Nonlocal Navier-Stokes problem with a small parameter
© Shakhmurov; licensee Springer. 2013
Received: 5 March 2013
Accepted: 12 April 2013
Published: 26 April 2013
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.
with independent of , , λ and f.
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 vector-valued Sobolev-Lions spaces are the main tools implemented to carry out the analysis.
2 Notations, definitions and background
is bounded in , (see, e.g., ). UMD spaces include, e.g., , spaces and Lorentz spaces , .
where is a sequence of independent symmetric -valued random variables on Ω. The smallest C for which the above estimate holds is called an R-bound of the collection G and denoted by .
which implies 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 in with bound if is independent of t, is dense in E and for all , , where M does not depend on t and λ.
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.
E is a UMD space and A is an R-positive operator in E for ;
, , , .
withindependent of, λ and f.
Further, by continuing this process n-times, we obtain the assertion.
From Theorem 3.1 we obtain the following. □
withindependent of f, and λ.
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. □
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.
E is a UMD space and A is an R-positive operator in E;
- (2)m is a positive integer , , and
with independent of , λ and f.
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 [, Theorem 2.1].
In a similar way as in [, §1.8.2, Theorem 2], we obtain the following lemma.
Then from (4.6)-(4.9) we obtain (4.4). □
Now we can represent a more general result for nonhomogeneous problem (4.2).
E is a UMD space and A is an R-positive operator in E;
, , , , .
Finally, from (4.18) and (4.20) we obtain (4.10). □
Now, we can prove the main result of this section.
where and .
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).
The function satisfying equation (1.6) a.e. on G is called the stronger solution of problem (1.6).
with C independent of u, where B is an open ball in and denotes the norm of u in or .
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 , we show the following.
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.
withindependent of f and ε.
uniformly in . □
6 Existence and uniqueness for the Navier-Stokes equation with parameters
To prove the main result, we need the following result which are obtained in a similar way as in [, Theorem 2].
Lemma 6.1 For any, the domainis the complex interpolation space.
Lemma 6.2 For each, the operatorextends uniquely to a uniformly bounded linear operator fromto.
By reasoning as in , we obtain the following. □
By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato , we obtain the following. □
for some ;
as for all α with uniformly with respect to the parameter ε.
as for some β with uniformly in ε.
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. □
Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA Texas A&M University-Kingsville-2012.
- Beirâo da Veiga H: Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 2006, 5(4):907-918.MATHMathSciNetView ArticleGoogle Scholar
- Hamouda M, Temam R: Some Singular Perturbation Problems Related to the Navier-Stokes Equations, Advances in Deterministic and Stochastic Analysis. World Scientific, Hackensack; 2007:197-227.Google Scholar
- Iftimie D, Planas G: Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity 2006, 19(4):899-918. 10.1088/0951-7715/19/4/007MATHMathSciNetView ArticleGoogle Scholar
- Lions J-L 1. In Mathematical Topics in Fluid Mechanics. The Clarendon Press Oxford University Press, New York; 1996. Incompressible models, Oxford Science PublicationsGoogle Scholar
- Temam R, Wang X: Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 2002, 179(2):647-686. 10.1006/jdeq.2001.4038MATHMathSciNetView ArticleGoogle Scholar
- Xiao Y, Xin Z: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 2007, 60(7):1027-1055. 10.1002/cpa.20187MATHMathSciNetView ArticleGoogle Scholar
- Amann H: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2000, 2: 16-98. 10.1007/s000210050018MATHMathSciNetView ArticleGoogle Scholar
- Caffarelli L, Kohn R, Nirenberg L: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 1982, 35: 771-831. 10.1002/cpa.3160350604MATHMathSciNetView ArticleGoogle Scholar
- Cannone M: A generalization of a theorem by Kato on Navier-Stokes equations. Rev. Mat. Iberoam. 1997, 13(3):515-541.MATHMathSciNetView ArticleGoogle Scholar
- Desch W, Hieber M, Prüss J:-theory of the Stokes equation in a half-space. J. Evol. Equ. 2001., 2001: Article ID 1Google Scholar
- Giga Y:Domains of fractional powers of the Stokes operator in spaces. Arch. Ration. Mech. Anal. 1985, 89: 251-265. 10.1007/BF00276874MATHMathSciNetView ArticleGoogle Scholar
- Giga Y, Miyakava T:Solutions in of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 1985, 89: 267-281. 10.1007/BF00276875MATHView ArticleGoogle Scholar
- Giga Y, Sohr H:Abstract estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 1991, 102: 72-94. 10.1016/0022-1236(91)90136-SMATHMathSciNetView ArticleGoogle Scholar
- Giga Y:Solutions for semilinear parabolic equation and regularity of weak solutions of the Navier-Stokes systems. J. Differ. Equ. 1986, 61: 186-212.MATHMathSciNetView ArticleGoogle Scholar
- Galdi GP: An Introduction to the Mathematical Theory of the Navier-Stokes Equations I: Linearized Steady Problems. 2nd edition. Springer, Berlin; 1998.Google Scholar
- Fefferman C, Constantin P: Direction of vorticity and the problem of global regularity for the 3-d Navier-Stokes equations. Indiana Univ. Math. J. 1993, 42: 775-789. 10.1512/iumj.1993.42.42034MATHMathSciNetView ArticleGoogle Scholar
- Fujiwara D, Morimoto H:An -theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 1977, 24: 685-700.MATHMathSciNetGoogle Scholar
- Fujita H, Kato T: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 1964, 16: 269-315. 10.1007/BF00276188MATHMathSciNetView ArticleGoogle Scholar
- Fabes EB, Lewas JE, Riviere NM: Boundary value problems for the Navier-Stokes equations. Am. J. Math. 1977, 99: 626-668. 10.2307/2373933MATHView ArticleGoogle Scholar
- Hopf E: Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr. 1950-51, 4: 213-231. 10.1002/mana.3210040121MATHMathSciNetView ArticleGoogle Scholar
- Heywood JG: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 1980, 29: 639-681. 10.1512/iumj.1980.29.29048MATHMathSciNetView ArticleGoogle Scholar
- Kato T, Fujita H: On the nonstationary Navier-Stokes system. Rend. Semin. Mat. Univ. Padova 1962, 32: 243-260.MATHMathSciNetGoogle Scholar
- Masmoudi N: Examples of singular limits in hydrodynamics. Handb. Differ. Equ. III. In Evolutionary Equations. Elsevier, Amsterdam; 2007:195-276.Google Scholar
- Leray J: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 1934, 63: 193-248. 10.1007/BF02547354MATHMathSciNetView ArticleGoogle Scholar
- Ladyzhenskaya OA: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York; 1969.MATHGoogle Scholar
- Solonnikov V: Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 1977, 8: 467-529. 10.1007/BF01084616MATHView ArticleGoogle Scholar
- Sobolevskii PE: Study of Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces. Sov. Math. Dokl. 1964, 5: 720-723.Google Scholar
- Shakhmurov VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl. 2006, 327(2):1182-1201.MathSciNetView ArticleGoogle Scholar
- Teman R: Navier-Stokes Equations. North-Holland, Amsterdam; 1984.Google Scholar
- Triebel H: Interpolation Theory. Function Spaces. Differential Operators. North-Holland, Amsterdam; 1978.Google Scholar
- Wiegner M: Navier-Stokes equations a neverending challenge. Jahresber. Dtsch. Math.-Ver. 1999, 101: 1-25.MATHMathSciNetGoogle Scholar
- Weissler FB:The Navier-Stokes initial value problem in . Arch. Ration. Mech. Anal. 1980, 74: 219-230. 10.1007/BF00280539MATHMathSciNetView ArticleGoogle Scholar
- Weis L:Operator-valued Fourier multiplier theorems and maximal regularity. Math. Ann. 2001, 319: 735-758. 10.1007/PL00004457MATHMathSciNetView ArticleGoogle Scholar
- Kato T:Strong -solutions of the Navier-Stokes equation in , with applications to weak solutions. Math. Z. 1984, 187: 471-480. 10.1007/BF01174182MATHMathSciNetView ArticleGoogle Scholar
- Amann H 1. In Linear and Quasi-Linear Equations. Birkhäuser, Basel; 1995.Google Scholar
- Denk R, Hieber M, Pruss J: R -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 2005., 166: Article ID 788Google Scholar
- Dore C, Yakubov S: Semigroup estimates and non coercive boundary value problems. Semigroup Forum 2000, 60: 93-121. 10.1007/s002330010005MATHMathSciNetView ArticleGoogle Scholar
- Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel; 2003.Google Scholar
- Shakhmurov VB: Nonlinear abstract boundary value problems in vector-valued function spaces and applications. Nonlinear Anal., Theory Methods Appl. 2006, 67(3):745-762.MathSciNetView ArticleGoogle Scholar
- Shakhmurov VB: Linear and nonlinear abstract equations with parameters. Nonlinear Anal., Theory Methods Appl. 2010, 73: 2383-2397. 10.1016/j.na.2010.06.004MATHMathSciNetView ArticleGoogle Scholar
- Shakhmurov VB, Shahmurova A: Nonlinear abstract boundary value problems atmospheric dispersion of pollutants. Nonlinear Anal., Real World Appl. 2010, 11(2):932-951. 10.1016/j.nonrwa.2009.01.037MATHMathSciNetView ArticleGoogle Scholar
- Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl. 2004, 292(2):605-620. 10.1016/j.jmaa.2003.12.032MATHMathSciNetView ArticleGoogle Scholar
- Yakubov S, Yakubov Ya: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall/CRC, Boca Raton; 2000.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.