In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem (1.7)-(1.8).
First at all, we consider the BVP for the variable coefficient differential operator equation (DOE)
(3.1)
where and are linear operators in a Banach space E, are complex-valued functions, and λ is a complex parameter.
Maximal regularity properties for DOEs studied, e.g., in [1, 11, 12, 14, 16–24]. Nonlocal BVPs for PDE were studied in [25].
Let , , be roots of the equations
Let and and
Condition 3.1 Assume:
-
(1)
E is a UMD space and is a uniformly R-positive operator in E for ;
-
(2)
, , , for all , ;
-
(3)
, , , ;
-
(4)
, , , , .
Remark 3.1 Let , where are real-valued positive functions and. Then Condition 3.1 is satisfied.
Remark 3.2 The conditions , are given due to the nonlocality of the boundary conditions. For local boundary conditions these assumptions are not required.
From [[19], Theorem 4.1] we have the following.
Theorem 3.1 Suppose Condition 3.1 is satisfied. Then problem (3.1) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
Consider the differential operator in generated by problem (1.7)-(1.8), i.e.,
Let . From Theorem 3.1 we obtain the following.
Result 3.1 For there is a resolvent and the following uniform coercive estimate holds:
Let E be a Banach space and denote the class of E-valued system of function with norm
denote the E-valued solenoidal space and A be a positive operator in E. The spaces , will be denoted by and .
Consider the problem
(3.2)
where , and are linear operators in a Banach space E, are complex-valued functions, and λ is a complex parameter. From Theorem 3.1 we obtain the following result.
Result 3.2 Suppose Condition 3.1 is satisfied. Then problem (3.2) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
Consider the differential operator in generated by problem (3.2), for , i.e.,
From Result 3.2 we obtain the following uniform coercive estimate:
(3.3)
Consider the space
becomes a Banach space with this norm.
It is known that (see, e.g., [3, 5]) the vector field has a Helmholtz decomposition. In the following theorem we generalize this result for the E-valued function space .
Theorem 3.2 Let E be an UMD space and . Then has a Helmholtz decomposition, i.e., there exists a linear bounded projection operator from onto with null space . In particular, all have a unique decomposition with , so that
(3.4)
Moreover, , where .
For proving Theorem 3.2 we need some lemmas.
Consider the equation
(3.5)
Lemma 3.1 Let E be an UMD space, A a R-positive operator in E and . Then, for , problem (3.2) has a unique solution and the following coercive estimate holds:
(3.6)
Proof Consider the problem
(3.7)
where is an extension of the function on . Then, by using the Fourier inversion formula, operator-valued multiplier theorems in spaces, and by reasoning as in [[17], Theorem 3.2] we see that problem (3.7) has a unique solution for and the following coercive estimate holds:
This fact implies that the function u which is a restriction of on G is a solution of problem (3.5). The estimate (3.6) is obtained from the above estimate.
Let be a unit normal to the boundary Γ of the domain G and is a normal component of on Γ, i.e.,
Here and hereafter will denoted the conjugate of E, and (resp. ) denotes the duality pairing of functions on G (resp. Γ). □
By reasoning as in [[3], Lemma 2] we get the following.
Lemma 3.2 is dense in .
Proposition 3.1 There exists a unique bounded linear operator from , onto
such that
and the following estimate holds:
(3.8)
where
Proof For consider the linear form
(3.9)
By virtue of the trace theorem in , the interpolation of intersection and dual spaces (see, e.g., [[13], §1.8.2, 1.12.1, 1.11.2]) and by a localization argument we obtain the result that the operator is a bounded linear and surjective from onto
Hence, we can find for each an element so that
Therefore, from (3.9) we get
This implies the existence of an element
such that
and
Thus, we have proved the existence of the operator . The uniqueness follows from Lemma 3.2. □
Proposition 3.1 implies the following.
Result 3.3 Assume the conditions of Proposition 3.1 are satisfied. Then
and is a closed subspace of .
Let and . Consider the following problem:
(3.10)
Lemma 3.3 Let E be an UMD space, A a R-positive operator in E and . Then, for , problem (3.10) has a unique solution and the coercive estimate holds
(3.11)
Proof Consider the equation
(3.12)
By Lemma 3.1, problem (3.12) has a unique solution for and the following estimate holds:
(3.13)
Consider now the BVP
(3.14)
By using Theorem 3.1, Result 3.3, and Proposition 3.1 we conclude that problem (3.14) for has a unique solution and the following coercive estimate holds:
(3.15)
Then we conclude that problem (3.10) has a unique solution and (3.13), (3.15) imply the estimate (3.11). □
Result 3.4 For the case of we obtain from (3.9), (3.11), and (3.12) that the problem
(3.16)
has a unique solution for and the following estimate holds:
(3.17)
Result 3.5 For the case of we obtain from Lemma 3.1 and from (3.9), (3.10) that the problem
(3.18)
has a unique solution for and the following estimate holds:
(3.19)
By (3.9), . So, by (3.14) and Proposition 3.1 we get
(3.20)
From Results 3.4, 3.5, and estimate (3.17) we obtain
Result 3.6 The problem
(3.21)
has a unique solution for and the estimate holds
(3.22)
Consider the operator defined by
where w and υ are solutions of problems (3.18), (3.13), respectively. It is clear that we have the following.
Lemma 3.4 Let E be an UMD space and . Then is a closed subspace of .
Lemma 3.5 Let E be an UMD space and . Then the operator is a bounded linear operator in and if .
Proof The linearity of the operator P is clear by construction. Moreover, by Result 3.6 we have
(3.23)
If then by Result 3.3 we get . Moreover, by the estimate (3.20) we obtain , i.e., . □
Lemma 3.6 Assume E is an UMD space and . Then the conjugate of is defined as , and is bounded linear in .
Proof It is known (see, e.g., [13, 18]) that the dual space of is . Since is dense in we have only to show for any . But this is deriving by reasoning as in [[3], Lemma 5]. Moreover, by Lemma 3.5 the dual operator is a bounded linear in . □
Let
From Lemmas 3.5, 3.6 we obtain the following.
Result 3.7 Assume E is an UMD space and . Then any element uniquely can be expressed as a sum of elements of and .
In a similar way as Lemmas 6, 7 of [3] we obtain, respectively, the following.
Lemma 3.7 Assume E is an UMD space and . Then
Lemma 3.8 Assume E is an UMD space and . Then
Now we are ready to prove Theorem 3.2.
Proof of Theorem 3.2 From Lemmas 3.7, 3.8 we get . Then, by construction of , we have . By Lemmas 3.3, 3.5, we obtain the estimate (3.4). Moreover, by Lemma 3.4, is a close subspace of . Then it is known that the dual space of the quotient space is . In view of first assertion we have and by Lemma 3.8 we obtain the second assertion. □
Theorem 3.3 Let Condition 3.1 hold. Then problem (1.7)-(1.8) has a unique solution for , , , and the following coercive uniform estimate holds:
(3.24)
Proof By virtue of Result 3.2, we find that problem (3.2) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
By applying the operator to problem (1.7)-(1.8) we get the Stokes problem (1.9)-(1.10). It is clear that
where is the Stokes operator and B is a operator generated by problem (3.2) for . Then by Theorem 3.2 we obtain the assertion. □
Result 3.8 From Result 3.2 we find that is a positive operator in and −O generate a bounded holomorphic semigroup for .
In a similar way as in [6] we show the following.
Proposition 3.2 The following estimate holds:
for and .
Proof From the estimate (3.3) we see that the operator O is positive in , i.e., for , the following estimate holds:
where the constant M is independent of λ. Then, by using the Danford integral and operator calculus (see, e.g., in [12]), we obtain the assertion. □