lemma

3.3 allow to conclude that

A(x, t)DS

).DT

) * A(x, t)DS

(u).DT

(u) weakly in L

(Q),

as ε tends to 0. And A(x, t)DS

(u).DT

(u) = S

(u)A(x, t)Du.Du a.e. in

? Since suppS

⊂ [−K, K], we have S

)Φ

) = S

)Φ

)) a.e.

in Q. As a consequence of (19) and (36), it follows that for any 1 ≤ q <

+∞, S

)Φ

) converges to S

(u)Φ(T

(u)) strongly in L

(Q), as ε

tends to 0. The term S

(u)Φ(T

(u)) is denoted by S

(u)Φ(u).

? Since S

∈ W

1,∞

(IR) with suppS

⊂ [−K, K], we have

)Φ

)Du

= Φ

))DS

) a.e. in Q, we have, DS

) con-

verges to DS

(u) weakly in L

(Q)

as ε tends to 0, while Φ

)) is

uniformly bounded with respect to ε and converges a.e. in Q to Φ(T

(u))

as ε tends to 0. Therefore

)Φ

)Du

* Φ(T

(u))DS

(u) weakly in L

(Q).

? Due to (20) and (35), we have f

(x, t, T

n+1

))S

) converges to

f(x, t, T

n+1

(u))S

(u) strongly in L

(Q), as ε tends to 0.

As a consequence of the above convergence result, we are in a position

to pass to the limit as ε tends to 0 in equation (67) and to conclude that u

satisﬁes (28).

It remains to show that b

(x, u) satisﬁes the initial condition (29). To

this end, ﬁrstly remark that, S ∈ W

2,∞

(IR) such that S

has a com-

pact support, as a consequence of (17) we have b

(x, u

) is bounded in

(0, T ; H

(Ω)). Secondly, (67) and the above considerations on the be-

havior of the terms of this equation show that

∂b

(x, u

)

∂t

is bounded

in L

(Q) + L

(0, T ; H

−1

(Ω)). As a consequence, an Aubin’s type lemma

(see, e.g, ,

Corollary 4) implies that b

(x, u

) lies in a compact set of

([0, T ]; W

−1,s

(Ω)) for any s < inf



N−1



. It follows that, on one

hand, b

(x, u

)(t = 0) = b

(x, u

) converges to b

(x, u)(t = 0) strongly

in W

−1,s

(Ω). On the other hand, (21) and the smoothness of S im-

ply that b

(x, u

) converges to b

(x, u)(t = 0) strongly in L

(Ω) for all

q < +∞.Then we conclude that b

(x, u)(t = 0) = b

(x, u

) in Ω. As a

conclusion of step 1-step 7, the proof of theorem 3.1 is complete.

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4. Comparison principle and uniqueness result

This section is concerned with a comparison principle (and a uniqueness

result) for renormalized solutions in the case where f(x, t, u) is independent

of u. We establish the following theorem.

Theorem 4.1. Assume that assumptions (4), (5), (6), (7) and (11) hold

true and moreover that

For any K > 0, there exists a positive real number β

> 0, such that



∂b(x, z

)

∂s

−

∂b(x, z

)

∂s



≤ β



− z



(68)

for almost every x in Ω, and for every z

and every z

such that |z

| ≤ K

and |z

| ≤ K.

Φ is a locally lipschitz-continuous function on IR. (69)

Let then u

and u

be renormalized solutions corresponding to the data

, u

) and (f

, u

) for problem (i = 1, 2)

∂b(x, u

)

∂t

− div



A(x, t)Du

+ Φ(u

)



= f

(x, t) in Ω × (0, T ), (70)

b(x, u

)(t = 0) = b(x, u

) in Ω, (71)

= 0 on ∂Ω ×(0, T ), (72)

, f

∈ L

(Ω ×(0, T )). (73)

If these data satisfying f

≤ f

and u

≤ u

almost every where, we have

≤ u

almost every where.

Sketch of the Proof of theorem 4.1 . Here we give just an idea on how u

≤

can be obtained following the outlines of .

The proof is divided into two Steps. In Step 1, we deﬁne a smooth

approximation S

of T

, and we consider tow renormalized solutions u

and

of (70)-(72) for the data (f

, u

) and (f

, u

) respectively we plug the

test function



(x, u

) −b

(x, u

))



in the diﬀerence of equations

(14) for u

and u

in which we have taken S = S

In Step 2, we investigate the behavior of the diﬀerent terms in the

estimate obtained in step 1 (estimates (76)) as σ tends to 0 and when n

tends to +∞.

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166

? Step 1 . Remark that when Φ is locally-continuous on IR the following

derivation is licit for any function S and u satisfying the conditions men-

tioned in Deﬁnition 2.1.

div



(u)Φ(u)



− S

(u)Φ(u)Du = S

(u)Φ

(u)Du = div(Φ

(u)) (74)

Where Φ

= (Φ

S,1

, Φ

S,2

, ··· , Φ

S,N

) with

S,i

(r) =

S,i

(t)S

(t) dt.

Let us now introduce a speciﬁc choice of function S in (14). For all

n > 0, let S

∈ C

(IR) be the function deﬁned by S

(r) = 1 for |r| ≤

n ; S

(r) = n + 1 − |r| for n ≤ |r| ≤ n + 1 and S

(r) = 0 for |r| ≥ n + 1. It

yields, taking S = S

in (14)

∂b

(x, u

)

∂t

− div



)A(x, t)Du



+ S

)A(x, t)Du

(75)

− div



)



= f

) in D

(Q) ;

for i = 1, 2 and where b

(x, r) =

∂b(x, s)

∂s

(s) ds.

We use



(x, u

)−b

(x, u

)



as a test function in the diﬀerence

of equations (75) for u

and u

∂



(x, u

) − b

(x, u

)



∂t

; T



(x, u

) − b

(x, u

)



ids dt

(76)

= B

+ C

+ D

for any σ > 0, n > 0, and where

Ω

)A(t, x)Du

− S

)A(t, x)Du

(77)



(x, u

) −b

(x, u

)



dx ds dt

Ω

)A(x, t)Du



(x, u

) − b

(x, u

)



dx ds dt

(78)

−

Ω

)A(x, t)Du



(x, u

) − b

(x, u

)



dx ds dt

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167

Ω

) − Φ

)



(x, u

) − b

(x, u

)



dx ds dt

(79)

Ω

) − f

)



(x, u

) − b

(x, u

)



dx ds dt.

(80)

In the sequel we pass to the limit in (76) when σ tends to 0 and then n

tends to +∞. Upon application of lemma 2.4 of ,

the ﬁrst term in the right

hand side of (76) is derived as

∂



(x, u

) − b

(x, u

)



∂t

; T



(x, u

) − b

(x, u

)



ids dt

(81)



(x, u

) − b

(x, u

)



dx dt

−

Ω



(x, u

) − b

(x, u

)



where

(t) =

(s) ds.

Due to the assumption u

≤ u

a.e. in Ω and the monotone character

of b

(x, .) and T

(.) , we have

Ω



(x, u

) − b

(x, u

)



dx = 0 (82)

It follows from (76), (81) and (82) that



(x, u

) − b

(x, u

)



dx dt + A

= B

+ C

+ D

(83)

for any σ > 0 and any n > 0.

? Step 2. In this step, we study the behaviors of the terms A

, B

, C

and D

when σ tends to 0 and n → +∞. More precisely, we prove the

following Lemma

Lemma 4.1. We have

lim

n→+∞

lim

σ→0

≥ 0, (84)

lim

n→+∞

lim

σ→0

= 0, (85)

lim

σ→0

= 0 for all n, (86)

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168

lim

n→+∞

lim

σ→0

≤ 0. (87)

Proof of Lemma 4.1. The lemma is proved in .

In view of estimates (82), (83), (84), (85), (86) and (87) we have



b(x, u

) − b(x, u

)



dx dt ≤ 0,

so that b(x, u

) ≤ b(x, u

) a.e. in Q which in turn implies that u

≤ u

a.e.

in Q, theorem 4.1 will be then established.

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