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
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:
We consider, then, the stationary abstract Stokes problem with small parameters
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
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 , ,
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
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).
Next we show the smoothness of problem (3.1). The main result is the following.
Proof A solution of equation (1.7) is given by
It is sufficient to show that the operator-functions
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
By using the well-known inequality
From (3.4) and (3.5) we have the uniform estimate
Due to R-positivity of the operator A, the set
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
Let E be a Banach space. Consider the space
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. Thenhas a Helmholtz decomposition, i.e., there exists a bounded linear projection operatorfromontowith the null space. In particular, allhas the unique decompositionwith, so that
To prove Theorem 4.1, we need some lemmas. Consider the problem
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
Now consider the system of equations
Proof Problem (4.5) can be expressed as the following system:
By reasoning as in [, Lemma 2], we get the following lemma.
Consider the problem
From Lemma 4.2 we obtain the following results.
where φ is a solution of problem (4.8).
Proof The linearity of the operator P is clear by construction. Moreover, by Result 4.1 we have
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.
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. □
Then by Lemma 4.2 we obtain the assertion.
In a similar way to that in  we show the following.
Proposition 4.1The 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).
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.
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
are unknown solutions and
is a given function;
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:
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
7 Infinite system of Stokes equations with small parameters
Consider the IVB for the following infinite system of instationary Stokes equations with small parameters:
with the norm
It is easy to see that
The author declares that they have no competing interests.
All results belong to VS.
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