Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

1.2.2 The Sobolev space Wm,p(I)

Proposition 1.2.1 ([26])

Let a be an open domain in le, Then the distribution T 2 D^'(a) is in L^P(a) if there exists a function f 2 L^P(a) such that

(T, (P) = I

f(x)cp(x)dx, for all cp 2 D(a),

where 1 < p < oo, and it's well-known that f is unique.

Definition 1.2.2 Let m 2 N and p 2 [0, oo] . The W^m,P(a) is the space of all f 2 L^P(a), defined

as

W^m'^P(a) = 2 L^P(a), such that O^af 2 L^P(a) for all a 2 N^tm such that

jj (1.17)

= j=1ai < m, where, a^s = @~2

2 ::@~n

n g:

Theorem 1.2.4 ([9])

W^m,P(a) is a Banach space with their usual norm

mi,vm^,pm= E jj~m

11a^sf11Lp , 1 < p < oo, for all f 2 W^rd'(a). (1.18)

Definition 1.2.3 Denote by Wr'^P(a) the closure of D(a) in W^m,P(a).

Definition 1.2.4 When p = 2, we prefer to denote by W^{m 2}(a) = H^t (a) and Wc7² (a) = H^m0 (a) supplied with the norm

0 11/11H-(n) = (11.9711L2)2) jj~m

1 2

;

(1.19)

which do at H^tm (a) a real Hilbert space with their usual scalar product

(u, v) (n) = E J O^auo^avdx (1.20)

101<m n

Theorem 1.2.5 ([42], Proposition 1.2.1)

1) H^t (a) supplied with inner product (.,.)H.(n) is a Hilbert space.

2) If m > m^', H^tm (a) y H^m' (a), with continuous imbedding .

Lemma 1.2.6 ([26])

Since D(a) is dense in H^m₀ (a), we identify a dual H' (a) of H^m₀ (a) in a weak subspace on a, and we have

D(a) y H^m0 (a) y L² (a) y H^' (a) y D⁰(a),

Lemma 1.2.7 (Sobolev-Poincaré's inequality) If

m - 2,

2 ~ q ~

2m m > 3

q ~ 2, m = 1,2,

then

kuk_q ~ C(q, ) VuM₂ , (1.21)

for all u 2 H1 0 (1).

The next results are fundamental in the study of partial differential equations

Theorem 1.2.6 ([9] Theorem 1.3.1)

Assume that is an open domain in R^N (N ~ 1), with smooth boundary F. Then,

(i) if 1 p m, we have W¹' c L^q(l), for every q 2 [p, p^*] , where p^* = mp .

m ~ p

(ii) if p = m we have W^1' C L^q(1), for every q 2 [p, oc). (iii) if p > m we have W^1' C L^°°(l) fl C^0'(l), where a = - m p

.

Theorem 1.2.7 ([9] Theorem 1.3.2)

If 1 is a bounded, the embedding (ii) and (iii) of theorem 1.1.4 are compacts. The embedding (i) is compact for all q 2 [p, p^*).

Remark 1.2.2 ([26])

For all çü 2 H²(1), LIço 2 L²(1) and for F sufficiently smooth, we have

ko(t)MH2(~) C _{k~co(t)ML2(~)} . (1.22)

Proposition 1.2.2 ([43], Green's formula) For all u 2 H²(~), v 2 H¹(1) we have

f-

ZLuvdx = ~

fVuVvdx - @~

@u vda, (1.23) @~

@u

where is a normal derivation of u at F.

@~

1.2.3 The LP (0, T, X) spaces

Definition 1.2.5 Let X be a Banach space, denote by L^p(0, T, X) the space of measurable functions

f : ]0,T[ -- X t' f(t)

such that

(

T I 0

11 f (t)111 dt)

= 11I11LP(0,T,x) < oc, for 1 < p < oo. (1.24)

If p = oo,

11I11L.0(0,T,x) = sup tE]0,T[

ess 11f(t)11x - (1.25)

Theorem 1.2.8 ([42])

The space L^p(0, T, X) is complete.

We denote by D' (0, T, X) the space of distributions in ]0, T[ which take its values in X, and let us define

D' (0,T, X) = r (D ]0,T[, X) ,

where r (0, (p) is the space of the linear continuous applications of _q to (p. Since u 2 D' (0, T, X) , we define the distribution derivation as

au

at ((p) = u (4)

t ' Vcp 2 D (]0,T[) , (1.26)

d

and since u 2 L^p (0, T, X) , we have

u((p) =

T I 0

u(t)(p(t)dt, Vcp 2 D (]0,T[) . (1.27)

We will introduce some basic results on the L^p(0, T, X) space. These results, will be very useful in the other chapters of this thesis.

Lemma 1.2.8 ([26] Lemma 1.2 )

0 f

Let f 2 L^p(0, T, X) and _@t2 L^p(0, T, X), (1 < p < oo) , then, the function f is continuous from [0, 7¹] to X.i.e. f 2 C¹(0, T, X).

Lemma 1.2.9 ([26])

Let çü = ]0, T[x an open bounded domain in RxRⁿ, and let _g,1, g are two functions in L (]0, T[, L^q(c)), 1 < q < 1 such that

Mg,LMLq(0,T,Lq(~)) ~ C, V,LL 2 N (1.28)

and

g,1 --p g in çü,

then

g,1 - g in L (ço).

Theorem 1.2.9 ([9], Proposition 1.4.17) L^1(0, T, X) equipped with the norm k.kLp(0,T,x), 1 < p < 1 is a Banach space. Proposition 1.2.3 ([14]) 1 Let X be a reflexive Banach space, X^' it's dual, and 1 p < oc, 1 q < oc, p dual of L^P(0, T, X) is identify algebraically and topologically with L^q(0, T, X^').

+

1 q

= 1. Then the

Proposition 1.2.4 ([9])

Let X, Y be to Banach space, X c Y with continuous embedding, then we have L^P(0, T, X) c L^P(0, T, Y ) with continuous embedding.

The following compactness criterion will be useful for nonlinear evolution problems, especially in the limit of the non linear terms.

Proposition 1.2.5 ([26]).

Let B0, B, B1 be Banach spaces with B0 C B C B1, assume that the embedding B0 ,! B is compact and B ,! B1 are continuous. Let 1 < p < oc, 1 < q < oc, assume further that B0 and B1 are reflexive.

Define

W ~ {u 2 L^° (0, T, B0) : u^' 2 L^q (0, T, B1)}. (1.29)

Then, the embedding W ,! L^p (0, T, B) is compact.