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.
KeywordsStokes operators Navier-Stokes equations differential equations with small parameters semigroups of operators boundary value problems differential-operator equations maximal regularity
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.
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