Time periodic problems for Navier-Stokes equations in domains with cylindrical outlets to infinity ; Navjė-Stokso lygčių periodiniai laiko atžvilgiu uždaviniai srityse su cilindriniais išėjimais į begalybę

Sujets

• Mathematics
• Volterra integral equation

Informations

• Publié par: vilnius_gediminas_technical_university
• Publié le: 01 janvier 2008
• Nombre de lectures: 12
• Langue: English

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VILNIUS GEDIMINAS TECHNINICAL UNIVERSITETY
INSTITUTE OF MATHEMATICS AND INFORMATICS
Vaidas KEBLIKAS
TIME PERIODIC PROBLEMS FOR
NAVIER STOKES EQUATIONS IN DOMAINS
WITH CYLINDRICAL OUTLETS TO
INFINITY
Doctoral Dissertation
Physical Sciences, Mathematics (01P)
Vilnius 2008
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VILNIAUS GEDIMINO TECHNIKOS UNIVERSITETAS
MATEMATIKOS IR INFORMATIKOS INSTITUTAS
Vaidas KEBLIKAS
˙ ˇNAVJE STOKSO LYGCIU ˛ PERIODINIAI
LAIKO ATŽVILGIU UŽDAVINIAI SRITYSE
˙SU CILINDRINIAIS IŠEJIMAIS I˛ BEGALYBE ˛
Daktaro disertacija
Fiziniai mokslai, matematika (01P)
Vilnius 2008
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The Dissertation was prepared at Institute of Mathematics and Informatics in
2003–2008.
Scientific supervisor
Prof Dr Habil Konstantinas PILECKAS (Institute of mathematics and
informatics, physical sciences, mathematics –01P).
http://leidykla.vgtu.lt
VGTU leidyklos TECHNIKA 1490 M mokslo literaturos¯ knyga
ISBN 978 9955 28 213 6
c? Keblikas V., 2008
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Abstract
The research area of current PhD thesis is the analysis of the time peri-
odic Navier–Stokes equations in domains with cylindrical outlets to infinity
(system of pipes). The objects of investigation is so called non-stationary
Poiseuille solution in the straight cylinder, Stokes and Navier-Stokes equa-
tions in domains with cylindrical outlets to infinity. First in this thesis is
proved the existence and uniqueness of the non-stationary Poiseuille solution
in Hiolder spaces. Then the existence and uniqueness of the time periodic
Stokes problem obtained in weighted Sobolev spaces. Finally, the existence
of time periodic solutions to Navier-Stokes problem in weighted Sobolev
spaces is proved. The weight-function describes asymptotical behavior of
solutions, then jxj ! 1. The obtained results are theoretical. However,
they could be used to solve practical problems of fluid dynamics.
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Reziume˙
Disertacijoje nagrinejami˙ Navje-Stokso˙ lygčiu˛ periodiniai laiko atžvilgiu
uždaviniai srityse su cilindriniais išejimais˙ i˛ begalybę. Pagrindiniai tyrimo
objektai yra taip vadinami Puazelio sprendiniai tiesiame cilindre ir Stokso,
beiNavje-Stokso˙ lygčiu˛sistemoscilindru˛sistemoje. Pirmiausiadarbe˛roi do-
mas Puazelio sprendinio egzistavimas ir vienatis Hiolderio erdvese.˙ Tada
˛roi domas periodinis Stokso uždavinio sprendinio egzistavimas svorinese˙ So-
bolevoerdvese.˙ Irgaliausiai, i˛rodytasperiodiniosprendinioNavje-Stokso˙ už-
daviniuiegzistavimassvorinese˙ Sobolevoerdvese.˙ Svorine˙ funkcijaapibudina¯
sprendiniu˛ nykimo greiti˛, kaijxj!1. Gauti rezultatai yra teoriniai, tačiau
gali buti pritaikyti skysčiu˛ dinamikos praktiniams uždaviniams spręsti.
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Contents
Abstract i
Reziume˙ iii
Introduction 1
1 Function spaces and auxiliary results 7
1.1 Function spaces and main notations . . . . . . . . . . . . . . . . . 7
1.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Non stationary Poiseuille solution 17
2.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 17
2.2 Construction of an approximate solution . . . . . . . . . . . . . . 20
2.3 Uniform estimates for the solution to integral equation(2:17) . . . 23
2.4 Existence of a solution to problem (2.11) . . . . . . . . . . . . . . 28
3 Time periodic Stokes problem 37
3.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 37
3.2 Solvability of problem (3.1) with zero fluxes . . . . . . . . . . . . 38
3.3 Estimates of the solution in weighted function spaces . . . . . . . . 44
3.4 Time periodic Stokes problem with nonzero fluxes . . . . . . . . . 50
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vi Contents
4 Two dimensional time periodic Navier Stokes problem 55
4.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 55
4.2 A priori estimates of the solution . . . . . . . . . . . . . . . . . . . 60
4.3 Estimates of nonlinear terms . . . . . . . . . . . . . . . . . . . . . 64
4.4 Weighted estimates of the solution . . . . . . . . . . . . . . . . . . 66
4.5 Solvability of problem (4.8) . . . . . . . . . . . . . . . . . . . . . 70
5 Three dimensional time periodic Navier Stokes problem 79
5.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 79
5.2 Estimates of nonlinear terms . . . . . . . . . . . . . . . . . . . . . 82
5.3 Solvability of (5.3) problem . . . . . . . . . . . . . . . . . . . . . 83
General conclusions 89
References 91
List of V. Keblikas published works on the topic of the dissertation 99
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Introduction
Topicality of the problem
The research area of this work is the analysis of time periodic Navier–Stokes
equations in domains with cylindrical outlets to infinity. In this thesis the exis
tence and uniqueness of the non stationary Poiseuille solution to the Navier–Stokes
equations is proved. Time periodic problems for Stokes and Navier–Stokes equa
tions are studied in domains with cylindrical outlets to infinity. The existence of
the solutions to these problems is proved in weighted function spaces.
Actuality
Mathematical models of fluid dynamics are systems of linear and nonlinear
partial differential equations, known as Navier–Stokes equations. The rigorous
mathematical analysis of Navier Stokes equations started at the beginning of the
XX century from works of the famous French mathematician J. Leray. This analy
sis consists of studies concerning the correct formulations of initial boundary value
problems for Navier Stokes equations, proofs of the existence and uniqueness of
solutions in different functional spaces, investigation of solutions regularity, con
struction of asymptotics, etc. Such questions have been studied in many papers
and monographs [16], [27], [85].
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