Abstract
Initial nonlocal boundary value problems for a NavierStokes equation with a small parameter is considered. The uniform maximal regularity properties of the corresponding stationary Stokes operator, wellposedness of a nonstationary Stokes problem and the existence, uniqueness and uniformly estimates for the solution of the NavierStokes problem are established.
MSC: 35Q30, 76D05, 34G10, 35J25.
Keywords:
Stokes operators; NavierStokes equations; differential equations with small parameters; semigroups of operators; boundary value problems; differentialoperator equations; maximal regularity1 Introduction
Consider the following NavierStokes problem with a parameter:
where
, are complex numbers, ε is a small positive parameter,
represent the unknown velocity and pressure, respectively,
represents a given external force and a denotes the initial velocity. This problem is characterized by nonlocality of boundary conditions and by presence of a small term ε which corresponds to the inverse of Reynolds number Re very large for the NavierStokes equations. From both the theoretical and computational points of view, singularly perturbed problems and asymptotic behavior of the NavierStokes equations with small viscosity when the boundary is either characteristic or noncharacteristic have been well studied; see, e.g., [16]. In the present work, we established a uniform time of existence and estimates for solutions of problem (1.1)(1.3). It is clear that for , choosing the boundary conditions locally and , problem (1.1)(1.3) is reduced to the classical NavierStokes problem
Note that the existence of weak or strong solutions and regularity properties of classical NavierStokes problems were extensively studied, e.g., in [13,5,733]. There is extensive literature on the solvability of the initial value problem for the NavierStokes equation ( see, e.g., [25] for further papers cited there ). Hopf [20] proved the existence of a global weak solution of (1.4) using the FaedoGalerkin approximation and an energy inequality. Another approach to problem (1.4) is to use semigroup theory. Kato and Fujita [18,22,34] and Sobolevskii [27] transformed equation (1.4) into an evolution equation in the Hilbert space . They proved the existence of a unique global strong solution for any squaresummable initial velocity when . On the other hand, when they proved the existence of a unique local strong solution if the initial velocity has some regularity. Other contributions in this field have also assumed some regularity of the initial velocity corresponding to the Stokes problem; see, for example, Solonnikov [26] and Heywood [21]. Afterward, Giga and Sohr [13] improved this result in two directions. First, they generalized the result of Solonnikov for spaces with different exponents in space and time, and the estimate obtained was global in time. Here, first at all, we consider the nonlocal (boundary value problem) BVP for the following differential operator equation (DOE) with small parameters:
where A is a linear operator in a Banach space E, , are complex numbers, are positive and λ is a complex parameter. We show that problem (1.5) has a unique solution for and with sufficiently large , and the following coercive uniform estimate holds:
with independent of , , λ and f.
Further, we consider the nonlocal BVP for the stationary Stokes system with small parameters
where
Then we consider the initial nonlocal BVP for the following nonstationary Stokes equation with small parameters:
Problem (1.7) can be expressed as the abstract parabolic problem with a parameter
where is a stationary parameter depending on the Stokes operator in a solenoidal space defined by
We prove that the operator is positive in uniformly with respect to parameters and also is a generator of a holomorphic semigroup. Then, by using maximal regularity theorems (see, e.g., [33,35]) for abstract parabolic equations (1.8), we obtain that for every , , there is a unique solution of problem (1.8) and the following uniform estimate holds:
with independent of f and ε. Afterwords, by using the above uniform coercive estimate, we derive local uniform existence and uniform a priori estimates of a solution of problem (1.1)(1.3).
Modern analysis methods, particularly abstract harmonic analysis, the operator theory, the interpolation of Banach spaces, the theory of semigroups of linear operators, embedding and trace theorems in vectorvalued SobolevLions spaces are the main tools implemented to carry out the analysis.
2 Notations, definitions and background
Let E be a Banach space and denotes the space of strongly measurable Evalued functions that are defined on the measurable subset with the norm
The Banach space E is called a UMDspace if the Hilbert operator
is bounded in , (see, e.g., [36]). UMD spaces include, e.g., , spaces and Lorentz spaces , .
Let ℂ be the set of complex numbers and
A linear operator A is said to be ψpositive in a Banach space E with bound if is dense on E and for any , , where I is the identity operator in E, is the space of bounded linear operators in E. It is known [[30], §1.15.1] that there exist the fractional powers of a positive operator A. Let denote the space with the norm
Let ℕ denote the set of natural numbers. A set is called Rbounded (see, e.g., [36]) 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 an Rbound of the collection G and denoted by .
A set is called uniform Rbounded if there is a constant C independent of such that for all and , ,
The ψpositive operator A is said to be Rpositive in a Banach space E if the set , , is Rbounded.
The operator is said to be ψpositive in E uniformly in with bound if is independent of 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 let be continuously and densely embedded into E. Let Ω be a measurable set in and m be a positive integer. Let denote the space consisting of all functions that have the generalized derivatives , with the norm
For , , , the space will be denoted by .
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 .
3 Boundary value problems for abstract elliptic equations
In this section, we consider problem (1.5). We derive the maximal regularity properties of this problem.
It should be noted that BVPs for DOEs were studied, e.g., in [3538] and [6,26,27,3943]. For references, see [35,43]. Let and . First, we prove the following theorem.
Theorem 3.1Let the following conditions be satisfied:
(1) Eis a UMD space andAis anRpositive operator inEfor;
Then problem (1.5) has a unique solutionforandwith sufficiently large. Moreover, the following coercive uniform estimate holds:
Proof Let us consider the BVP
where , are defined by equalities (3.1)(3.2). For the investigation (3.4), we consider the following BVP for ordinary DOE:
where are boundary conditions of type (1.5) on . By virtue of [[40], Theorem 3.2], we obtain that problem (3.3) has a unique solution for , , with sufficiently large , and the following coercive uniform estimate holds:
Since , problem (3.2) can be expressed as the following problem:
where is the differential operator in generated by problem (3.3), i.e.,
By virtue of [[35], Theorem 4.5.2], provided , . Hence, by virtue of [[40], Theorem 3.2] and [[41], Theorem 3.1], the operator is uniformly Rpositive in F. Then, by applying again [[40], Theorem 3.2], we get that for , and sufficiently large , problem (3.5) has a unique solution , and the following coercive uniform estimate holds:
The estimate (3.4) implies the uniform estimate
By using (3.4) and (3.7), we obtain that problem (3.7) has a unique solution for , with sufficiently large , and the coercive uniform estimate holds
Further, by continuing this process ntimes, we obtain the assertion.
From Theorem 3.1 we obtain the following. □
Corollary 3.1Let, . For, and forwith sufficiently large, there is a unique solutionuof problem (1.5) and the following uniform coercive estimate holds:
Proof Let us put and in Theorem 3.1. It is known that the operator is Rpositive in (see, e.g., [36]). So, the estimate (3.1) implies Corollary 3.1.
Consider the differential operator generated by problem (1.5), i.e.,
From Theorem 3.1 we obtain the following. □
Result 3.1 For , there is a resolvent of the operator satisfying the following uniform estimate:
It is clear that the solution u of problem (1.5) depends on parameters , i.e., . In view of Theorem 3.1, we established estimates for the solution of (1.5) uniformly in .
4 Regularity properties of solutions for DOEs with parameters
In this section, we show the separability properties of problem (1.5) in Sobolev spaces . The main result is the following theorem.
Theorem 4.1Let the following conditions be satisfied:
(1) Eis a UMD space andAis anRpositive operator inE;
(2) mis a positive integer, , and
Then problem (3.1)(3.2) has a unique solution for , , with sufficiently large , and the following coercive uniform estimate holds:
with independent of , λ and f.
Consider first the following nonlocal BVP for an ordinary DOE with a small parameter:
where , , , are complex numbers, t is positive, λ is a complex parameter and A is a linear operator in E. Let .
To prove the main result, we need the following result in [[37], Theorem 2.1].
Theorem ALetEbe a UMD space, Abe aψpositive 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, ψ, m, αandp) such that for everyand,
In a similar way as in [[43], §1.8.2, Theorem 2], we obtain the following lemma.
Lemma 4.1Letmandjbe integer numbers, , , , . Then, for, the transformationis bounded linear fromontoand the following inequality holds:
Consider at first the homogeneous problem of (4.2)
Let
Lemma 4.2LetAbe anRpositive operator in a UMD spaceEand
Then problem (4.3) has a unique solutionfor, , , and the coercive uniform estimate holds
Proof In a similar way as in [[40], Theorem 3.1], we obtain the representation of the solution of (4.3)
where and are uniformly bounded operators. Then, in view of positivity of A, we obtain from (4.5)
By changing the variable and in view of Theorem A, we obtain
By using the estimate (4.8), by virtue of Theorem A, we get the uniform estimate
Then from (4.6)(4.9) we obtain (4.4). □
Now we can represent a more general result for nonhomogeneous problem (4.2).
Theorem 4.2Assume that the following conditions are satisfied:
(1) Eis a UMD space andAis anRpositive operator inE;
Then the operatoris an isomorphism fromontoforwith large enough. Moreover, the uniform coercive estimate holds
Proof The uniqueness of a solution of problem (4.2) is obtained from Lemma 4.2. Let us define
We will show that problem (4.2) has a solution for , and , where is the restriction on of the solution of the equation
and is a solution of the problem
A solution of equation (4.11) is given by
where . It follows from the above expression that
It is sufficient to show that the operatorfunctions
are uniform Fourier multipliers in . Actually, due to the positivity of A, we have
It is clear to observe that
Due to Rpositivity of the operator A, the sets
are Rbounded. Then, in view of the Kahane contraction principle, from the product properties of the collection of Rbounded operators (see, e.g., [36] Lemma 3.5, Proposition 3.4), we obtain
By [[33], Theorem 3.4] it follows that and are the uniform collection of multipliers in . Then in view of (4.13) we obtain that problem (4.11) has a solution and the uniform coercive estimate holds
Let be the restriction of u on . The estimate (4.16) implies that . By virtue of Lemma 4.1, we get
Hence, . Thus, by virtue of Lemma 4.2, problem (4.12) has a unique solution that belongs to the space and
Moreover, from (4.16) we obtain
Therefore, by Lemma 4.1 and by estimate (4.17), we obtain
So, in view of Lemma 4.1 and estimates (4.17)(4.19), we get
Finally, from (4.18) and (4.20) we obtain (4.10). □
Now, we can prove the main result of this section.
Proof of Theorem 4.1 Let . It is clear to see that
Let us consider the BVP
where are defined by equalities (1.5). Problem (4.21) can be expressed as the following BVP for an ordinary DOE:
where are boundary conditions of type (3.2), is the operator acting in and X defined by
Since and X are UMD spaces, (see, e.g., [[35], Theorem 4.5.2]) by virtue of Theorem 4.2, we obtain that problem (4.22) has a unique solution for and with sufficiently large . Moreover, the coercive uniform estimates holds
From (4.23) we obtain that problem (4.22) has a unique solution
Moreover, the uniform coercive estimates hold
By applying Theorem 4.2 for and , we get the following uniform estimate:
From estimates (4.24)(4.25) we conclude the corresponding claim for problem (4.21). Then, by continuing this process ntimes, we obtain the assertion. □
5 Nonlocal initialboundary value problems for the Stokes system with small parameters
In this section, we show the uniform maximal regularity properties of the nonlocal initial value problem for nonstationary Stokes equations (1.6).
The function satisfying equation (1.6) a.e. on G is called the stronger solution of problem (1.6).
Let , be the Sobolev space of order s such that . For , let denote the closure of in , where
It is known that ( see, e.g., Fujiwara and Morimoto [17]) a vector field has the Helmholtz decomposition, i.e., all can be uniquely decomposed as with , , where is a projection operator from to and , , so that
with C independent of u, where B is an open ball in and denotes the norm of u in or .
Then problem (1.6) can be reduced to the following BVP:
Consider the parameterdependent Stokes operator generated by problem (5.1), i.e.,
From Corollary 3.1 we get that the operator is positive and also is a generator of a bounded holomorphic semigroup for .
In a similar way as in [18], we show the following.
Proposition 5.1The following estimate holds:
Proof From Result 3.1 we obtain that the operator is uniformly positive in , i.e., for , , the following estimate holds:
where the constant M is independent of λ and ε. Then, by using the Danford integral and operator calculus as in [18], we obtain the assertion. □
Now consider problem (1.7). The main theorem in this section is the following.
Theorem 5.1Let, and. Then there is a unique solutionof problem (1.7) forand. Moreover, the following uniform estimate holds:
Proof Problem (1.7) can be expressed as the following abstract parabolic problem with a small parameter:
If we put and in Theorem 3.1, then the Result 3.1 implies that the operator is uniformly positive and generates bounded holomorphic semigroup in uniformly in . Moreover, by using [[41], Theorem 3.1] we get that operator is Rpositive in E. Since E is a UMD space, in a similar way as in [[33], Theorem 4.2], we obtain that for and , there is a unique solution of problem (5.3) so that the following uniform estimate holds:
From (5.4) for all , we get the following estimate:
6 Existence and uniqueness for the NavierStokes equation with parameters
In this section, we study the NavierStokes problem (1.1)(1.3) in the space . Problem (1.1)(1.3) can be expressed as
where
We consider equation (6.1) in an integral form
To prove the main result, we need the following result which are obtained in a similar way as in [[11], Theorem 2].
Lemma 6.1For any, the domainis the complex interpolation space.
Lemma 6.2For each, the operatorextends uniquely to a uniformly bounded linear operator fromto.
Proof Since is a positive operator, it has fractional powers . From Lemma 6.1, it follows that the domain is continuously embedded in for any , where is the vectorvalued Bessel space. Then, by using the duality argument and due to uniform positivity of , we obtain the following uniformly in ε estimate:
By reasoning as in [12], we obtain the following. □
Lemma 6.3Let. Then the following estimate holds:
uniformly inεwith some constantprovided that, , and
Proof Assume that . Since is continuously embedded in , and since is the same as , by the Sobolev embedding theorem, we obtain that the operator
is bounded, where
By the duality argument then, we get that the operator is bounded from to , where
Consider first the case . Since is bilinear in u, υ, it suffices to prove the estimate on a dense subspace. Therefore, assume that u and υ are smooth. Since , we get
Taking , using the uniform boundedness of from to and Lemma 6.2 for all , we obtain
By assumption we can take r and η such that
Since is continuously embedded in , by the Sobolev embedding, we get
i.e., we have the required result for . In particular, we get the following uniform estimate:
Similarly, we obtain
for and . The above two estimates show that the map is a uniform bounded operator from to and from to . By using Lemma 6.1 and the interpolation of Banach spaces [[30], §1.3.2] for , we obtain
By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato [18], we obtain the following. □
Theorem 6.1Let, . Letbe a real number andsuch that
Suppose that, and thatis continuous onand satisfies
Then there isindependent ofεand a local solution of (6.2) such that
(3) asfor allαwithuniformly with respect to the parameterε.
Moreover, the solution of (5.2) is unique if
(5) asfor someβwithuniformly inε.
Proof We introduce the following iteration scheme:
By estimating the term in (6.4) and by using Proposition 5.1 for , we get
uniformly in ε with
where and is the beta function. Here we suppose . By induction assume that satisfies the following estimate:
We will estimate by using (6.2). To estimate the term , we suppose
so that the numbers θ, σ, δ satisfy the assumptions of Lemma 6.3. Using Lemma 6.3 and (6.5), we get the following uniform estimate:
Therefore, we obtain
with
Since we get the uniform estimates with respect to the parameter ε, the remaining part of the proof is the same as in [[12], Theorem 2.3], so this part is omitted. □
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TXUSA Texas A&M UniversityKingsville2012.
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