221
+ loT h p(x)lux(x, t)IP-2ux(X, t)Vx(X, t)dxdt
== loT h{Q(X,t)Z(X,t) - Re(x,t,U)}v(x,t)dxdt + h uo(x)v(x,O)dx
(12)
for some function z(x, t) which satisfies that
z(x,t) E P(u(x,t)) a.e. x E I and t E (O,T).
(13)
For several purposes it will be useful to take the solution u as a test function. So, for t
fixed, the integrals
must be finite. Then a natural "energy space" associated to (P) is the one defined by
v == {w E L2(1): WX E P(I: pH,
where P(I : p) is the weighted-Lebesgue space
P(I: p) == {v:1I v IILP(I:p)== [hp(x)lv(xWdxj~ < oo}.
It is easy to see that V is a separable and reflexive Banach space with the norm
II u IIv==1I u lIu(l) + I I U X IILP(I:p) .
Any weak solution must satisfy u(·, t) E V for a.e. t E (0, T). It is not difficult to see
that in that case lux(-' t)IP-2ux (-, t) E p' (I : p), with p' == p/(p -1). We also remark that
because of the physical modelling of the problem we shall restrict our study to the class
of bounded functions.
Definition 1 . By a bounded weak solution of problem (P) we mean a function u E
C([O,Tj: L2(1)) n Loo(l x (O,T)) such that u E P(O,T: V), Re(-,·,u) E £1(1 x (O,T))
and there exist z E Loo(l x (0, T)) satisfying (13) and the identity (12) holds for any
v E P(O, T : V) n Loo(l x (0, T)) such that Vt E p' (0, T : V').
The main purpose of this section is to prove the following result
Theorem 1 For any Uo E L 00 (I) there exist at least one bounded weak solution u of (P).
The proof of the above theorem can be carried out by means of different methods. Here
we shall present two different type of techniques: (i) a compactness abstract method, and
(ii) a regularization method.
2.1. Existence via a compactness abstract method.
+ loT h p(x)lux(x, t)IP-2ux(X, t)Vx(X, t)dxdt
== loT h{Q(X,t)Z(X,t) - Re(x,t,U)}v(x,t)dxdt + h uo(x)v(x,O)dx
(12)
for some function z(x, t) which satisfies that
z(x,t) E P(u(x,t)) a.e. x E I and t E (O,T).
(13)
For several purposes it will be useful to take the solution u as a test function. So, for t
fixed, the integrals
must be finite. Then a natural "energy space" associated to (P) is the one defined by
v == {w E L2(1): WX E P(I: pH,
where P(I : p) is the weighted-Lebesgue space
P(I: p) == {v:1I v IILP(I:p)== [hp(x)lv(xWdxj~ < oo}.
It is easy to see that V is a separable and reflexive Banach space with the norm
II u IIv==1I u lIu(l) + I I U X IILP(I:p) .
Any weak solution must satisfy u(·, t) E V for a.e. t E (0, T). It is not difficult to see
that in that case lux(-' t)IP-2ux (-, t) E p' (I : p), with p' == p/(p -1). We also remark that
because of the physical modelling of the problem we shall restrict our study to the class
of bounded functions.
Definition 1 . By a bounded weak solution of problem (P) we mean a function u E
C([O,Tj: L2(1)) n Loo(l x (O,T)) such that u E P(O,T: V), Re(-,·,u) E £1(1 x (O,T))
and there exist z E Loo(l x (0, T)) satisfying (13) and the identity (12) holds for any
v E P(O, T : V) n Loo(l x (0, T)) such that Vt E p' (0, T : V').
The main purpose of this section is to prove the following result
Theorem 1 For any Uo E L 00 (I) there exist at least one bounded weak solution u of (P).
The proof of the above theorem can be carried out by means of different methods. Here
we shall present two different type of techniques: (i) a compactness abstract method, and
(ii) a regularization method.
2.1. Existence via a compactness abstract method.