Keywords:Stokes systems; Navier-Stokes equations; abstract differential equations with variable coefficients; semigroups of operators; boundary value problems; maximal regularity
We consider the initial and boundary value problem (BVP) for the following Stokes type equation with variable coefficients:
represent the unknown functions. Moreover, , , and are E-valued functions. This problem is characterized by the presence of abstract operator functions and complex-valued variable coefficients in the principal part. Moreover, boundary conditions are nonlocal, generally. The existence, uniqueness, and coercive estimates of maximal regular solution of problem (1.1)-(1.2) are obtained. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing E and operators A, we can obtain maximal regularity properties for numerous class of Stokes type problems with variable coefficients. For , , problem (1.1)-(1.2) is reduced to nonlocal Stokes problem
where ℂ is the set of complex numbers. Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems were extensively studied, e.g., in [1-10]. There is an extensive literature on the solvability of the initial value problems (IVPs) for the Stokes equation (see, e.g., [1,2,10] and further papers cited there). Solonnikov  proved that for every , the instationary Stokes problem
Then Giga and Sohr  improved the result of Sollonikov for spaces with different exponents in space and time, i.e., they proved that for every 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 the global - estimates (1.6), Giga and Sohr used abstract parabolic semigroup theory in UMD spaces. The estimate (1.6) allows one to study the existence of solution and regularity properties of the corresponding Navier-Stokes problem (see, e.g., ). Consider first at all, the stationary version of problem (1.1)-(1.3), i.e., consider the abstract Stokes problem
λ is a complex number. By applying the corresponding projection transformation P, (1.7)-(1.8) can be reduced to the following problem:
We prove that is a positive operator and is a generator of an analytic semigroup in . In other words, we consider the instationary Stokes problem (1.1)-(1.3) and prove the well-posedness of this problem. We prove that there is a unique solution of problem (1.1)-(1.3) for , , and the following estimate holds:
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 .
2 Definitions and background
The Banach space E is called an UMD space if the Hilbert operator
is bounded in , (see, e.g., ). UMD spaces include e.g., spaces, and Lorentz spaces , .
A linear operator A is said to be ψ-positive in a Banach space E with bound if is dense on E and for any , , where I is the identity operator in E, and is the space of bounded linear operators in E. It is known [, §1.15.1] that there exist fractional powers of a positive operator A. Let denote the space with norm
Let and be two Banach spaces. By , , , will be denoted the interpolation spaces obtained from by the K-method [, §1.3.2].
Let ℕ denote 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 continuously and densely embedded into E. Let Ω be a domain in and m is a positive integer. denotes the space of all functions that have generalized derivatives with the norm
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 .
Theorem A1Suppose the following conditions are satisfied:
(1) Eis a UMD space andAis anR-positive operator inE;
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.
Theorem A4LetEbe a Banach space, Abe aφ-positive operator inEwith boundM, . Letmbe a positive integer, and. Then, for, an operatorgenerates a semigroupwhich is holomorphic for. Moreover, there exists a positive constantC (depending only onM, φ, m, αandp) such that for everyand,
3 The stationary Stokes system with variable coefficients
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)
Condition 3.1 Assume:
From [, Theorem 4.1] we have the following.
Consider the problem
From Result 3.2 we obtain the following uniform coercive estimate:
Consider the space
Theorem 3.2LetEbe an UMD space and. Thenhas a Helmholtz decomposition, i.e., there exists a linear bounded projection operatorfromontowith null space. In particular, allhave a unique decompositionwith, so that
For proving Theorem 3.2 we need some lemmas.
Consider the equation
Proof Consider the problem
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 [, Theorem 3.2] we see that problem (3.7) has a unique solution for and the following coercive estimate holds:
By reasoning as in [, Lemma 2] we get the following.
and the following estimate holds:
By virtue of the trace theorem in , the interpolation of intersection and dual spaces (see, e.g., [, §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
Therefore, from (3.9) we get
This implies the existence of an element
Proposition 3.1 implies the following.
Result 3.3 Assume the conditions of Proposition 3.1 are satisfied. Then
Proof Consider the equation
Consider now the BVP
From Results 3.4, 3.5, and estimate (3.17) we obtain
Result 3.6 The problem
where w and υ are solutions of problems (3.18), (3.13), respectively. It is clear that we have the following.
Proof The linearity of the operator P is clear by construction. Moreover, by Result 3.6 we have
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.
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. □
In a similar way as in  we show the following.
Proposition 3.2The following estimate holds:
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. □
4 Well-posedness of instationary Stokes problems with variable coefficients
In this section, we will show the well-posedness of problem (1.1)-(1.2).
Proof Problem (1.1)-(1.2) can be expressed as the following abstract parabolic problem:
If we put then by Proposition 3.2, operator O is positive and generates a bounded holomorphic semigroup in . Moreover, by using [, Theorem 3.1] we see that the operator O is R-positive in E. Since E is a UMD space, in a similar way as in [, Theorem 4.2] we see that for all and there is a unique solution of problem (4.2) so that the following estimate holds:
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.
Consider the stationary Stokes problem
are unknown functions;
From Theorem 3.3 we obtain the following.
Theorem 5.1Let the following conditions be satisfied:
Proof Let . By virtue of [, Theorem 4.5.2] is an UMD space. Consider the operator A which is defined by
Problem (5.1)-(5.3) can be rewritten in the form of (1.7)-(1.8) for , , where and are functions with values in . In view of [, Theorem 8.2] the problem
Consider now the instationary Stokes problem
From Theorem 4.1 and Theorem 5.1 we obtain the following.
The author declares that they have no competing interests.
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