Our approach is to consider an approximate problem first. Given numbers
and given a function satisfying (1.20), consider the system
(2.2)
in
(2.3)
(2.4)
equipped with the following boundary and initial conditions:
for any , ,
for any and , ,
(2.5)
for any , ,
for any , ,
for any , ,
for any , ,
for any , ,
for any , ,
for any and, finally,
(2.6)
for any . Note that in this section h is given and , p and η are to be found (h and η are therefore not related by (1.4)).
Assume for a moment that is in fact a smooth solution of problem (2.2)-(2.6) above. Then
(2.7)
for , , and
(2.8)
for , solve the following problem:
(2.9)
in , where ,
(2.10)
(2.11)
in ,
(2.12)
for any , with the boundary and initial conditions listed below:
(2.13)
for any , , ,
(2.14)
for any , , ,
(2.15)
for any , , ,
for any , ,
(2.16)
for any , and, finally,
(2.17)
for any and for any .
We continue this section by making precise the meaning of the solution of problem (2.9)-(2.17). To this end, we first define
(2.18)
where
(2.19)
and
Throughout this and the next sections, we assume
(2.20)
The function spaces we use are rather familiar, and we adopt the notation of [15].
Definition 2.1 We call a weak solution of the initial boundary value problem (2.9)-(2.17) if the following two properties are fulfilled:
-
(1)
, , , and .
-
(2)
satisfies the system of differential equations, that is,
(2.21)
for every , , .
Remark 2.1 Note that
(2.22)
for every test function with .
At the end of this section, we discuss the existence and uniqueness of weak solutions to problem (2.9)-(2.17) for given h, , and satisfying (1.20) and (2.20).
Theorem 2.1 Assume (2.1), (2.20), and let satisfying (1.7) and (1.8) be given. Hereafter denotes h defined by (1.4) with the given , and we assume that (1.20) holds. Then there exists a unique solution
of problem (2.9)-(2.17) in the sense of Definition 2.1 such that
(2.23)
where the function ℓ in (2.23) is defined by (2.10) for , i.e.
Proof of Theorem 2.1 By analogy with the approach taken in [[5], the proof of Theorem 6.3], we can construct our weak solution by the implicit time discretisation method. The paper [5] presents all the technicalities of the proof, therefore we omit the proof here. □
We conclude this section by some interpolation inequalities which are needed especially for estimating the nonlinear item. For this purpose, we complete the notations of function spaces (2.18) and (2.19) by appropriate weighted spaces. Let
(2.24)
and
(2.25)
Proposition 2.1 (i) Let φ be any function in such that on or on . Then, for any and for any number θ with
there exists a constant
such that
(2.26)
Moreover, if such that on or on for almost all , then the following holds for any :
(2.27)
-
(ii)
Let now
and
fulfil the condition
Then there exists a constant
such that
(2.28)
If, moreover, such that on and on for almost all , then the following holds for any :
(2.29)
Proof (i) The form of Nirenberg-Gagliardo inequality (2.26) can be found, e.g. in [[16], Theorem 2.2]. Then (2.27) follows from (2.26) for by integration over .
(ii) To prove (2.28), we define and transform the integrals on the domain D into integrals on the cylinder . Then the weighted interpolation inequality (2.28) is equivalent to an unweighted interpolation inequality in for given on , which yields the assertion (cf. also [[17], Theorem 19.9] for ). Inequality (2.29) again follows by integration of (2.28) over with . □