Open Access

Nonlocal Navier-Stokes problem with a small parameter

Boundary Value Problems20132013:107

DOI: 10.1186/1687-2770-2013-107

Received: 5 March 2013

Accepted: 12 April 2013

Published: 26 April 2013

Abstract

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 L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq1_HTML.gif 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 L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq2_HTML.gif regularity

1 Introduction

Consider the following Navier-Stokes problem with a parameter:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ3_HTML.gif
(1.3)
where
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equa_HTML.gif
α k j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq3_HTML.gif, β k j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq4_HTML.gif are complex numbers, ε is a small positive parameter,
u = ( u 1 ( x , t ) , u 2 ( x , t ) , , u n ( x , t ) ) , φ = φ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equb_HTML.gif
represent the unknown velocity and pressure, respectively,
f = ( f 1 ( x , t ) , f 2 ( x , t ) , , f n ( x , t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equc_HTML.gif
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., [16]. 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 ε = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq5_HTML.gif, choosing the boundary conditions locally and m k j = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq6_HTML.gif, problem (1.1)-(1.3) is reduced to the classical Navier-Stokes problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ4_HTML.gif
(1.4)
Note that the existence of weak or strong solutions and regularity properties of classical Navier-Stokes problems were extensively studied, e.g., in [13, 5, 733]. There is extensive literature on the solvability of the initial value problem for the Navier-Stokes equation ( see, e.g., [25] for further papers cited there ). Hopf [20] 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 [27] transformed equation (1.4) into an evolution equation in the Hilbert space L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq7_HTML.gif. They proved the existence of a unique global strong solution for any square-summable initial velocity when n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq8_HTML.gif. On the other hand, when n = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq9_HTML.gif 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 [26] and Heywood [21]. Afterward, Giga and Sohr [13] 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:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ5_HTML.gif
(1.5)
where A is a linear operator in a Banach space E, α k j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq3_HTML.gif, β k j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq4_HTML.gif are complex numbers, ε k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq10_HTML.gif are positive and λ is a complex parameter. We show that problem (1.5) has a unique solution u W 2 , q ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq11_HTML.gif for f W m , q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq12_HTML.gif and λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif with sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif, and the following coercive uniform estimate holds:
k = 1 n i = 0 m + 2 ε k i m + 2 | λ | 1 i 2 i u x k i L q ( G ; E ) + A u L q ( G ; E ) C f W m , q ( G ; E ) , m 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equd_HTML.gif

with C ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq15_HTML.gif independent of ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq16_HTML.gif, ε 2 , , ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq17_HTML.gif, λ and f.

Further, we consider the nonlocal BVP for the stationary Stokes system with small parameters
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ6_HTML.gif
(1.6)
where
u = ( u 1 ( x , t ) , u 2 ( x , t ) , , u n ( x , t ) ) , f = ( f 1 ( x , t ) , f 2 ( x , t ) , , f n ( x , t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Eque_HTML.gif
Then we consider the initial nonlocal BVP for the following nonstationary Stokes equation with small parameters:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ7_HTML.gif
(1.7)
Problem (1.7) can be expressed as the abstract parabolic problem with a parameter
d u d t + O ε , q u = f ( t ) , u ( 0 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ8_HTML.gif
(1.8)
where O ε , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq18_HTML.gif is a stationary parameter depending on the Stokes operator in a solenoidal space L σ q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq19_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equf_HTML.gif
We prove that the operator O ε , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq18_HTML.gif is positive in L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq20_HTML.gif uniformly with respect to parameters ε = ( ε 1 , ε 2 , , ε n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq21_HTML.gif and also is a generator of a holomorphic semigroup. Then, by using L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq1_HTML.gif-maximal regularity theorems (see, e.g., [33, 35]) for abstract parabolic equations (1.8), we obtain that for every f L p ( 0 , T ; L q ( Ω ; R n ) ) = B ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq22_HTML.gif, p , q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq23_HTML.gif, there is a unique solution ( u , φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq24_HTML.gif of problem (1.8) and the following uniform estimate holds:
u t B ( p , q ) + k = 1 n ε k 2 u x k 2 B ( p , q ) + φ B ( p , q ) C f B ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equg_HTML.gif

with C = C ( T , Ω , p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq25_HTML.gif 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

Let E be a Banach space and L p ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq26_HTML.gif denotes the space of strongly measurable E-valued functions that are defined on the measurable subset Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq27_HTML.gif with the norm
f L p = f L p ( Ω ; E ) = ( Ω f ( x ) E p d x ) 1 p , 1 p < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equh_HTML.gif
The Banach space E is called a UMD-space if the Hilbert operator
( H f ) ( x ) = lim ε 0 | x y | > ε f ( y ) x y d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equi_HTML.gif

is bounded in L p ( R , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq28_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq29_HTML.gif (see, e.g., [36]). UMD spaces include, e.g., L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq2_HTML.gif, l p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq30_HTML.gif spaces and Lorentz spaces L p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq31_HTML.gif, p , q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq32_HTML.gif.

Let be the set of complex numbers and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equj_HTML.gif
A linear operator A is said to be ψ-positive in a Banach space E with bound M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq33_HTML.gif if D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq34_HTML.gif is dense on E and ( A + λ I ) 1 B ( E ) M ( 1 + | λ | ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq35_HTML.gif for any λ S ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq36_HTML.gif, 0 ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq37_HTML.gif, where I is the identity operator in E, B ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq38_HTML.gif is the space of bounded linear operators in E. It is known [[30], §1.15.1] that there exist the fractional powers A θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq39_HTML.gif of a positive operator A. Let E ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq40_HTML.gif denote the space D ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq41_HTML.gif with the norm
u E ( A θ ) = ( u p + A θ u p ) 1 p , 1 p < , 0 < θ < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equk_HTML.gif
Let denote the set of natural numbers. A set G B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq42_HTML.gif is called R-bounded (see, e.g., [36]) if there is a positive constant C such that for all T 1 , T 2 , , T m G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq43_HTML.gif and u 1 , u 2 , , u m E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq44_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq45_HTML.gif,
Ω j = 1 m r j ( y ) T j u j E 2 d y C Ω j = 1 m r j ( y ) u j E 1 d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equl_HTML.gif

where { r j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq46_HTML.gif is a sequence of independent symmetric { 1 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq47_HTML.gif-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 R ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq48_HTML.gif.

A set G h B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq49_HTML.gif is called uniform R-bounded if there is a constant C independent of h Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq50_HTML.gif such that for all T 1 ( h ) , T 2 ( h ) , , T m ( h ) G h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq51_HTML.gif and u 1 , u 2 , , u m E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq52_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq53_HTML.gif,
Ω j = 1 m r j ( y ) T j ( h ) u j E 2 d y C Ω j = 1 m r j ( y ) u j E 1 d y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equm_HTML.gif

which implies that sup h Q R ( G h ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq54_HTML.gif.

The ψ-positive operator A is said to be R-positive in a Banach space E if the set L A = { ξ ( A + ξ ) 1 : ξ S ψ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq55_HTML.gif, 0 ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq37_HTML.gif, is R-bounded.

The operator A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq56_HTML.gif is said to be ψ-positive in E uniformly in t σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq57_HTML.gif with bound M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq33_HTML.gif if D ( A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq58_HTML.gif is independent of t, D ( A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq58_HTML.gif is dense in E and ( A ( t ) + λ ) 1 M ( 1 + | λ | ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq59_HTML.gif for all λ S ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq36_HTML.gif, 0 ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq60_HTML.gif, where M does not depend on t and λ.

Let E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq61_HTML.gif and E be two Banach spaces, and let E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq61_HTML.gif be continuously and densely embedded into E. Let Ω be a measurable set in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq62_HTML.gif and m be a positive integer. Let W m , p ( Ω ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq63_HTML.gif denote the space consisting of all functions u L p ( Ω ; E 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq64_HTML.gif that have the generalized derivatives m u x k m L p ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq65_HTML.gif, with the norm
u W m , p ( Ω ; E 0 , E ) = u L p ( Ω ; E 0 ) + k = 1 n m u x k m L p ( Ω ; E ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equn_HTML.gif

For n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq66_HTML.gif, Ω = ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq67_HTML.gif, a , b N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq68_HTML.gif, the space W m , p ( Ω ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq63_HTML.gif will be denoted by W m , p ( a , b ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq69_HTML.gif.

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 C α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq70_HTML.gif.

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.

It should be noted that BVPs for DOEs were studied, e.g., in [3538] and [6, 26, 27, 3943]. For references, see [35, 43]. Let α k j = α k j m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq71_HTML.gif and β k j = β k j m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq72_HTML.gif. First, we prove the following theorem.

Theorem 3.1 Let the following conditions be satisfied:
  1. (1)

    E is a UMD space and A is an R-positive operator in E for 0 ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq37_HTML.gif;

     
  2. (2)

    q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq73_HTML.gif, η k = ( 1 ) m 1 α k 1 β k 2 ( 1 ) m 2 α k 2 β k 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq74_HTML.gif, 0 < t k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq75_HTML.gif, k = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq76_HTML.gif.

     
Then problem (1.5) has a unique solution u W 2 , q ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq11_HTML.giffor f L q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq77_HTML.gifand λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gifwith sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif. Moreover, the following coercive uniform estimate holds:
k = 1 n j = 0 2 ε k j 2 | λ | 1 j 2 j u x k j L q ( G ; E ) + A u L q ( G ; E ) C f L q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ9_HTML.gif
(3.1)

with C ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq15_HTML.gifindependent of ε 1 , ε 2 , , ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq78_HTML.gif, λ and f.

Proof Let us consider the BVP
ε 1 2 u x 1 2 ε 2 2 u x 2 2 + ( A + λ ) u ( x 1 , x 2 ) = f ( x 1 , x 2 ) , L k j ε u = 0 , k , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ10_HTML.gif
(3.2)
where x 1 , x 2 G 2 = ( 0 , b 1 ) × ( 0 , b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq79_HTML.gif, L k j ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq80_HTML.gif are defined by equalities (3.1)-(3.2). For the investigation (3.4), we consider the following BVP for ordinary DOE:
L u = ε 2 d 2 u d x 2 2 + ( A + λ ) u ( x 2 ) = f ( x 2 ) , y ( 0 , b 2 ) , L 2 j ε 2 u = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ11_HTML.gif
(3.3)
where L 2 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq81_HTML.gif are boundary conditions of type (1.5) on ( 0 , b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq82_HTML.gif. By virtue of [[40], Theorem 3.2], we obtain that problem (3.3) has a unique solution u W 2 , q ( 0 , b 2 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq83_HTML.gif for f L q ( 0 , b 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq84_HTML.gif, λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif, with sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif, and the following coercive uniform estimate holds:
j = 0 2 ε 2 j 2 | λ | 1 j 2 u ( j ) L q ( 0 , b 2 ; E ) + A u L q ( 0 , b 2 ; E ) C f L q ( 0 , b 2 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ12_HTML.gif
(3.4)
Since L q ( 0 , b 1 ; L q ( 0 , b 2 ; E ) ) = L q ( G 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq85_HTML.gif, problem (3.2) can be expressed as the following problem:
ε 1 d u d x 1 2 + ( B ε 2 + λ ) u ( x 1 ) = f ( x 1 ) , L k 1 ε 1 u = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ13_HTML.gif
(3.5)
where B ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq86_HTML.gif is the differential operator in L q ( 0 , b 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq87_HTML.gif generated by problem (3.3), i.e.,
D ( B ε 2 ) = W 2 , q ( 0 , b 2 ; E ( A ) , E , L 2 k ε 2 ) , B ε 2 u = ε 2 d 2 u d x 2 2 + A u ( x 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equo_HTML.gif
By virtue of [[35], Theorem 4.5.2], F = L q ( 0 , b 2 ; E ) UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq88_HTML.gif provided E UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq89_HTML.gif, q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq90_HTML.gif. Hence, by virtue of [[40], Theorem 3.2] and [[41], Theorem 3.1], the operator B ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq86_HTML.gif is uniformly R-positive in F. Then, by applying again [[40], Theorem 3.2], we get that for f L q ( 0 , b 1 ; F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq91_HTML.gif, λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif and sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq92_HTML.gif, problem (3.5) has a unique solution u W 2 , q ( 0 , b 1 ; D ( B ) , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq93_HTML.gif, and the following coercive uniform estimate holds:
j = 0 2 ε 1 j 2 | λ | 1 j 2 u ( j ) L q ( 0 , b 1 ; F ) + B ε 2 u L q ( 0 , b 1 ; F ) C f L q ( 0 , b 1 ; F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ14_HTML.gif
(3.6)
The estimate (3.4) implies the uniform estimate
j = 0 2 ε 2 j 2 | λ | 1 j 2 u ( j ) L q ( 0 , b 1 ; F ) C B ( ε 2 ) u L q ( 0 , b 1 ; F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ15_HTML.gif
(3.7)
By using (3.4) and (3.7), we obtain that problem (3.7) has a unique solution u W 2 , q ( G 2 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq94_HTML.gif for f L q ( G 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq95_HTML.gif, λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif with sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq92_HTML.gif, and the coercive uniform estimate holds
k = 1 2 j = 0 2 ε k j 2 | λ | 1 j 2 j u x k j L q ( G 2 ; E ) + A u L q ( G 2 ; E ) C f L q ( G 2 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equp_HTML.gif

Further, by continuing this process n-times, we obtain the assertion.

From Theorem 3.1 we obtain the following. □

Corollary 3.1 Let 0 < ε k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq96_HTML.gif, ( 1 ) m k 1 α k 1 β k 2 ( 1 ) m k 2 α k 2 β k 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq97_HTML.gif. For f L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq98_HTML.gif, q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq73_HTML.gifand for λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gifwith sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif, there is a unique solution u of problem (1.5) and the following uniform coercive estimate holds:
k = 1 n i = 0 2 | λ | 1 i 2 ε k i 2 i u x k i q C f q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equq_HTML.gif

with C = C ( q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq99_HTML.gifindependent of f, ε k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq10_HTML.gifand λ.

Proof Let us put E = R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq100_HTML.gif and A = ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq101_HTML.gif in Theorem 3.1. It is known that the operator A = ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq101_HTML.gif is R-positive in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq62_HTML.gif (see, e.g., [36]). So, the estimate (3.1) implies Corollary 3.1.

Consider the differential operator Q ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq102_HTML.gif generated by problem (1.5), i.e.,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equr_HTML.gif

From Theorem 3.1 we obtain the following. □

Result 3.1 For λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif, there is a resolvent ( Q ε + λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq103_HTML.gif of the operator Q ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq102_HTML.gif satisfying the following uniform estimate:
k = 1 n i = 0 2 | λ | 1 i 2 ε k i 2 i x k i ( Q ε + λ ) 1 B ( L q ( G ; E ) ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equs_HTML.gif

It is clear that the solution u of problem (1.5) depends on parameters ε = ( ε 1 , ε 2 , , ε n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq104_HTML.gif, i.e., u = u ( x , ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq105_HTML.gif. In view of Theorem 3.1, we established estimates for the solution of (1.5) uniformly in ε 1 , ε 2 , , ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq106_HTML.gif.

4 Regularity properties of solutions for DOEs with parameters

In this section, we show the separability properties of problem (1.5) in Sobolev spaces W m , q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq107_HTML.gif. The main result is the following theorem.

Theorem 4.1 Let the following conditions be satisfied:
  1. (1)

    E is a UMD space and A is an R-positive operator in E;

     
  2. (2)
    m is a positive integer q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq73_HTML.gif, 0 < t k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq75_HTML.gif, and
    η k = ( 1 ) m 1 α k 1 β k 2 ( 1 ) m 2 α k 2 β k 1 0 , k = 1 , 2 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equt_HTML.gif
     
Then problem (3.1)-(3.2) has a unique solution u W 2 + m , q ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq108_HTML.gif for f W m , q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq109_HTML.gif, λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif, with sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif, and the following coercive uniform estimate holds:
k = 1 n i = 0 m + 2 ε k i m + 2 | λ | 1 i m + 2 i u x k i W m , q ( G ; E ) + A u L q ( G ; E ) C f W m , q ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ16_HTML.gif
(4.1)

with C = C ( q , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq110_HTML.gif independent of ε 1 , ε 2 , , ε n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq78_HTML.gif, λ and f.

Consider first the following nonlocal BVP for an ordinary DOE with a small parameter:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ17_HTML.gif
(4.2)

where σ i = i 2 + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq111_HTML.gif, m k { 0 , m + 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq112_HTML.gif, α k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq113_HTML.gif, β k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq114_HTML.gif are complex numbers, t is positive, λ is a complex parameter and A is a linear operator in E. Let A λ = A + λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq115_HTML.gif.

To prove the main result, we need the following result in [[37], Theorem 2.1].

Theorem A Let E be a UMD space, A be a ψ-positive operator in E with bound M, 0 ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq37_HTML.gif. Let m be a positive integer, 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq116_HTML.gif, and α ( 1 2 p , 1 2 p + m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq117_HTML.gif. Then, for λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq118_HTML.gif, an operator A λ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq119_HTML.gifgenerates a semigroup e x A λ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq120_HTML.gifwhich is holomorphic for x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq121_HTML.gif. Moreover, there exists a positive constant C (depending only on M, ψ, m, α and p) such that for every u ( E , E ( A m ) ) α m 1 2 m p , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq122_HTML.gifand λ S ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq36_HTML.gif,
0 A λ α e x A λ 1 2 u p d x C [ u ( E , E ( A m ) ) α m 1 2 m p , p p + | λ | α p 1 2 u E p ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equu_HTML.gif

In a similar way as in [[43], §1.8.2, Theorem 2], we obtain the following lemma.

Lemma 4.1 Let m and j be integer numbers, 0 j m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq123_HTML.gif, θ j = p j + 1 p m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq124_HTML.gif, 0 < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq125_HTML.gif, x 0 [ 0 , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq126_HTML.gif. Then, for u W p m ( 0 , b ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq127_HTML.gif, the transformation u u ( j ) ( x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq128_HTML.gifis bounded linear from W p m ( 0 , b ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq129_HTML.gifonto ( E 0 , E ) θ j , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq130_HTML.gifand the following inequality holds:
t θ j u ( j ) ( x 0 ) ( E 0 , E ) θ j , p C ( t u ( m ) L p ( 0 , b ; E ) + u L p ( 0 , b ; E 0 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equv_HTML.gif
Consider at first the homogeneous problem of (4.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ18_HTML.gif
(4.3)
Let
E k = ( E ( A ) , E ) θ k , p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equw_HTML.gif
Lemma 4.2 Let A be an R-positive operator in a UMD space E and
0 < t 1 , η = ( 1 ) m 1 α 1 β 2 ( 1 ) m 2 α 2 β 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equx_HTML.gif
Then problem (4.3) has a unique solution u W m + 2 , p ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq131_HTML.giffor f k E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq132_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq29_HTML.gif, λ S ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq36_HTML.gif, and the coercive uniform estimate holds
i = 0 m + 2 t i m + 2 | λ | 1 i m + 2 u ( i ) L p ( 0 , 1 ; E ) + A u L p ( 0 , 1 ; E ) M k = 1 2 ( f k E k + | λ | 1 θ k f k E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ19_HTML.gif
(4.4)
Proof In a similar way as in [[40], Theorem 3.1], we obtain the representation of the solution of (4.3)
u ( x ) = t 1 2 p { e x t 1 2 A λ 1 2 [ C 11 + d 11 ( λ , t ) ] + e ( 1 x ) t 1 2 A λ 1 2 [ C 12 + d 12 ( λ , t ) ] } A λ m 1 2 f 1 + t 1 2 p { e x t 1 2 A λ 1 2 [ C 21 + d 21 ( λ , t ) ] + e ( 1 x ) t 1 2 A λ 1 2 [ C 22 + d 22 ( λ , t ) ] } A λ m 2 2 f 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ20_HTML.gif
(4.5)
where C i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq133_HTML.gif and d i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq134_HTML.gif are uniformly bounded operators. Then, in view of positivity of A, we obtain from (4.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ21_HTML.gif
(4.6)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ22_HTML.gif
(4.7)
By changing the variable x t 1 2 = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq135_HTML.gif and in view of Theorem A, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ23_HTML.gif
(4.8)
By using the estimate (4.8), by virtue of Theorem A, we get the uniform estimate
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ24_HTML.gif
(4.9)

Then from (4.6)-(4.9) we obtain (4.4). □

Now we can represent a more general result for nonhomogeneous problem (4.2).

Theorem 4.2 Assume that the following conditions are satisfied:
  1. (1)

    E is a UMD space and A is an R-positive operator in E;

     
  2. (2)

    η = ( 1 ) m 1 α 1 β 2 ( 1 ) m 2 α 2 β 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq136_HTML.gif, θ k = m k m + 2 + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq137_HTML.gif, k = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq138_HTML.gif, 0 < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq125_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq29_HTML.gif.

     
Then the operator u { ( L t + λ ) u , L 1 t u , L 2 t u } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq139_HTML.gifis an isomorphism from W m + 2 , p ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq140_HTML.gifonto W m , p ( 0 , 1 ; E ) × E 1 × E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq141_HTML.giffor λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gifwith large enough ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif. Moreover, the uniform coercive estimate holds
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ25_HTML.gif
(4.10)
Proof The uniqueness of a solution of problem (4.2) is obtained from Lemma 4.2. Let us define
f ¯ ( x ) = { f ( x ) , if  x [ 0 , 1 ] , 0 , if  x [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equy_HTML.gif
We will show that problem (4.2) has a solution u W m + 2 , p ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq142_HTML.gif for f W m , p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq143_HTML.gif, f k E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq132_HTML.gif and u = u 1 + u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq144_HTML.gif, where u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq145_HTML.gif is the restriction on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq146_HTML.gif of the solution of the equation
( L t + λ ) u = f ¯ ( x ) , x R = ( , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ26_HTML.gif
(4.11)
and u 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq147_HTML.gif is a solution of the problem
( L t + λ ) u = 0 , L k t u = f k L k t u 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ27_HTML.gif
(4.12)
A solution of equation (4.11) is given by
u ( x ) = F 1 L 1 ( λ , t , ξ ) F f ¯ = 1 2 π e i ξ x L 1 ( λ , t , ξ ) ( F f ¯ ) ( ξ ) d ξ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equz_HTML.gif
where L ( λ , t , ξ ) = A + t ξ 2 + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq148_HTML.gif. It follows from the above expression that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ28_HTML.gif
(4.13)
It is sufficient to show that the operator-functions
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaa_HTML.gif
are uniform Fourier multipliers in L p ( R ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq149_HTML.gif. Actually, due to the positivity of A, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ29_HTML.gif
(4.14)
It is clear to observe that
ξ d d ξ Ψ t λ ( ξ ) = 2 t ξ 2 A L 2 ( λ , t , ξ ) = [ 2 t ξ 2 L 1 ( λ , t , ξ ) ] A L 1 ( λ , t , ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equab_HTML.gif
Due to R-positivity of the operator A, the sets
{ 2 t ξ 2 [ A + t ξ 2 + λ ] 1 } , { A [ A + t ξ 2 + λ ] 1 } , ξ R { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equac_HTML.gif
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., [36] Lemma 3.5, Proposition 3.4), we obtain
R { ξ i d i d ξ i Ψ t λ ( ξ ) : ξ R { 0 } } C 1 , R { ξ i d i d ξ i σ t λ ( ξ ) : ξ R { 0 } } C 2 , i = 0 , 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ30_HTML.gif
(4.15)
By [[33], Theorem 3.4] it follows that Ψ t , λ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq150_HTML.gif and σ t λ ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq151_HTML.gif are the uniform collection of multipliers in L p ( R ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq149_HTML.gif. Then in view of (4.13) we obtain that problem (4.11) has a solution u W m + 2 , p ( R ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq152_HTML.gif and the uniform coercive estimate holds
j = 0 m + 2 t j m + 2 | λ | 1 j m + 2 u ( j ) L p ( R ; E ) + A u L p ( R ; E ) C f ¯ L p ( R ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ31_HTML.gif
(4.16)
Let u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq145_HTML.gif be the restriction of u on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq153_HTML.gif. The estimate (4.16) implies that u 1 W m + 2 , p ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq154_HTML.gif. By virtue of Lemma 4.1, we get
u 1 ( m k ) ( ) ( E ( A ) ; E ) θ k , p , k = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equad_HTML.gif
Hence, L k t u 1 E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq155_HTML.gif. Thus, by virtue of Lemma 4.2, problem (4.12) has a unique solution u 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq156_HTML.gif that belongs to the space W m + 2 , p ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq157_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ32_HTML.gif
(4.17)
Moreover, from (4.16) we obtain
j = 0 m + 2 t j m + 2 | λ | 1 j m + 2 u 1 ( j ) L p ( 0 , 1 ; E ) + A u 1 L p ( 0 , 1 ; E ) C f W m , p ( 0 , 1 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ33_HTML.gif
(4.18)
Therefore, by Lemma 4.1 and by estimate (4.17), we obtain
t θ k u 1 ( m k ) ( ) E k C u 1 W t m + 2 , p ( 0 , 1 ; E ( A ) , E ) C f W m , p ( 0 , 1 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ34_HTML.gif
(4.19)
So, in view of Lemma 4.1 and estimates (4.17)-(4.19), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ35_HTML.gif
(4.20)

Finally, from (4.18) and (4.20) we obtain (4.10). □

Now, we can prove the main result of this section.

Proof of Theorem 4.1 Let G 2 = ( 0 , b 1 ) × ( 0 , b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq158_HTML.gif. It is clear to see that
W m , q ( G 2 ; E ) = W m , q ( 0 , b 1 ; X 0 , X ) = W m , q ( 0 , b 1 ; X ) L q ( 0 , b 1 ; X 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equae_HTML.gif

where X 0 = W m , q ( 0 , b 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq159_HTML.gif and X = L q ( 0 , b 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq160_HTML.gif.

Let us consider the BVP
ε 1 2 u x 1 2 ε 2 2 u x 2 2 + ( A + λ ) u ( x 1 , x 2 ) = f ( x 1 , x 2 ) , L k j ε u = 0 , k , j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ36_HTML.gif
(4.21)
where L k j ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq80_HTML.gif are defined by equalities (1.5). Problem (4.21) can be expressed as the following BVP for an ordinary DOE:
L u = ε 1 d 2 u d x 1 2 + ( B ε 2 + λ ) u ( x 2 ) = f ( x 2 ) , x 1 ( 0 , b 1 ) , L k 1 ε 1 u = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ37_HTML.gif
(4.22)
where L k 1 ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq161_HTML.gif are boundary conditions of type (3.2), B ε 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq86_HTML.gif is the operator acting in X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq162_HTML.gif and X defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaf_HTML.gif
Since X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq162_HTML.gif and X are UMD spaces, (see, e.g., [[35], Theorem 4.5.2]) by virtue of Theorem 4.2, we obtain that problem (4.22) has a unique solution u W m + 2 , q ( 0 , b 1 ; D ( B ε 2 ) , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq163_HTML.gif for f W m , q ( 0 , b 1 ; X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq164_HTML.gif and λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq13_HTML.gif with sufficiently large ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq14_HTML.gif. Moreover, the coercive uniform estimates holds
i = 0 m + 2 ε 1 i m + 2 | λ | 1 i m + 2 u ( i ) L q ( 0 , b 1 ; X ) + B ε 2 u L q ( 0 , b 1 ; X ) C f W m , q ( 0 , b 1 ; X ) , i = 0 2 ε 1 i 2 | λ | 1 i 2 u ( i ) L q ( 0 , b 1 ; X 0 ) + B ε 2 u L q ( 0 , b 1 ; X 0 ) C f L q ( 0 , b 1 ; X 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ38_HTML.gif
(4.23)
From (4.23) we obtain that problem (4.22) has a unique solution
u W m + 2 , q ( G 2 ; E ( A ) , E ) for  W m , q ( G 2 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equag_HTML.gif
Moreover, the uniform coercive estimates hold
i = 0 m + 2 ε 1 i m + 2 | λ | 1 i m + 2 u ( i ) L q ( 0 , b 1 ; X ) + B ε 2 u L q ( 0 , b 1 ; X ) C f W m , q ( G 2 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ39_HTML.gif
(4.24)
By applying Theorem 4.2 for f k = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq165_HTML.gif and E = X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq166_HTML.gif, we get the following uniform estimate:
j = 0 m + 2 ε 2 i m + 2 | λ | 1 i m + 2 u ( i ) X + A u X C B ε 2 u W m , q ( 0 , b 2 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ40_HTML.gif
(4.25)

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 u W σ 2 , q ( G , L k j ε ) = { u W 2 , q ( G ; R n ) , L k j ε u = 0 , div u = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq167_HTML.gif satisfying equation (1.6) a.e. on G is called the stronger solution of problem (1.6).

Let W s , q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq168_HTML.gif, 0 < s < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq169_HTML.gif be the Sobolev space of order s such that W 0 , q ( G ) = L q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq170_HTML.gif. For q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq73_HTML.gif, let X q = L σ q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq171_HTML.gif denote the closure of C 0 σ ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq172_HTML.gif in L p ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq173_HTML.gif, where
C 0 σ ( G ) = { u C 0 ( G ) , div u = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equah_HTML.gif
It is known that ( see, e.g., Fujiwara and Morimoto [17]) a vector field u L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq174_HTML.gif has the Helmholtz decomposition, i.e., all u L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq174_HTML.gif can be uniquely decomposed as u = u 0 + φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq175_HTML.gif with u 0 L σ q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq176_HTML.gif, u 0 = P q u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq177_HTML.gif, where P q = P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq178_HTML.gif is a projection operator from L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq179_HTML.gif to L σ q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq180_HTML.gif and φ L loc q ( G ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq181_HTML.gif, φ L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq182_HTML.gif, so that
φ q C u q , φ L q ( G B ) C u q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equai_HTML.gif

with C independent of u, where B is an open ball in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq62_HTML.gif and u p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq183_HTML.gif denotes the norm of u in L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq179_HTML.gif or L q ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq184_HTML.gif.

Then problem (1.6) can be reduced to the following BVP:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ41_HTML.gif
(5.1)
Consider the parameter-dependent Stokes operator O ε = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq185_HTML.gif O ε , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq18_HTML.gif generated by problem (5.1), i.e.,
D ( O ε ) = ( W σ 2 , q ( G ; L k j ) ) n , O ε u = P ε u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaj_HTML.gif

From Corollary 3.1 we get that the operator O ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq186_HTML.gif is positive and also is a generator of a bounded holomorphic semigroup S ε ( t ) = exp ( O ε t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq187_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq188_HTML.gif.

In a similar way as in [18], we show the following.

Proposition 5.1 The following estimate holds:
O ε α S ε ( t ) C t α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equak_HTML.gif

uniformly in ε = ( ε 1 , ε 2 , , ε n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq104_HTML.giffor α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq189_HTML.gifand t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq188_HTML.gif.

Proof From Result 3.1 we obtain that the operator O ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq190_HTML.gif is uniformly positive in L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq191_HTML.gif, i.e., for λ S ψ , ϰ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq192_HTML.gif, 0 < ψ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq193_HTML.gif, the following estimate holds:
( O ε + λ ) 1 M | λ | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equal_HTML.gif

where the constant M is independent of λ and ε. Then, by using the Danford integral and operator calculus as in [18], we obtain the assertion. □

Now consider problem (1.7). The main theorem in this section is the following.

Theorem 5.1 Let 0 < ε k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq96_HTML.gif, ( 1 ) m k 1 α k 1 β k 2 ( 1 ) m k 2 α k 2 β k 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq194_HTML.gifand p , q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq23_HTML.gif. Then there is a unique solution ( u , φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq24_HTML.gifof problem (1.7) for f L p ( 0 , T ; L q ( G ; R n ) ) = B ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq195_HTML.gifand a B p , q 2 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq196_HTML.gif. Moreover, the following uniform estimate holds:
u t B ( p , q ) + k = 1 n ε k 2 u x k 2 B ( p , q ) + φ B ( p , q ) C f B ( p , q ) + a B p , q 2 2 p ( G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ42_HTML.gif
(5.2)

with C = C ( T , G , p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq197_HTML.gifindependent of f and ε.

Proof Problem (1.7) can be expressed as the following abstract parabolic problem with a small parameter:
d u d t + O ε u = f ( t ) , u ( 0 ) = a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ43_HTML.gif
(5.3)
If we put E = L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq198_HTML.gif and A = ϰ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq101_HTML.gif in Theorem 3.1, then the Result 3.1 implies that the operator O ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq199_HTML.gif is uniformly positive and generates bounded holomorphic semigroup in L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq179_HTML.gif uniformly in ε k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq10_HTML.gif. Moreover, by using [[41], Theorem 3.1] we get that operator O ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq186_HTML.gif is R-positive in E. Since E is a UMD space, in a similar way as in [[33], Theorem 4.2], we obtain that for f L p ( 0 , T ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq200_HTML.gif and a ( D ( O ε ) , E ) 1 p , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq201_HTML.gif, there is a unique solution u W 1 , p ( 0 , T , D ( O ε ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq202_HTML.gif of problem (5.3) so that the following uniform estimate holds:
d u d t L p ( 0 , T ; E ) + O ε u L p ( 0 , T ; E ) C ( f L p ( 0 , T ; E ) + a ( D ( A ε ) , E ) 1 p , p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ44_HTML.gif
(5.4)
From (5.4) for all u W σ 2 , q ( G , L k j ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq203_HTML.gif, we get the following estimate:
k = 1 n ε k 2 u x k 2 q C A ε u q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equam_HTML.gif

uniformly in ε = ( ε 1 , ε 2 , , ε n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq104_HTML.gif. □

6 Existence and uniqueness for the Navier-Stokes equation with parameters

In this section, we study the Navier-Stokes problem (1.1)-(1.3) in the space X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq204_HTML.gif. Problem (1.1)-(1.3) can be expressed as
d u d t + O ε u = F u + P f , u ( 0 ) = 0 , t > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ45_HTML.gif
(6.1)
where
F u = P ( u , ) u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equan_HTML.gif
We consider equation (6.1) in an integral form
u ( t ) = S ε ( t ) a + 0 t S ε ( t s ) [ F u ( s ) + P f ( s ) ] d s , t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ46_HTML.gif
(6.2)

To prove the main result, we need the following result which are obtained in a similar way as in [[11], Theorem 2].

Lemma 6.1 For any 0 α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq205_HTML.gif, the domain D ( O ε α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq206_HTML.gifis the complex interpolation space [ X q , D ( O ε ) ] α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq207_HTML.gif.

Lemma 6.2 For each k = 1 , 2 , , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq76_HTML.gif, the operator u O ε 1 2 P ( x k ) u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq208_HTML.gifextends uniquely to a uniformly bounded linear operator from L q ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq179_HTML.gifto X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq204_HTML.gif.

Proof Since O ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq186_HTML.gif is a positive operator, it has fractional powers O ε α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq209_HTML.gif. From Lemma 6.1, it follows that the domain D ( O ε α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq210_HTML.gif is continuously embedded in X q H q 2 α ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq211_HTML.gif for any α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq212_HTML.gif, where H q 2 α ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq213_HTML.gif is the vector-valued Bessel space. Then, by using the duality argument and due to uniform positivity of O ε 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq214_HTML.gif, we obtain the following uniformly in ε estimate:
O ε 1 2 P ( x k ) u L q ( G ; R n ) C u X q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ47_HTML.gif
(6.3)

By reasoning as in [12], we obtain the following. □

Lemma 6.3 Let 0 δ < 1 2 + n 2 ( 1 1 q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq215_HTML.gif. Then the following estimate holds:
ε O ε δ P ( u , ) υ q M O ε θ u q O ε σ u q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equao_HTML.gif
uniformly in ε with some constant M = M ( δ , θ , q , σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq216_HTML.gifprovided that θ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq217_HTML.gif, σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq218_HTML.gif, σ + δ > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq219_HTML.gifand
θ + σ + δ > n 2 q + 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equap_HTML.gif
Proof Assume that 0 < ν < n 2 ( 1 1 q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq220_HTML.gif. Since D ( O ε α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq206_HTML.gif is continuously embedded in X q H q 2 α ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq221_HTML.gif, and since L q ( G ; R n ) X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq222_HTML.gif is the same as X s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq223_HTML.gif, by the Sobolev embedding theorem, we obtain that the operator
O ε , q ν : X q D ( O ε , q ν ) X s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaq_HTML.gif
is bounded, where
1 s = 1 q 2 ν n , 1 q + 1 q = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equar_HTML.gif
By the duality argument then, we get that the operator u O ε , q ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq224_HTML.gif is bounded from X s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq225_HTML.gif to X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq204_HTML.gif, where
1 s = 1 1 s = 1 q + 2 ν n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equas_HTML.gif
Consider first the case δ > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq226_HTML.gif. Since P ( u , ) υ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq227_HTML.gif is bilinear in u, υ, it suffices to prove the estimate on a dense subspace. Therefore, assume that u and υ are smooth. Since div u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq228_HTML.gif, we get
( u , ) υ = k = 1 n x k ( u k υ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equat_HTML.gif
Taking ν = δ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq229_HTML.gif, using the uniform boundedness of O ε , q ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq230_HTML.gif from X s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq225_HTML.gif to X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq204_HTML.gif and Lemma 6.2 for all ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq231_HTML.gif, we obtain
ε O ε δ P ( u , ) υ q = ε O ε , q 1 2 ν k = 1 n P x k ( u k υ ) q | u | | υ | s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equau_HTML.gif
By assumption we can take r and η such that
1 r 1 q 2 θ n , 1 η 1 q 2 σ n , 1 r + 1 η = 1 s , r > 1 , η < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equav_HTML.gif
Since D ( O ε , q α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq232_HTML.gif is continuously embedded in X q H q 2 α ( G ; R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq211_HTML.gif, by the Sobolev embedding, we get
| u | | υ | s u r υ η M O ε , q θ u r O ε , q σ υ η , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaw_HTML.gif
i.e., we have the required result for δ > 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq226_HTML.gif. In particular, we get the following uniform estimate:
ε O ε 1 2 P ( u , ) υ q M O ε , q θ u r O ε , q σ υ η , θ + β n 2 q , β > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equax_HTML.gif
Similarly, we obtain
ε P ( u , ) υ q C u r υ η C O ε , q θ u r O ε , q β + 1 2 υ η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equay_HTML.gif
for 1 r + 1 η = 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq233_HTML.gif and δ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq234_HTML.gif. The above two estimates show that the map υ P ( u , ) υ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq235_HTML.gif is a uniform bounded operator from D ( O ε β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq236_HTML.gif to D ( O ε 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq237_HTML.gif and from D ( O ε β + 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq238_HTML.gif to X q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq204_HTML.gif. By using Lemma 6.1 and the interpolation of Banach spaces [[30], §1.3.2] for 0 δ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq239_HTML.gif, we obtain
ε P ( u , ) υ q C O ε , q θ u r O ε , q σ υ η . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equaz_HTML.gif

By using Lemma 6.3 and the iteration argument, by reasoning as in Fujita and Kato [18], we obtain the following. □

Theorem 6.1 Let 0 < ε k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq96_HTML.gif, ( 1 ) m k 1 α k 1 β k 2 ( 1 ) m k 2 α k 2 β k 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq194_HTML.gif. Let γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq240_HTML.gifbe a real number and δ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq241_HTML.gifsuch that
n 2 q 1 2 γ , γ < δ < 1 | γ | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equba_HTML.gif
Suppose that a D ( O ε γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq242_HTML.gif, and that O ε δ P f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq243_HTML.gifis continuous on ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq244_HTML.gifand satisfies
O ε δ P f ( t ) = o ( t γ + δ 1 ) as t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbb_HTML.gif
Then there is T ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq245_HTML.gifindependent of ε and a local solution of (6.2) such that
  1. (1)

    u C ( [ 0 , T ] ; D ( O ε γ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq246_HTML.gif, u ( 0 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq247_HTML.gif;

     
  2. (2)

    u C ( ( 0 T ] ; D ( O ε α ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq248_HTML.gif for some T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq249_HTML.gif;

     
  3. (3)

    O ε α u ( t ) = o ( t γ α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq250_HTML.gif as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq251_HTML.gif for all α with γ < α < 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq252_HTML.gif uniformly with respect to the parameter ε.

     
Moreover, the solution of (5.2) is unique if
  1. (4)

    u C ( ( 0 T ] ; D ( O ε β ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq253_HTML.gif;

     
  2. (5)

    O ε α u ( t ) = o ( t γ β ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq254_HTML.gif as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq251_HTML.gif for some β with β > | γ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq255_HTML.gif uniformly in ε.

     
Proof We introduce the following iteration scheme:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ48_HTML.gif
(6.4)
By estimating the term u 0 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq256_HTML.gif in (6.4) and by using Proposition 5.1 for γ α < 1 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq257_HTML.gif, we get
O ε α u 0 ( t ) O ε α S ε ( t ) a + 0 t O ε α + δ S ε ( t s ) O ε δ P f ( s ) d s O ε α S ε ( t ) a + C α + δ 0 t ( t s ) ( α + δ ) O ε δ P f ( s ) d s M α t γ α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbc_HTML.gif
uniformly in ε with
M α = sup 0 < t T , ε > 0 t α γ O ε α + δ S ε ( t ) a + C α + δ N B ( 1 δ α , γ + α ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbd_HTML.gif
where N = sup 0 < t T t 1 γ δ O ε δ P f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq258_HTML.gif and B ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq259_HTML.gif is the beta function. Here we suppose γ + δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq260_HTML.gif. By induction assume that u m ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq261_HTML.gif satisfies the following estimate:
O ε α u m ( t ) M α m t γ α , γ α < 1 δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equ49_HTML.gif
(6.5)
We will estimate O ε α u m + 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq262_HTML.gif by using (6.2). To estimate the term O ε δ F u m ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_IEq263_HTML.gif, we suppose
θ + σ + δ = 1 + γ , γ < θ < 1 δ , γ < σ < 1 δ , θ > 0 , σ > 0 , δ + σ > 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Eqube_HTML.gif
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:
O ε δ F u m ( s ) C M θ m M σ m s γ + δ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbf_HTML.gif
Therefore, we obtain
O ε α u m ( t ) M α t γ α + M α + δ 0 t ( t s ) ( α + δ ) O ε δ F u m ( s ) d s M α m + 1 t γ α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbg_HTML.gif
with
M α m + 1 = M α + M α + δ M B ( 1 δ α , γ + δ ) M θ m M σ m . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-107/MediaObjects/13661_2013_Article_368_Equbh_HTML.gif

Since we get the uniform estimates with respect to the parameter ε, the remaining part of the proof is the same as in [[12], Theorem 2.3], so this part is omitted. □

Declarations

Acknowledgements

Dedicated to International Conference on the Theory, Methods and Applications of Nonlinear Equations in Kingsville, TX-USA Texas A&M University-Kingsville-2012.

Authors’ Affiliations

(1)
Department of Mechanical Engineering, Okan University, Akfirat
(2)
Institute of Mathematics and Mechanics, Azerbaijan National Akademy of Science

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