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.
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:
and are linear operators in a Banach space E, , are complex numbers, and are complex-valued functions. Here represents a given, a denotes the initial data and
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
has a unique solution so that
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:
Let denote the operator generated by problem (1.9)-(1.10), i.e., let be a Stokes operator in the E-valued solenoidal space defined by
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
Let E be a Banach space. denotes the space of strongly measurable E-valued functions that are defined on the measurable subset with the norm
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 , ,
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 .
A set is called uniform R-bounded in h, if there is a constant C independent on such that
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.
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
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].
Let denote a Liouville-Lions type space, i.e.,
denote the E-valued solenoidal space, i.e., the closure of in , where
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;
(2) andmis a positive integer such that , , , , is a fixed positive number;
(3) is a region such that there exists a bounded linear extension operator from to .
Then the embedding is continuous and for all the following uniform estimate holds:
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 .
Theorem A2Suppose all conditions of Theorem A1are satisfied and . Moreover, let Ω be a bounded region and . Then the embedding
Theorem A3Suppose all conditions of Theorem A1satisfied and . Then the embedding
is continuous and there exists a positive constant such that for all the uniform estimate holds
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 operator generates a semigroup which is holomorphic for . Moreover, there exists a positive constantC (depending only onM, φ, m, αandp) such that for every and ,
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)
where and are linear operators in a Banach space E, are complex-valued functions, and λ is a complex parameter.
Let , , be roots of the equations
Let and and
Condition 3.1 Assume:
(1) E is a UMD space and is a uniformly R-positive operator in E for ;
(2) , , , for all , ;
(3) , , , ;
(4) , , , , .
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.
Theorem 3.1Suppose Condition 3.1 is satisfied. Then problem (3.1) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
Consider the differential operator in generated by problem (1.7)-(1.8), i.e.,
Let . From Theorem 3.1 we obtain the following.
Result 3.1 For there is a resolvent and the following uniform coercive estimate holds:
Let E be a Banach space and denote the class of E-valued system of function with norm
denote the E-valued solenoidal space and A be a positive operator in E. The spaces , will be denoted by and .
Consider the problem
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.
Result 3.2 Suppose Condition 3.1 is satisfied. Then problem (3.2) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
Consider the differential operator in generated by problem (3.2), for , i.e.,
From Result 3.2 we obtain the following uniform coercive estimate:
Consider the space
becomes a Banach space with this norm.
Theorem 3.2LetEbe an UMD space and . Then has a Helmholtz decomposition, i.e., there exists a linear bounded projection operator from onto with null space . In particular, all have a unique decomposition with , so that
Moreover, , where .
For proving Theorem 3.2 we need some lemmas.
Consider the equation
Lemma 3.1LetEbe an UMD space, AaR-positive operator inEand . Then, for , problem (3.2) has a unique solution and the following coercive estimate holds:
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:
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.
Let be a unit normal to the boundary Γ of the domain G and is a normal component of on Γ, i.e.,
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 .
Proposition 3.1There exists a unique bounded linear operator from , onto
and the following estimate holds:
Proof For consider the linear form
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
Hence, we can find for each an element so that
Therefore, from (3.9) we get
This implies the existence of an element
Thus, we have proved the existence of the operator . The uniqueness follows from Lemma 3.2. □
Proposition 3.1 implies the following.
Result 3.3 Assume the conditions of Proposition 3.1 are satisfied. Then
and is a closed subspace of .
Let and . Consider the following problem:
Lemma 3.3LetEbe an UMD space, AaR-positive operator inEand . Then, for , problem (3.10) has a unique solution and the coercive estimate holds
Proof Consider the equation
By Lemma 3.1, problem (3.12) has a unique solution for and the following estimate holds:
Consider now the BVP
By using Theorem 3.1, Result 3.3, and Proposition 3.1 we conclude that problem (3.14) for has a unique solution and the following coercive estimate holds:
Then we conclude that problem (3.10) has a unique solution and (3.13), (3.15) imply the estimate (3.11). □
Result 3.4 For the case of we obtain from (3.9), (3.11), and (3.12) that the problem
has a unique solution for and the following estimate holds:
Result 3.5 For the case of we obtain from Lemma 3.1 and from (3.9), (3.10) that the problem
has a unique solution for and the following estimate holds:
By (3.9), . So, by (3.14) and Proposition 3.1 we get
From Results 3.4, 3.5, and estimate (3.17) we obtain
Result 3.6 The problem
has a unique solution for and the estimate holds
Consider the operator defined by
where w and υ are solutions of problems (3.18), (3.13), respectively. It is clear that we have the following.
Lemma 3.4LetEbe an UMD space and . Then is a closed subspace of .
Lemma 3.5LetEbe an UMD space and . 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 3.6 we have
If then by Result 3.3 we get . Moreover, by the estimate (3.20) we obtain , i.e., . □
Lemma 3.6AssumeEis 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.
Lemma 3.7AssumeEis an UMD space and . Then
Lemma 3.8AssumeEis an UMD space and . Then
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. □
Theorem 3.3Let Condition 3.1 hold. Then problem (1.7)-(1.8) has a unique solution for , , , and the following coercive uniform estimate holds:
Proof By virtue of Result 3.2, we find that problem (3.2) has a unique solution for and for sufficiently large . Moreover, the following coercive uniform estimate holds:
By applying the operator to problem (1.7)-(1.8) we get the Stokes problem (1.9)-(1.10). It is clear that
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.
Proposition 3.2The following estimate holds:
for and .
Proof From the estimate (3.3) we see that the operator O is positive in , i.e., for , the 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).
Theorem 4.1Then, for , , and , , there is a unique solution of problem (1.1)-(1.2) and the following estimate holds:
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
where represents a given and
are unknown functions;
, are complex numbers, , are complex-valued functions, .
Let , . Now will denote the space of all p-summable scalar-valued functions with mixed norm i.e., the space of all measurable functions f defined on , for which
Analogously, denotes the Sobolev space with corresponding mixed norm.
denotes the class of vector function
and , where is the anisotropic Sobolev space with mixed norm.
From Theorem 3.3 we obtain the following.
Theorem 5.1Let the following conditions be satisfied:
(1) Ω is a domain in with sufficiently smooth boundary∂Ω, , for each , , for each with , and , ;
(2) for each , ;
(3) for , , , , let
(4) for each the local BVPs in local coordinates corresponding to
has a unique solution for all and for with ;
(5) , , , for all , , , .
Then, problem (5.1)-(5.3) has a unique solution for , , and the following coercive uniform estimate holds:
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
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.
Consider now the instationary Stokes problem
where , are data and solution vector-functions, respectively. □
From Theorem 4.1 and Theorem 5.1 we obtain the following.
Theorem 5.2Assume all conditions of Theorem 5.1 are satisfied . Then, for , , and , there is a unique solution of problem (5.6)-(5.9) and the following estimate holds:
The author declares that they have no competing interests.
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