Initial nonlocal boundary value problems for a Navier-Stokes equation with a small parameter is considered. The uniform maximal regularity properties of the corresponding stationary Stokes operator, well-posedness of a nonstationary Stokes problem and the existence, uniqueness and uniformly estimates for the solution of the Navier-Stokes problem are established.
MSC: 35Q30, 76D05, 34G10, 35J25.
Keywords:Stokes operators; Navier-Stokes equations; differential equations with small parameters; semigroups of operators; boundary value problems; differential-operator equations; maximal regularity
Consider the following Navier-Stokes problem with a 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 Navier-Stokes equations. From both the theoretical and computational points of view, singularly perturbed problems and asymptotic behavior of the Navier-Stokes equations with small viscosity when the boundary is either characteristic or non-characteristic have been well studied; see, e.g., [1-6]. 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 Navier-Stokes problem
Note that the existence of weak or strong solutions and regularity properties of classical Navier-Stokes problems were extensively studied, e.g., in [1-3,5,7-33]. There is extensive literature on the solvability of the initial value problem for the Navier-Stokes equation ( see, e.g.,  for further papers cited there ). Hopf  proved the existence of a global weak solution of (1.4) using the Faedo-Galerkin approximation and an energy inequality. Another approach to problem (1.4) is to use semigroup theory. Kato and Fujita [18,22,34] and Sobolevskii  transformed equation (1.4) into an evolution equation in the Hilbert space . They proved the existence of a unique global strong solution for any square-summable 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  and Heywood . Afterward, Giga and Sohr  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:
Further, we consider the nonlocal BVP for the stationary Stokes system with small parameters
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
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:
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 vector-valued Sobolev-Lions spaces are the main tools implemented to carry out the analysis.
2 Notations, definitions 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 ℂ be the set of complex numbers and
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, is the space of bounded linear operators in E. It is known [, §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 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. Let denote the space consisting of all functions that have the generalized derivatives , with the norm
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.
Theorem 3.1Let the following conditions be satisfied:
Proof Let us consider the BVP
where are boundary conditions of type (1.5) on . By virtue of [, Theorem 3.2], we obtain that problem (3.3) has a unique solution for , , with sufficiently large , and the following coercive uniform estimate holds:
By virtue of [, Theorem 4.5.2], provided , . Hence, by virtue of [, Theorem 3.2] and [, Theorem 3.1], the operator is uniformly R-positive in F. Then, by applying again [, 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
Further, by continuing this process n-times, we obtain the assertion.
From Theorem 3.1 we obtain the following. □
Proof Let us put and in Theorem 3.1. It is known that the operator is R-positive in (see, e.g., ). So, the estimate (3.1) implies Corollary 3.1.
From Theorem 3.1 we obtain the following. □
4 Regularity properties of solutions for DOEs with parameters
Theorem 4.1Let the following conditions be satisfied:
(1) Eis a UMD space andAis anR-positive operator inE;
Consider first the following nonlocal BVP for an ordinary DOE with a small parameter:
To prove the main result, we need the following result in [, Theorem 2.1].
Theorem ALetEbe a UMD space, Abe aψ-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, ψ, m, αandp) such that for everyand,
In a similar way as in [, §1.8.2, Theorem 2], we obtain the following lemma.
Consider at first the homogeneous problem of (4.2)
Lemma 4.2LetAbe anR-positive operator in a UMD spaceEand
Proof In a similar way as in [, Theorem 3.1], we obtain the representation of the solution of (4.3)
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 anR-positive operator inE;
Proof The uniqueness of a solution of problem (4.2) is obtained from Lemma 4.2. Let us define
A solution of equation (4.11) 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. Then, in view of the Kahane contraction principle, from the product properties of the collection of R-bounded operators (see, e.g.,  Lemma 3.5, Proposition 3.4), we obtain
By [, 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
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.
Let us consider the BVP
Since and X are UMD spaces, (see, e.g., [, 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
From estimates (4.24)-(4.25) we conclude the corresponding claim for problem (4.21). Then, by continuing this process n-times, we obtain the assertion. □
5 Nonlocal initial-boundary 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).
It is known that ( see, e.g., Fujiwara and Morimoto ) 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
Then problem (1.6) can be reduced to the following BVP:
In a similar way as in , we show the following.
Proposition 5.1The following estimate holds:
where the constant M is independent of λ and ε. Then, by using the Danford integral and operator calculus as in , we obtain the assertion. □
Now consider problem (1.7). The main theorem in this section is the following.
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 [, Theorem 3.1] we get that operator is R-positive in E. Since E is a UMD space, in a similar way as in [, Theorem 4.2], we obtain that for and , there is a unique solution of problem (5.3) so that the following uniform estimate holds:
6 Existence and uniqueness for the Navier-Stokes equation with parameters
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 [, Theorem 2].
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 vector-valued 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 , we obtain the following. □
is bounded, where
By assumption we can take r and η such that
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 [, §1.3.2] for , we obtain
By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato , we obtain the following. □
Moreover, the solution of (5.2) is unique if
Proof We introduce the following iteration scheme:
uniformly in ε with
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
Since we get the uniform estimates with respect to the parameter ε, the remaining part of the proof is the same as in [, Theorem 2.3], so this part is omitted. □
The author declares that they have no competing interests.
Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA Texas A&M University-Kingsville-2012.
Iftimie, D, Planas, G: Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity. 19(4), 899–918 (2006). Publisher Full Text
Temam, R, Wang, X: Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differ. Equ.. 179(2), 647–686 (2002). Publisher Full Text
Xiao, Y, Xin, Z: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun. Pure Appl. Math.. 60(7), 1027–1055 (2007). Publisher Full Text
Amann, H: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech.. 2, 16–98 (2000). Publisher Full Text
Caffarelli, L, Kohn, R, Nirenberg, L: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math.. 35, 771–831 (1982). Publisher Full Text
Giga, Y: Domains of fractional powers of the Stokes operator in spaces. Arch. Ration. Mech. Anal.. 89, 251–265 (1985). Publisher Full Text
Giga, Y, Miyakava, T: Solutions in of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal.. 89, 267–281 (1985). Publisher Full Text
Giga, Y, Sohr, H: Abstract estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal.. 102, 72–94 (1991). Publisher Full Text
Fefferman, C, Constantin, P: Direction of vorticity and the problem of global regularity for the 3-d Navier-Stokes equations. Indiana Univ. Math. J.. 42, 775–789 (1993). Publisher Full Text
Fujita, H, Kato, T: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal.. 16, 269–315 (1964). Publisher Full Text
Fabes, EB, Lewas, JE, Riviere, NM: Boundary value problems for the Navier-Stokes equations. Am. J. Math.. 99, 626–668 (1977). Publisher Full Text
Hopf, E: Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr.. 4, 213–231 (1950-51). Publisher Full Text
Heywood, JG: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J.. 29, 639–681 (1980). Publisher Full Text
Leray, J: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.. 63, 193–248 (1934). Publisher Full Text
Solonnikov, V: Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math.. 8, 467–529 (1977). Publisher Full Text
Weissler, FB: The Navier-Stokes initial value problem in . Arch. Ration. Mech. Anal.. 74, 219–230 (1980). Publisher Full Text
Weis, L: Operator-valued Fourier multiplier theorems and maximal regularity. Math. Ann.. 319, 735–758 (2001). Publisher Full Text
Kato, T: Strong -solutions of the Navier-Stokes equation in , with applications to weak solutions. Math. Z.. 187, 471–480 (1984). Publisher Full Text
Dore, C, Yakubov, S: Semigroup estimates and non coercive boundary value problems. Semigroup Forum. 60, 93–121 (2000). Publisher Full Text
Shakhmurov, VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Theory Methods Appl.. 73, 2383–2397 (2010). Publisher Full Text
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: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl.. 292(2), 605–620 (2004). Publisher Full Text