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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research Article

Parameter-dependent Stokes problems in vector-valued spaces and applications

Veli B Shakhmurov

Author Affiliations

Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul, 34959, Turkey

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

Boundary Value Problems 2013, 2013:172  doi:10.1186/1687-2770-2013-172


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/172


Received:8 March 2013
Accepted:28 June 2013
Published:23 July 2013

© 2013 Shakhmurov; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The stationary and instationary Stokes equations with operator coefficients in abstract function spaces are studied. The problems are considered in the whole space, and equations include small parameters. The uniform separability of these problems is established.

MSC: 35Q30, 76D05, 34G10, 35J25.

Keywords:
Stokes systems; Navier-Stokes equations; differential equations with small parameters; semigroups of operators; boundary value problems; differential-operator equations; maximal <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M1">View MathML</a> regularity

Dedication

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

1 Introduction

We consider the initial value problem (IVP) for the following Stokes equation with small parameter:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M2">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M3">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M4">View MathML</a>, A is a linear operator in a Banach space E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5">View MathML</a> are a small positive parameters. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M7">View MathML</a> are E-valued unknown solutions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M8">View MathML</a> is a given function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M9">View MathML</a> is an initial date. This problem is characterized by the presence of an abstract operator A and small terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5">View MathML</a> which correspond to the inverse of Reynolds number Re very large. We prove that problem (1.1)-(1.2) has a unique strong maximal regular solution u on a time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M11">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M15">View MathML</a>, problem (1.1)-(1.2) is reduced to the Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M16">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M17">View MathML</a>

(1.4)

where ℂ is the set of complex numbers and b is a positive constant.

Note that the existence of weak or strong solutions and regularity properties for the classical Stokes problems has been extensively studied, e.g., in [1-10]. There is an extensive literature on the solvability of the IVPs for the Stokes equation (see, e.g., [1,3,10] and further papers cited there). Solonnikov [8] proved that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M19">View MathML</a>, the time-dependent Stokes problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M20">View MathML</a>

(1.5)

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M21">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M22">View MathML</a>

Then Giga and Sohr [3] improved the result of Solonnikov for spaces with different exponents in space and time, i.e., they proved that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M23">View MathML</a> there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M24">View MathML</a> of problem (1.5) so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M25">View MathML</a>

(1.6)

Moreover, the estimate obtained was global in time, i.e., the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M26">View MathML</a> is independent of T and f. To derive global <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M27">View MathML</a> estimates (1.6), Giga and Sohr used the abstract parabolic semigroup theory in UMD-type Banach spaces. Estimate (1.6) allows to study the existence of a solution and regularity properties of the corresponding Navier-Stokes problem (see, e.g., [5]).

In this paper, first we consider the following differential operator equation (DOE) with small parameters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M28">View MathML</a>

(1.7)

where A is a linear operator in a Banach space E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M29">View MathML</a> are positive and λ is a complex parameter.

We show that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M31">View MathML</a>, problem (1.7) has a unique solution u belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M32">View MathML</a> and the uniform coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M34">View MathML</a> is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M35">View MathML</a>, λ and f.

We consider, then, the stationary abstract Stokes problem with small parameters

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M36">View MathML</a>

(1.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M37">View MathML</a> is data and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M38">View MathML</a> is a solution. By applying the projection transformation P, equation (1.8) can be reduced to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M39">View MathML</a>

(1.9)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40">View MathML</a> denote the operator generated by problem (1.9), i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40">View MathML</a> is a Stokes operator in solenoidal space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M42">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M43">View MathML</a>

We prove that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M40">View MathML</a> is uniformly positive and also is a generator of an analytic semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M45">View MathML</a>. Finally, the instationary Stokes problem (1.1)-(1.3) is considered and the well-posedness of this problem is derived. In application we show the separability properties of the anisotropic stationary Stokes operator in mixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M46">View MathML</a> spaces and maximal regularity properties of infinity system of instationary Stokes equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M1">View MathML</a> spaces.

2 Notations and background

Let E be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M48">View MathML</a> denote the space of strongly measurable E-valued functions that are defined on the measurable subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M49">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M50">View MathML</a>

The Banach space E is called a UMD-space if the Hilbert operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M51">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M19">View MathML</a> (see, e.g., [11]). UMD spaces include, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M55">View MathML</a> spaces and Lorentz spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M57">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M59">View MathML</a> be two Banach spaces. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M60">View MathML</a> denotes the space of bounded linear operators from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M58">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M59">View MathML</a> endowed with the usual uniform operator topology. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M63">View MathML</a>, it is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M64">View MathML</a>. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M67">View MathML</a>, denotes interpolation spaces defined by the K method [[12], §1.3.1].

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M68">View MathML</a>

A linear operator A is said to be ψ-positive in a Banach space E with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M69">View MathML</a> if the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M70">View MathML</a> is dense on E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M71">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73">View MathML</a>, where I is the identity operator in E. It is known [[12], §1.15.1] that there exist the fractional powers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M74">View MathML</a> of the positive operator A. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M75">View MathML</a> denote the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M76">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M77">View MathML</a>

ℕ denotes the set of natural numbers. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M78">View MathML</a> is called R-bounded (see, e.g., [11]) if there is a positive constant C such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M81">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M82">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M83">View MathML</a> is a sequence of independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M84">View MathML</a>-valued random variables on Ω. The smallest C, for which the estimate above holds, is called an R-bound of the collection G and denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M85">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M86">View MathML</a> depending of parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M87">View MathML</a> is called uniform R-bounded with respect to h if there is a constant C, independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M87">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M81">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M92">View MathML</a>

It implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M93">View MathML</a>.

The ψ-positive operator A is said to be R-positive in a Banach space E if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73">View MathML</a>, is R-bounded.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M96">View MathML</a> is said to be ψ-positive in E uniformly with respect to t with bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M69">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M98">View MathML</a> is independent of t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M99">View MathML</a> is dense in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M100">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73">View MathML</a>, where M does not depend of t and λ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M103">View MathML</a> and E be two Banach spaces and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M103">View MathML</a> be continuously and densely embedded into E. Let Ω be a measurable set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M105">View MathML</a> and m be a positive integer. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106">View MathML</a> denotes the class of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M107">View MathML</a> that have the generalized derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M108">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M109">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M112">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M114">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M115">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M106">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M117">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M119">View MathML</a> denote an E-valued Liouville space of order s, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M120">View MathML</a>

where F and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M121">View MathML</a> denote the Fourier and inverse Fourier transforms, respectively.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M122">View MathML</a> be a Liouville-Lions type space, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M123">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M124">View MathML</a> we define the parameter-dependent norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M126">View MathML</a>

Sometimes we use one and the same symbol C without distinction 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M127">View MathML</a>.

3 Boundary value problems for abstract elliptic equations

In this section, we derive the maximal regularity properties of problem (1.7).

BVPs for DOEs were studied, e.g., in [9,11,13-19]. For references, see, e.g., [19]. From [[18], Theorem 4.1] we have the following result.

Theorem 3.1LetEbe aUMDspace and letAbe anR-positive operator inEfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M73">View MathML</a>. Then problem (1.7) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M129">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M130">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>. Moreover, the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M132">View MathML</a>

(3.1)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M34">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M134">View MathML</a>, λandf.

Consider the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M135">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136">View MathML</a> generated by problem (3.1), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M137">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M138">View MathML</a>. From Theorem 3.1 we obtain the following.

Result 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, there is a resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M140">View MathML</a> satisfying the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M141">View MathML</a>

Next we show the smoothness of problem (3.1). The main result is the following.

Theorem 3.2Assume thatEis aUMDspace, Ais anR-positive operator inE, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>andmis a positive integer.

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, problem (3.1) has a unique solutionuthat belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M145">View MathML</a>and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M146">View MathML</a>

(3.2)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M147">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M148">View MathML</a>, λandf.

Proof A solution of equation (1.7) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M149">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M151">View MathML</a>. It follows from the expression above that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M152">View MathML</a>

(3.3)

It is sufficient to show that the operator-functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M153">View MathML</a>

are uniform Fourier multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136">View MathML</a>. Actually, due to positivity of A, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M155">View MathML</a>

(3.4)

It is clear to observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M156">View MathML</a>

Due to R-positivity of the operator A, the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M157">View MathML</a>

are R-bounded. So, in view of Kahane’s contraction principle and from the product properties of the collection of R-bounded operators (see, e.g., [11], Lemma 3.5, Proposition 3.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M158">View MathML</a>

Namely, the R-bounds of sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M159">View MathML</a> are independent of ε and λ. Next, let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M160">View MathML</a>. It is clear to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M161">View MathML</a>

By using the well-known inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M162">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M164">View MathML</a>, we get the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M165">View MathML</a>

(3.5)

From (3.4) and (3.5) we have the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M166">View MathML</a>

Due to R-positivity of the operator A, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M167">View MathML</a>

is R-bounded. By using this fact, in view of (3.4) and Kahane’s contraction principle, we obtain the R-boundedness of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M168">View MathML</a>. In a similar way, we obtain the uniform estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M169">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M170">View MathML</a>

By the aid of the estimates above, due to R-positivity of the operator A, in view of estimate (3.4), by virtue of Kahane’s contraction principle, from the additional and product properties of the collection of R-bounded operators, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M172">View MathML</a> and independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M173">View MathML</a>-valued random variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M176">View MathML</a>, we obtain the uniform estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M177">View MathML</a>

The same estimates are obtained for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M178">View MathML</a> in a similar way. Hence, by virtue of [[11], Theorem 3.4] it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M180">View MathML</a> are the uniform collection of multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M181">View MathML</a>. Then, by using equality (3.3), we obtain the assertion. □

4 The stationary Stokes system with small parameters

In this section we derive the maximal regularity properties of the stationary abstract Stokes problem (1.8).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M119">View MathML</a> denote the E-valued Liouville space of order s such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M184">View MathML</a>. It is known that if E is a UMD space, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M185">View MathML</a> for a positive integer m (see, e.g., [[20], §15]). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a> let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M187">View MathML</a> denote the space of an E-valued system of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M188">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M189">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M190">View MathML</a> denotes the E-valued solenoidal space, i.e., closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M191">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M192">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M193">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M194">View MathML</a>

Let E be a Banach space. Consider the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M195">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M196">View MathML</a> becomes a Banach space with this norm.

It is known that (see, e.g., Fujiwara and Morimoto [4]) the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M197">View MathML</a> has a Helmholtz decomposition. In the following theorem we generalize this result for an E-valued function space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>.

Theorem 4.1LetEbe aUMDspace and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M200">View MathML</a>has a Helmholtz decomposition, i.e., there exists a bounded linear projection operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M203">View MathML</a>with the null space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M204">View MathML</a>. In particular, all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M200">View MathML</a>has the unique decomposition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M206">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M208">View MathML</a>so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M209">View MathML</a>

(4.1)

for any open ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M210">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M212">View MathML</a>.

To prove Theorem 4.1, we need some lemmas. Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M213">View MathML</a>

(4.2)

Lemma 4.1LetEbe aUMDspace, letAbe anR-positive operator inE, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M215">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, problem (4.2) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M218">View MathML</a>and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M219">View MathML</a>

(4.3)

Proof By using the Fourier transform, we see that estimate (4.3) is equivalent to the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M220">View MathML</a>

(4.4)

To prove (4.4) it is sufficient to show that the operator functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M221">View MathML</a>

are multipliers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136">View MathML</a> uniformly in λ and ε. This fact is derived as the step in the proof of Theorem 3.2. □

Now consider the system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M223">View MathML</a>

(4.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M38">View MathML</a> is a solution of (4.5).

We define in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M226">View MathML</a> the following parameter-dependent norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M227">View MathML</a>

Lemma 4.2LetEbe aUMDspace, letAbe anR-positive operator inE, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M215">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, problem (4.5) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M232">View MathML</a>and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M233">View MathML</a>

(4.6)

Proof Problem (4.5) can be expressed as the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M234">View MathML</a>

(4.7)

By Lemma 4.1 we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, equation (4.7) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M237">View MathML</a> and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M238">View MathML</a>

Hence, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M239">View MathML</a> is a unique solution of problem (4.5) and (4.3) implies (4.6). □

By reasoning as in [[6], Lemma 2], we get the following lemma.

Lemma 4.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M240">View MathML</a>is dense in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M241">View MathML</a>.

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M242">View MathML</a>

(4.8)

From Lemma 4.2 we obtain the following results.

Result 4.1 Let E be a UMD space, let A be an R-positive operator in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, problem (4.8) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M246">View MathML</a> and the following coercive uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M247">View MathML</a>

Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M248">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M249">View MathML</a>

where φ is a solution of problem (4.8).

Result 4.2 Let E be a UMD space, let A be an R-positive operator in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M251">View MathML</a> is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>.

Lemma 4.4LetEbe aUMDspace, letAbe anR-positive operator inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M254">View MathML</a>is a bounded linear operator in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M256">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M257">View MathML</a>.

Proof The linearity of the operator P is clear by construction. Moreover, by Result 4.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M258">View MathML</a>

(4.9)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M259">View MathML</a>, then by Lemma 4.2 we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M260">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M256">View MathML</a>. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M262">View MathML</a> denote the dual space of E.

Lemma 4.5Assume thatEis aUMDspace and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263">View MathML</a>. Then the conjugate of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201">View MathML</a>is defined as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M266">View MathML</a>and is bounded linear in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M267">View MathML</a>.

Proof It is known (see, e.g., [13,20]) that the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M136">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M269">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M270">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M271">View MathML</a>, we only have to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M272">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M273">View MathML</a>. But this is derived by reasoning as in [[4], Lemma 5]. Moreover, by Lemma 4.4, the dual operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M274">View MathML</a> is bounded linear in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M275">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M276">View MathML</a>

 □

From Lemmas 4.4, 4.5 we obtain the following result.

Result 4.3 Assume that E is a UMD space, A is an R-positive operator in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then any element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M278">View MathML</a> uniquely can be expressed as the sum of elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M280">View MathML</a>.

In a similar way to Lemmas 6, 7 of [4] we obtain, respectively, the following lemmas.

Lemma 4.6AssumeEis aUMDspace and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M282">View MathML</a>

Lemma 4.7AssumeEis aUMDspace and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M263">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M284">View MathML</a>

Now we are ready to prove Theorem 4.1.

Proof of Theorem 4.1 From Lemmas 4.6, 4.7 we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M285">View MathML</a>. Then, by construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M287">View MathML</a>. By Lemmas 4.2, 4.4, we obtain estimate (4.1). Moreover, by Result 4.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M280">View MathML</a> is a close subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>. It is known that the dual space of the quotient space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M290">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M291">View MathML</a>. By the first assertion we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M292">View MathML</a>, and by Lemma 4.7 we obtain the second assertion. □

Theorem 4.2LetEbe aUMDspace, letAbe anR-positive operator inE, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M142">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M72">View MathML</a>, problem (1.8) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M297">View MathML</a>and the uniform coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M298">View MathML</a>

(4.10)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M147">View MathML</a>independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M148">View MathML</a>, λandf.

Proof By applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M201">View MathML</a> to equation (1.8), we get problem (1.9). It is clear to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M302">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M303">View MathML</a> is the Stokes operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M304">View MathML</a> is an operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a> generated by problem (4.5) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M306">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M307">View MathML</a>

 □

Then by Lemma 4.2 we obtain the assertion.

Result 4.4 From Theorem 4.2 we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M135">View MathML</a> is a positive operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a> and it also generates a bounded holomorphic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M310">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M311">View MathML</a>.

In a similar way to that in [21] we show the following.

Proposition 4.1The following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M312">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M313">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M314">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M311">View MathML</a>.

Proof From Theorem 4.2 we obtain that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M316">View MathML</a> is uniformly positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a>, i.e., the following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M318">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M320">View MathML</a>, where the constant M is independent of λ and ε. Hence, by using Danford integral and operator calculus (see, e.g., in [11]) we obtain the assertion. □

5 Well-posedness of the instationary parameter-dependent Stokes problem

In this section, we show the uniform well-posedness of problem (1.1)-(1.2).

Theorem 5.1Assume thatEis aUMDspace, Ais anR-positive operator inEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M321">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M322">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M324">View MathML</a>, there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M325">View MathML</a>of problem (1.1)-(1.2) and the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M326">View MathML</a>

(5.1)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M327">View MathML</a>independent offandε.

Proof Problem (1.1)-(1.2) can be expressed as the following parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M328">View MathML</a>

(5.2)

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M329">View MathML</a>, then by Proposition 4.1 operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M316">View MathML</a> is uniformly positive and generates a bounded holomorphic semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M198">View MathML</a> uniformly in ε. Moreover, by using [[15], Theorem 3.1] we get that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M332">View MathML</a> is R-positive in E uniformly in ε. Since E is a UMD space, in a similar way to that in [[22], Theorem 4.2] we obtain that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M334">View MathML</a>, there is a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M335">View MathML</a> of problem (5.2) such that the following uniform estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M336">View MathML</a>

(5.3)

 □

From estimates (4.10) and (5.3) we obtain the assertion.

Result 5.1 It should be noted that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M337">View MathML</a>, then we obtain maximal regularity properties of an abstract Stokes problem without any parameters in principal part.

Remark 5.2 There are a lot of positive operators in concrete Banach spaces. Therefore, putting in (1.8) and (1.1) concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of A, by virtue of Theorem 4.2 and Theorem 5.1, we can obtain the maximal regularity properties of a different class of stationary and instationary Stokes problems which occur in numerous physics and engineering problems.

6 Separability properties of anisotropic Stokes equations

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M338">View MathML</a> be an open connected set with compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M339">View MathML</a>-boundary Ω. Consider the BVP for the following anisotropic Stokes equations with parameters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M340">View MathML</a>

(6.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M341">View MathML</a>

(6.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M342">View MathML</a>

(6.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M343">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M344">View MathML</a> are complex-valued functions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M345">View MathML</a>

are unknown solutions and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M346">View MathML</a>

is a given function;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M347">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5">View MathML</a> are positive and λ is a complex parameter.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M352">View MathML</a> will denote the space of all p-summable scalar-valued functions with mixed norm (see, e.g., [[12], §1], i.e., the space of all measurable functions f defined on G, for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M353">View MathML</a>

Analogously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M354">View MathML</a> denotes the anisotropic Sobolev space with a corresponding mixed norm [[12], §10]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M355">View MathML</a>. From Theorem 4.2 we obtain the following result.

Theorem 6.1Let the following conditions be satisfied:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M356">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M357">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M358">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M359">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M360">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M361">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M362">View MathML</a>for eachj, βand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M363">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M364">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M366">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M367">View MathML</a>is normal toΩ;

(3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M369">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M371">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M372">View MathML</a>let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M373">View MathML</a>;

(4) for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M374">View MathML</a>, the local BVP in local coordinates corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M375">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M376">View MathML</a>

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M377">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M378">View MathML</a>, and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M379">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M380">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M382">View MathML</a>with sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M383">View MathML</a>problem (6.1)-(6.3) has a unique solutionubelonging to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M384">View MathML</a>and the uniform coercive estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M385">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M386">View MathML</a>. By virtue of [[11], Theorem 3.6], E is a UMD space. Consider the operator A in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M387">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M388">View MathML</a>

Problem (6.1)-(6.3) can be rewritten in the form (1.8), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M389">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M390">View MathML</a> are vector-functions with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M386">View MathML</a>. By virtue of [[11], Theorem 8.2] the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M392">View MathML</a>

has a unique solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M393">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M395">View MathML</a>. Moreover, the operator A is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M396">View MathML</a>, i.e., all the conditions of Theorem 4.2 hold. So, we obtain the assertion. □

7 Infinite system of Stokes equations with small parameters

Consider the IVB for the following infinite system of instationary Stokes equations with small parameters:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M397">View MathML</a>

(7.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M5">View MathML</a> are positive parameters. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M400">View MathML</a> are unknown solutions, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M401">View MathML</a> is a given function. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M402">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M403">View MathML</a> is a class of functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M404">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M405">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M406">View MathML</a>. From Theorem 5.1 we obtain the following.

Theorem 7.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M321">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M408">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M409">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M410">View MathML</a>there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M411">View MathML</a>of problem (7.1) belonging to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M412">View MathML</a>and the following uniform coercive estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M413">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M327">View MathML</a>independent offandε.

Proof Really, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M415">View MathML</a>, A and be an infinite matrix, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M416">View MathML</a>

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M417">View MathML</a>

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M418">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M419">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M420">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M421">View MathML</a> and independent symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M84">View MathML</a>-valued random variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M423">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M425">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M426">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M427">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M428">View MathML</a>, from the above we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M429">View MathML</a>

i.e., the operator A is R-positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/172/mathml/M430">View MathML</a>. Therefore, all the conditions of Theorem 5.1 hold and we obtain the assertion. □

Competing interests

The author declares that they have no competing interests.

Authors’ contributions

All results belong to VS.

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