- Research Article
- Open Access
Parameter-dependent Stokes problems in vector-valued spaces and applications
© Shakhmurov; licensee Springer 2013
- Received: 8 March 2013
- Accepted: 28 June 2013
- Published: 23 July 2013
The stationary and instationary Stokes equations with operator coefficients in abstract function spaces are studied. The problems are considered in the whole space, and equations include small parameters. The uniform separability of these problems is established.
MSC:35Q30, 76D05, 34G10, 35J25.
- Stokes systems
- Navier-Stokes equations
- differential equations with small parameters
- semigroups of operators
- boundary value problems
- differential-operator equations
- maximal regularity
Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA, Texas A&M University-Kingsville-2012
where ℂ is the set of complex numbers and b is a positive constant.
Moreover, the estimate obtained was global in time, i.e., the constant is independent of T and f. To derive global estimates (1.6), Giga and Sohr used the abstract parabolic semigroup theory in UMD-type Banach spaces. Estimate (1.6) allows to study the existence of a solution and regularity properties of the corresponding Navier-Stokes problem (see, e.g., ).
where A is a linear operator in a Banach space E, are positive and λ is a complex parameter.
where is independent of , λ and f.
We prove that the operator is uniformly positive and also is a generator of an analytic semigroup in . Finally, the instationary Stokes problem (1.1)-(1.3) is considered and the well-posedness of this problem is derived. In application we show the separability properties of the anisotropic stationary Stokes operator in mixed spaces and maximal regularity properties of infinity system of instationary Stokes equations in spaces.
is bounded in , (see, e.g., ). UMD spaces include, e.g., , spaces and Lorentz spaces , .
Let and be two Banach spaces. denotes the space of bounded linear operators from into endowed with the usual uniform operator topology. For , it is denoted by . Now , , , denotes interpolation spaces defined by the K method [, §1.3.1].
where is a sequence of independent symmetric -valued random variables on Ω. The smallest C, for which the estimate above holds, is called an R-bound of the collection G and denoted by .
It implies 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 of t, is dense in E and for all , , where M does not depend of t and λ.
For , , , the space is denoted by . For the space is denoted by .
where F and denote the Fourier and inverse Fourier transforms, respectively.
Sometimes we use one and the same symbol C without distinction 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 .
In this section, we derive the maximal regularity properties of problem (1.7).
with independent of , λ and f.
Let . From Theorem 3.1 we obtain the following.
Next we show the smoothness of problem (3.1). The main result is the following.
Theorem 3.2 Assume that E is a UMD space, A is an R-positive operator in E, and m is a positive integer.
with independent of , λ and f.
The same estimates are obtained for in a similar way. Hence, by virtue of [, Theorem 3.4] it follows that and are the uniform collection of multipliers in . Then, by using equality (3.3), we obtain the assertion. □
In this section we derive the maximal regularity properties of the stationary abstract Stokes problem (1.8).
becomes a Banach space with this norm.
It is known that (see, e.g., Fujiwara and Morimoto ) the vector field has a Helmholtz decomposition. In the following theorem we generalize this result for an E-valued function space .
for any open ball . Moreover, , .
are multipliers in uniformly in λ and ε. This fact is derived as the step in the proof of Theorem 3.2. □
where and is a solution of (4.5).
Hence, we get that is a unique solution of problem (4.5) and (4.3) implies (4.6). □
By reasoning as in [, Lemma 2], we get the following lemma.
Lemma 4.3 is dense in .
From Lemma 4.2 we obtain the following results.
where φ is a solution of problem (4.8).
Result 4.2 Let E be a UMD space, let A be an R-positive operator in E and . Then is a closed subspace of .
Lemma 4.4 Let E be a UMD space, let A be an R-positive operator in E and . Then the operator is a bounded linear operator in and if .
If , then by Lemma 4.2 we get that , i.e., . □
Let denote the dual space of E.
Lemma 4.5 Assume that E is a UMD space and . Then the conjugate of is defined as , and is bounded linear in .
Proof It is known (see, e.g., [13, 20]) that the dual space of is . Since is dense in , we only have to show for any . But this is derived by reasoning as in [, Lemma 5]. Moreover, by Lemma 4.4, the dual operator is bounded linear in .
From Lemmas 4.4, 4.5 we obtain the following result.
Result 4.3 Assume that E is a UMD space, A is an R-positive operator in E and . Then any element uniquely can be expressed as the sum of elements of and .
In a similar way to Lemmas 6, 7 of  we obtain, respectively, the following lemmas.
Now we are ready to prove Theorem 4.1.
Proof of Theorem 4.1 From Lemmas 4.6, 4.7 we get that . Then, by construction of , we have . By Lemmas 4.2, 4.4, we obtain estimate (4.1). Moreover, by Result 4.2, is a close subspace of . It is known that the dual space of the quotient space is . By the first assertion we have , and by Lemma 4.7 we obtain the second assertion. □
with independent of , λ and f.
Then by Lemma 4.2 we obtain the assertion.
Result 4.4 From Theorem 4.2 we get that is a positive operator in and it also generates a bounded holomorphic semigroup for .
In a similar way to that in  we show the following.
uniformly in for and .
for , , where the constant M is independent of λ and ε. Hence, by using Danford integral and operator calculus (see, e.g., in ) we obtain the assertion. □
In this section, we show the uniform well-posedness of problem (1.1)-(1.2).
with independent of f and ε.
From estimates (4.10) and (5.3) we obtain the assertion.
Result 5.1 It should be noted that if , then we obtain maximal regularity properties of an abstract Stokes problem without any parameters in principal part.
Remark 5.2 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.8) and (1.1) 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 4.2 and Theorem 5.1, we can obtain the maximal regularity properties of a different class of stationary and instationary Stokes problems which occur in numerous physics and engineering problems.
, are positive and λ is a complex parameter.
Analogously, denotes the anisotropic Sobolev space with a corresponding mixed norm [, §10]. Let . From Theorem 4.2 we obtain the following result.
for each and for each with and ;
for each j, β and , , for , , where is normal to ∂ Ω;
for , , , , let ;
- (4)for each , the local BVP in local coordinates corresponding to
has a unique solution for and for , . Moreover, the operator A is R-positive in , i.e., all the conditions of Theorem 4.2 hold. So, we obtain the assertion. □
Let . From Theorem 5.1 we obtain the following.
with independent of f and ε.
i.e., the operator A is R-positive in . Therefore, all the conditions of Theorem 5.1 hold and we obtain the assertion. □
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