- Open Access
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.
- Stokes 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.
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 .
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 .
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. □
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 . □
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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