- Open Access
Stokes operators with variable coefficients and applications
© Shakhmurov; licensee Springer. 2014
- Received: 30 September 2013
- Accepted: 21 April 2014
- Published: 2 May 2014
The stationary and instationary Stokes problems with variable coefficients in abstract spaces are considered. The problems contain abstract operators and nonlocal boundary conditions. The well-posedness of these problems is derived.
MSC:35Q30, 76D05, 34G10, 35J25.
- Stokes systems
- Navier-Stokes equations
- abstract differential equations with variable coefficients
- semigroups of operators
- boundary value problems
- maximal regularity
where is the class of E-valued -spaces and is a corresponding interpolation space. The estimate (1.11) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem. Finally, we give some application of this abstract Stokes problem to anisotropic Stokes equations and systems of equations. Note that the abstract Stokes problem with constant coefficients was studied in .
is bounded in , (see, e.g., ). UMD spaces include e.g. , spaces, and Lorentz spaces , .
Let and be two Banach spaces. By , , , will be denoted the interpolation spaces obtained from by the K-method [, §1.3.2].
where is a sequence of independent symmetric -valued random variables on Ω. The smallest C for which the above estimate holds is called a R-bound of the collection Φ and denoted by .
for all and , . It is implied that .
The ψ-positive operator A is said to be R-positive in a Banach space E if the set , is R-bounded.
The operator is said to be ψ-positive in E uniformly with respect to t with bound if is independent on t, is dense in E and for all , , where M does not depend on t and λ.
For , , the space will be denoted by . For the space is denoted by .
Let , , denote the E-valued Liouville space of order s such that . It is known that if E is a UMD space, then for positive integer m (see, e.g., [, §15].
Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, say α, we write .
The embedding theorems in vector-valued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use the following embedding theorems from .
E is a UMD space and A is an R-positive operator in E;
and m is a positive integer such that , , , , is a fixed positive number;
is a region such that there exists a bounded linear extension operator from to .
Remark 2.1 If is a region satisfying the strong l-horn condition (see [, §7]), , , then for there exists a bounded linear extension operator from to .
From [, Theorem 2.1] we obtain the following.
In this section, we derive the maximal regularity properties of the stationary abstract Stokes problem (1.7)-(1.8).
where and are linear operators in a Banach space E, are complex-valued functions, and λ is a complex parameter.
E is a UMD space and is a uniformly R-positive operator in E for ;
, , , for all , ;
, , , ;
, , , , .
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 [, Theorem 4.1] we have the following.
Let . From Theorem 3.1 we obtain the following.
denote the E-valued solenoidal space and A be a positive operator in E. The spaces , will be denoted by and .
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.
becomes a Banach space with this norm.
Moreover, , where .
For proving Theorem 3.2 we need some lemmas.
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.
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 [, Lemma 2] we get the following.
Lemma 3.2 is dense in .
Thus, we have proved the existence of the operator . The uniqueness follows from Lemma 3.2. □
Proposition 3.1 implies the following.
and is a closed subspace of .
Then we conclude that problem (3.10) has a unique solution and (3.13), (3.15) imply the estimate (3.11). □
From Results 3.4, 3.5, and estimate (3.17) we obtain
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 .
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 [, Lemma 5]. Moreover, by Lemma 3.5 the dual operator is a bounded linear in . □
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  we obtain, respectively, the following.
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. □
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  we show the following.
for and .
where the constant M is independent of λ. Then, by using the Danford integral and operator calculus (see, e.g., in ), we obtain the assertion. □
In this section, we will show the well-posedness of problem (1.1)-(1.2).
From the estimates (3.3) and (4.3) we obtain the assertion.
Remark 4.1 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.7)-(1.8) and (1.1)-(1.3) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, by virtue of Theorem 3.3 and Theorem 4.1 we can obtain the maximal regularity properties of different class of stationary and instationary Stokes problems, respectively, which occur in numerous physics and engineering problems.
Let us now show some application of Theorem 3.3 and Theorem 4.1.
, are complex numbers, , are complex-valued functions, .
Analogously, denotes the Sobolev space with corresponding mixed norm.
and , where is the anisotropic Sobolev space with mixed norm.
From Theorem 3.3 we obtain the following.
Ω is a domain in with sufficiently smooth boundary ∂ Ω, , for each , , for each with , and , ;
for each , ;
- (3)for , , , , let
- (4)for each the local BVPs in local coordinates corresponding to
, , , for all , , , .
has a unique solution for and arg , , and the operator A is R-positive in . Hence, all conditions of Theorem 3.2 are satisfied, i.e., we obtain the assertion.
where , are data and solution vector-functions, respectively. □
From Theorem 4.1 and Theorem 5.1 we obtain the following.
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