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.
Keywords: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
We consider the initial value problem (IVP) for the following Stokes equation with small parameter:
where , A is a linear operator in a Banach space E and are a small positive parameters. Here , are E-valued unknown solutions, is a given function and is an initial date. This problem is characterized by the presence of an abstract operator A and small terms which correspond to the inverse of Reynolds number Re very large. We prove that problem (1.1)-(1.2) has a unique strong maximal regular solution u on a time interval independent of . For , , , problem (1.1)-(1.2) is reduced to the Stokes problem
where ℂ is the set of complex numbers and b is a positive constant.
Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems has been extensively studied, e.g., in [1-10]. There is an extensive literature on the solvability of the IVPs for the Stokes equation (see, e.g., [1,3,10] and further papers cited there). Solonnikov  proved that for every , , the time-dependent Stokes problem
has a unique solution so that
Then Giga and Sohr  improved the result of Solonnikov for spaces with different exponents in space and time, i.e., they proved that for there is a unique solution of problem (1.5) so that
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., ).
In this paper, first we consider the following differential operator equation (DOE) with small parameters:
where A is a linear operator in a Banach space E, are positive and λ is a complex parameter.
We show that for , , problem (1.7) has a unique solution u belonging to and the uniform coercive estimate holds
where is independent of , λ and f.
We consider, then, the stationary abstract Stokes problem with small parameters
where is data and is a solution. By applying the projection transformation P, equation (1.8) can be reduced to the following problem:
Let denote the operator generated by problem (1.9), i.e., is a Stokes operator in solenoidal space defined by
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.
2 Notations and background
Let E be a Banach space and let denote the space of strongly measurable E-valued functions that are defined on the measurable subset with the norm
The Banach space E is called a UMD-space if the Hilbert operator
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].
A linear operator A is said to be ψ-positive in a Banach space E with bound if the domain is dense on E and for any , , where I is the identity operator in E. It is known [, §1.15.1] that there exist the fractional powers of the positive operator A. Let denote the space with the norm
ℕ denotes the set of natural numbers. A set is called R-bounded (see, e.g., ) if there is a positive constant C such that for all and , ,
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 .
A set depending of parameter is called uniform R-bounded with respect to h if there is a constant C, independent of such that for all and , ,
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.
Let and E be two Banach spaces and let be continuously and densely embedded into E. Let Ω be a measurable set in and m be a positive integer. denotes the class of all functions that have the generalized derivatives with the norm
For , , , the space is denoted by . For the space is denoted by .
Let , denote an E-valued Liouville space of order s, i.e.,
where F and denote the Fourier and inverse Fourier transforms, respectively.
Let be a Liouville-Lions type space, i.e.,
For we define the parameter-dependent norm in such that
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 .
3 Boundary value problems for abstract elliptic equations
In this section, we derive the maximal regularity properties of problem (1.7).
Theorem 3.1LetEbe aUMDspace and letAbe anR-positive operator inEfor . Then problem (1.7) has a unique solution for and . Moreover, the following uniform coercive estimate holds:
with independent of , λandf.
Consider the differential operator in generated by problem (3.1), i.e.,
Let . From Theorem 3.1 we obtain the following.
Result 3.1 For , there is a resolvent satisfying the uniform estimate
Next we show the smoothness of problem (3.1). The main result is the following.
Theorem 3.2Assume thatEis aUMDspace, Ais anR-positive operator inE, andmis a positive integer.
Then, for all , , problem (3.1) has a unique solutionuthat belongs to and the following uniform coercive estimate holds:
with independent of , λandf.
Proof A solution of equation (1.7) is given by
where and . It follows from the expression above that
It is sufficient to show that the operator-functions
are uniform Fourier multipliers in . Actually, due to positivity of A, we have
It is clear to observe that
Due to R-positivity of the operator A, the sets
are R-bounded. So, in view of Kahane’s contraction principle and from the product properties of the collection of R-bounded operators (see, e.g., , Lemma 3.5, Proposition 3.4), we obtain
Namely, the R-bounds of sets are independent of ε and λ. Next, let us consider . It is clear to see that
By using the well-known inequality
for and , we get the uniform estimate
From (3.4) and (3.5) we have the uniform estimate
Due to R-positivity of the operator A, the set
is R-bounded. By using this fact, in view of (3.4) and Kahane’s contraction principle, we obtain the R-boundedness of the set . In a similar way, we obtain the uniform estimates
By the aid of the estimates above, due to R-positivity of the operator A, in view of estimate (3.4), by virtue of Kahane’s contraction principle, from the additional and product properties of the collection of R-bounded operators, for , and independent symmetric -valued random variables , , , we obtain the uniform estimate
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. □
4 The stationary Stokes system with small parameters
In this section we derive the maximal regularity properties of the stationary abstract Stokes problem (1.8).
Let , denote the E-valued Liouville space of order s such that . It is known that if E is a UMD space, then for a positive integer m (see, e.g., [, §15]). For let denote the space of an E-valued system of functions with the norm
denotes the E-valued solenoidal space, i.e., closure of in , where
Let E be a Banach space. Consider the space
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 .
Theorem 4.1LetEbe aUMDspace and . Then has a Helmholtz decomposition, i.e., there exists a bounded linear projection operator from onto with the null space . In particular, all has the unique decomposition with , so that
for any open ball . Moreover, , .
To prove Theorem 4.1, we need some lemmas. Consider the problem
Lemma 4.1LetEbe aUMDspace, letAbe anR-positive operator inE, and . Then, for , , problem (4.2) has a unique solution and the following uniform coercive estimate holds:
Proof By using the Fourier transform, we see that estimate (4.3) is equivalent to the following estimate:
To prove (4.4) it is sufficient to show that the operator functions
are multipliers in uniformly in λ and ε. This fact is derived as the step in the proof of Theorem 3.2. □
Now consider the system of equations
where and is a solution of (4.5).
We define in the following parameter-dependent norm:
Lemma 4.2LetEbe aUMDspace, letAbe anR-positive operator inE, and . Then, for , , problem (4.5) has a unique solution and the following coercive uniform estimate holds:
Proof Problem (4.5) can be expressed as the following system:
By Lemma 4.1 we obtain that for , , equation (4.7) has a unique solution and the following uniform coercive estimate holds:
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 .
Consider the problem
From Lemma 4.2 we obtain the following results.
Result 4.1 Let E be a UMD space, let A be an R-positive operator in E and . Then, for , , problem (4.8) has a unique solution and the following coercive uniform estimate holds:
Consider the operator defined by
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 4.4LetEbe aUMDspace, letAbe anR-positive operator inEand . 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 4.1 we have
If , then by Lemma 4.2 we get that , i.e., . □
Let denote the dual space of E.
Lemma 4.5Assume thatEis aUMDspace 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.
Lemma 4.6AssumeEis aUMDspace and . Then
Lemma 4.7AssumeEis aUMDspace and . Then
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. □
Theorem 4.2LetEbe aUMDspace, letAbe anR-positive operator inE, . Then, for , , , problem (1.8) has a unique solution and the uniform coercive estimate holds
with independent of , λandf.
Proof By applying the operator to equation (1.8), we get problem (1.9). It is clear to see that
where is the Stokes operator and is an operator in generated by problem (4.5) for , i.e.,
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.
Proposition 4.1The following estimate holds
uniformly in for and .
Proof From Theorem 4.2 we obtain that the operator is uniformly positive in , i.e., the following estimate holds
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. □
5 Well-posedness of the instationary parameter-dependent Stokes problem
In this section, we show the uniform well-posedness of problem (1.1)-(1.2).
Theorem 5.1Assume thatEis aUMDspace, Ais anR-positive operator inEand . Then, for and , , there is a unique solution of problem (1.1)-(1.2) and the following uniform estimate holds:
with independent offandε.
Proof Problem (1.1)-(1.2) can be expressed as the following parabolic problem:
If we put , then by Proposition 4.1 operator is uniformly positive and generates a bounded holomorphic semigroup in uniformly in ε. Moreover, by using [, Theorem 3.1] we get that the operator is R-positive in E uniformly in ε. Since E is a UMD space, in a similar way to that in [, Theorem 4.2] we obtain that for and , there is a unique solution of problem (5.2) such that the following uniform estimate holds:
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.
6 Separability properties of anisotropic Stokes equations
Let be an open connected set with compact -boundary ∂Ω. Consider the BVP for the following anisotropic Stokes equations with parameters:
where and are complex-valued functions,
are unknown solutions and
is a given function;
, are positive and λ is a complex parameter.
If , , will denote the space of all p-summable scalar-valued functions with mixed norm (see, e.g., [, §1], i.e., the space of all measurable functions f defined on G, for which
Analogously, denotes the anisotropic Sobolev space with a corresponding mixed norm [, §10]. Let . From Theorem 4.2 we obtain the following result.
Theorem 6.1Let the following conditions be satisfied:
(1) for each and for each with and ;
(2) for eachj, βand , , for , , where is normal to∂Ω;
(3) for , , , , let ;
(4) for each , the local BVP in local coordinates corresponding to
has a unique solution for all , and for with . Then for , with sufficiently large problem (6.1)-(6.3) has a unique solutionubelonging to and the uniform coercive estimate holds
Proof Let . By virtue of [, Theorem 3.6], E is a UMD space. Consider the operator A in defined by
Problem (6.1)-(6.3) can be rewritten in the form (1.8), where , are vector-functions with values in . By virtue of [, Theorem 8.2] the problem
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. □
7 Infinite system of Stokes equations with small parameters
Consider the IVB for the following infinite system of instationary Stokes equations with small parameters:
where are positive parameters. Here , are unknown solutions, and is a given function. Let
is a class of functions
with the norm
Let . From Theorem 5.1 we obtain the following.
Theorem 7.1Let and . Then for , there is a unique solution of problem (7.1) belonging to and the following uniform coercive estimate holds:
with independent offandε.
Proof Really, let , A and be an infinite matrix, defined by
It is easy to see that
For all , , , and independent symmetric -valued random variables , , , we have
Since for , from the above we get
i.e., the operator A is R-positive in . Therefore, all the conditions of Theorem 5.1 hold and we obtain the assertion. □
The author declares that they have no competing interests.
All results belong to VS.
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