Abstract
Keywords:
Stokes systems; NavierStokes equations; abstract differential equations with variable coefficients; semigroups of operators; boundary value problems; maximal regularity1 Introduction
We consider the initial and boundary value problem (BVP) for the following Stokes type equation with variable coefficients:
where
and are linear operators in a Banach space E, , are complex numbers, and are complexvalued functions. Here represents a given, a denotes the initial data and
represent the unknown functions. Moreover, , , and are Evalued functions. This problem is characterized by the presence of abstract operator functions and complexvalued 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 [110]. 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 [8] proved that for every , the instationary Stokes problem
Then Giga and Sohr [2] 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
where
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 NavierStokes problem (see, e.g., [4]). 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 Evalued 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 wellposedness 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 Evalued 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 NavierStokes 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 [11].
2 Definitions and background
Let E be a Banach space. denotes the space of strongly measurable Evalued 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., [12]). UMD spaces include e.g., spaces, and Lorentz spaces , .
Let
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, and is the space of bounded linear operators in E. It is known [[13], §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 Kmethod [[13], §1.3.2].
Let ℕ denote the set of natural numbers. A set is called Rbounded (see, e.g., [14]) 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 Rbound of the collection Φ and denoted by .
A set is called uniform Rbounded in h, if there is a constant C independent on such that
for all and , . It is implied that .
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 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 Evalued 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], §15].
Let denote a LiouvilleLions type space, i.e.,
denote the Evalued 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 vectorvalued spaces play a key role in the theory of DOEs. For estimating lower order derivatives we use the following embedding theorems from [16].
Theorem A_{1}Suppose the following conditions are satisfied:
(1) Eis a UMD space andAis anRpositive 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 fromto.
Then the embeddingis continuous and for allthe following uniform estimate holds:
Remark 2.1 If is a region satisfying the strong lhorn condition (see [[4], §7]), , , then for there exists a bounded linear extension operator from to .
Theorem A_{2}Suppose all conditions of Theorem A_{1}are satisfied and. Moreover, let Ω be a bounded region and. Then the embedding
is compact.
Theorem A_{3}Suppose all conditions of Theorem A_{1}satisfied and. Then the embedding
is continuous and there exists a positive constantsuch that for allthe uniform estimate holds
From [[17], Theorem 2.1] we obtain the following.
Theorem A_{4}LetEbe a Banach space, Abe aφpositive 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)
where and are linear operators in a Banach space E, are complexvalued functions, and λ is a complex parameter.
Maximal regularity properties for DOEs studied, e.g., in [1,11,12,14,1624]. Nonlocal BVPs for PDE were studied in [25].
Let , , be roots of the equations
Condition 3.1 Assume:
(1) E is a UMD space and is a uniformly Rpositive operator in E for ;
Remark 3.1 Let , where are realvalued 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 [[19], Theorem 4.1] we have the following.
Theorem 3.1Suppose Condition 3.1 is satisfied. Then problem (3.1) has a unique solutionforand for sufficiently large. Moreover, the following coercive uniform estimate holds:
Consider the differential operatoringenerated 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 Evalued system of function with norm
denote the Evalued 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 complexvalued 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.
It is known that (see, e.g., [3,5]) the vector field has a Helmholtz decomposition. In the following theorem we generalize this result for the Evalued function 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
Lemma 3.1LetEbe an UMD space, AaRpositive operator inEand. Then, for, problem (3.2) has a unique solutionand the following coercive estimate holds:
Proof Consider the problem
where is an extension of the function on . Then, by using the Fourier inversion formula, operatorvalued multiplier theorems in spaces, and by reasoning as in [[17], 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 [[3], Lemma 2] we get the following.
Proposition 3.1There exists a unique bounded linear operatorfrom, onto
such that
and the following estimate holds:
where
Proof For consider the linear form
By virtue of the trace theorem in , the interpolation of intersection and dual spaces (see, e.g., [[13], §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
such that
and
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
Let and . Consider the following problem:
Lemma 3.3LetEbe an UMD space, AaRpositive operator inEand. Then, for, problem (3.10) has a unique solutionand 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. Thenis a closed subspace of.
Lemma 3.5LetEbe an UMD space and. Then the operatoris a bounded linear operator inandif.
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 ofis 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 [[3], Lemma 5]. Moreover, by Lemma 3.5 the dual operator is a bounded linear in . □
Let
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 [3] 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 solutionfor, , , 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 [6] we show the following.
Proposition 3.2The following estimate holds:
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 [12]), we obtain the assertion. □
4 Wellposedness of instationary Stokes problems with variable coefficients
Let
In this section, we will show the wellposedness of problem (1.1)(1.2).
Theorem 4.1Then, for, , and, , there is a unique solutionof 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 [[17], Theorem 3.1] we see that the operator O is Rpositive in E. Since E is a UMD space, in a similar way as in [[13], 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.
5 Application
Consider the stationary Stokes problem
are unknown functions;
, are complex numbers, , are complexvalued functions, .
Let , . Now will denote the space of all psummable scalarvalued 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
with norm
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 inwith sufficiently smooth boundary∂Ω, , for each, , for eachwith, and, ;
(4) for eachthe local BVPs in local coordinates corresponding to
has a unique solutionfor alland forwith;
Then, problem (5.1)(5.3) has a unique solutionfor, , and the following coercive uniform estimate holds:
Proof Let . By virtue of [[18], 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 [[14], Theorem 8.2] the problem
has a unique solution for and arg , , and the operator A is Rpositive 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 vectorfunctions, 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 solutionof problem (5.6)(5.9) and the following estimate holds:
Competing interests
The author declares that they have no competing interests.
References

Amann, H: On the strong solvability of the NavierStokes equations. J. Math. Fluid Mech.. 2, 16–98 (2000). Publisher Full Text

Giga, Y, Sohr, H: Abstract estimates for the Cauchy problem with applications to the NavierStokes equations in exterior domains. J. Funct. Anal.. 102, 72–94 (1991). Publisher Full Text

Fujiwara, D, Morimoto, H: An theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math.. 24, 685–700 (1977)

Fujita, H, Kato, T: On the NavierStokes initial value problem I. Arch. Ration. Mech. Anal.. 16, 269–315 (1964). Publisher Full Text

Farwing, R, Sohr, H: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn.. 46(4), 607–643 (1994). Publisher Full Text

Kato, T: Strong solutions of the NavierStokes equation in , with applications to weak solutions. Math. Z.. 187, 471–480 (1984). Publisher Full Text

Ladyzhenskaya, OA: The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York (1969)

Solonnikov, V: Estimates for solutions of nonstationary NavierStokes equations. J. Sov. Math.. 8, 467–529 (1977). Publisher Full Text

Sobolevskii, PE: Study of NavierStokes equations by the methods of the theory of parabolic equations in Banach spaces. Sov. Math. Dokl.. 5, 720–723 (1964)

Teman, R: NavierStokes Equations, NorthHolland, Amsterdam (1984)

Shakhmurov, V: Coercive boundary value problems for regular degenerate differentialoperator equations. J. Math. Anal. Appl.. 292(2), 605–620 (2004). Publisher Full Text

Denk, R, Hieber, M, Pruss, J: Rboundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc.. 166, Article ID 788 (2005)

Triebel, H: Interpolation Theory. Function Spaces. Differential Operators, NorthHolland, Amsterdam (1978)

Ashyralyev, A: On wellposedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim.. 24(12), 1–15 (2003). Publisher Full Text

Triebel, H: Fractals and Spectra: Related to Fourier Analysis and Function Spaces, Birkhäuser, Basel (1997)

Shakhmurov, VB: Embedding theorems and maximal regular differential operator equations in Banachvalued function spaces. J. Inequal. Appl.. 4, 605–620 (2005)

Shakhmurov, VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Real World Appl.. 73, 2383–2397 (2010)

Amann, H: Linear and QuasiLinear Equations, Birkhäuser, Basel (1995)

Shakhmurov, VB, Shahmurova, A: Nonlinear abstract boundary value problems atmospheric dispersion of pollutants. Nonlinear Anal., Real World Appl.. 11(2), 932–951 (2010). Publisher Full Text

Shakhmurov, VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl.. 327(2), 1182–1201 (2006)

Shakhmurov, V: Parameter dependent Stokes problems in vectorvalued spaces and applications. Bound. Value Probl.. 2013, Article ID 172 (2013)

Weis, L: Operatorvalued Fourier multiplier theorems and maximal regularity. Math. Ann.. 319, 735–758 (2001). Publisher Full Text

Yakubov, S, Yakubov, Y: DifferentialOperator Equations. Ordinary and Partial Differential Equations, Chapman & Hall/CRC, Boca Raton (2000)

Lunardi, A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (2003)

Skubachevskii, AL: Nonlocal boundary value problems. J. Math. Sci.. 155(2), 199–334 (2008). Publisher Full Text