Table Of ContentLecture Notes in Mathematical Fluid Mechanics
Joanna Rencławowicz
Wojciech M. Zaja¸czkowski
The Large Flux
Problem to
the Navier-Stokes
Equations
Global Strong Solutions
in Cylindrical Domains
AdvancesinMathematicalFluidMechanics
Lecture Notes in Mathematical Fluid Mechanics
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Joanna Rencławowicz
Wojciech M. Zaja˛czkowski
The Large Flux Problem to
the Navier-Stokes Equations
Global Strong Solutions in Cylindrical
Domains
JoannaRencławowicz WojciechM.Zaja˛czkowski
InstituteofMathematics InstituteofMathematics
PolishAcademyofSciences PolishAcademyofSciences
Warsaw,Poland Warsaw,Poland
InstituteofMathematics
andCryptology
MilitaryUniversityofTechnology
Warsaw,Poland
ISSN2297-0320 ISSN2297-0339 (electronic)
AdvancesinMathematicalFluidMechanics
ISSN2510-1374 ISSN2510-1382 (electronic)
LectureNotesinMathematicalFluidMechanics
ISBN978-3-030-32329-5 ISBN978-3-030-32330-1 (eBook)
https://doi.org/10.1007/978-3-030-32330-1
Mathematics Subject Classification: 35Q30, 76D03, 76D05, 35A01, 35B65, 35B45, 35D30, 35D35,
35G61
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Contents
1 Introduction.......................................................... 1
2 Notation and Auxiliary Results ................................. 11
2.1 Spaces and Basic Notation..................................... 11
2.2 Auxiliary Problems, Results, and Green Functions.......... 17
2.3 Interpolation, Imbeddings, Trace Theorems, and the
Korn Inequality................................................. 24
3 Energy Estimate: Global Weak Solutions..................... 31
3.1 Weak Solutions ................................................. 34
3.2 Estimates for v ................................................. 47
t
4 Local Estimates for Regular Solutions......................... 59
4.1 A Priori Estimates for Function h=v, ..................... 64
x3
4.2 A Priori Estimates for Vorticity Component χ............... 72
4.3 Relating v and h: rot–div System ............................. 75
5 Global Estimates for Solutions to Problem on (v,p)....... 83
6 Global Estimates for Solutions to Problem on (h,q)....... 95
7 Estimates for h ..................................................... 99
t
8 Auxiliary Results: Estimates for (v,p)......................... 107
9 Auxiliary Results: Estimates for (h,q) ........................ 117
10 The Neumann Problem (3.6) in L -Weighted Spaces...... 129
2
11 The Neumann Problem (3.6) in L -Weighted Spaces...... 143
p
12 Existence of Solutions (v,p) and (h,q) ........................ 157
12.1 Existence of Weak Solutions................................... 157
12.2 Existence of Regular Solutions ................................ 164
v
vi Contents
References.................................................................. 169
Notation Index............................................................ 173
Name Index................................................................ 177
Subject Index ............................................................. 179
Chapter 1
Introduction
Abstract This chapter is an introduction, where we describe the problem
and define some key parameters to formulate main theorems: Theorems 1.1
and 1.2. We consider incompressible Navier-Stokes equations in cylindrical
domain Ω with large inflow and outflow on the bottom and the top of the
cylinderandtheslipboundaryconditionsonthelateralpartoftheboundary.
In order to prove the global existence of (v,p), where v is a velocity and p
a pressure, with arbitrary large flux d, we show first the existence of weak
solution,nextwefindconditionsguaranteeingregularityofweaksolutionfor
large time T, and finally we achieve the existence of global regular solutions.
In Theorem 1.1 we conclude that for sufficiently small parameter Λ (T),
2
which depends on data: norms of flux derivatives tangent to the bottom and
thetopofthecylinder,theinitialconditionh(0)=v (0),thederivativewith
,x3
respect to x of force and with the estimate for weak solutions denoted with
3
A, we have v,h∈W2,1(ΩT),∇p,∇q =∇p ∈L (ΩT). In Theorem 1.2, for
2 ,x3 2
sufficiently large time T and from decay property of the equations we prove
the existence of solutions in the interval [kT,(k+1)T], k ∈N , therefore we
0
can extend solutions step by step.
In this book, the motion of incompressible fluid described by the Navier-
Stokes equations with large inflow and outflow is considered. The aim is to
prove the existence of global regular solutions with arbitrary large flux. The
domain is a straight non-axially symmetric cylinder with arbitrary cross-
section. We assume the slip boundary conditions on the lateral part of the
boundary, whereas on the bottom and the top of the cylinder there are
some inflow and outflow fluxes. There is no restriction on the magnitude
ofthefluxandweadmitarbitrarylargeL normofinitialvelocity,butsome
2
homogeneity of the flux is necessary. Moreover, the initial velocity does not
change too much along the axis of cylinder and the inflow does not change
much along the directions perpendicular to this axis either with respect to
time.
© Springer NatureSwitzerland AG 2019 1
J. Rencl(cid:2)awowicz, W. M. Zajączkowski, The Large Flux Problem to the
Navier-Stokes Equations,Advances in Mathematical Fluid Mechanics,
https://doi.org/10.1007/978-3-030-32330-1 1
2 1 Introduction
Fig.1.1 Domain Ω
Physically, the problem can describe a motion of blood through straight
partsofarteries.Itseemsthatslipboundaryconditionsaremoreappropriate
than nonslip because the blood can slip on the boundary. The model is
the first step for further analysis of inflow-outflow problems and next, flows
around some obstacles, with large velocities. This can also be the starting
pointtodescribebloodmotionbecausewallsofarteriesareinfactelastic,so
the problem with free boundary seems to be more natural: this will be the
next step.
We consider the following initial boundary value problem to the Navier-
StokesequationsinacylindricaldomainΩ withboundary∂Ω =S =S ∪S
1 2
(Fig.1.1) and with inflow and outflow
v +v·∇v−divT(v,p)=f in ΩT =Ω×(0,T),
t
divv =0 in ΩT,
v·n¯ =0 on ST =S ×(0,T),
1 1
νn¯·D(v)·τ¯ +γv·τ¯ =0, α=1,2 on ST, (1.1)
α α 1
v·n¯ =d on ST =S ×(0,T),
2 2
n¯·D(v)·τ¯ =0, α=1,2 on ST,
α 2
v| =v(0) in Ω,
t=0
where the stress tensor has the form
T(v,p)=νD(v)−pI.
By ν > 0 we denote the constant viscosity coefficient, γ > 0 is the slip
coefficient,n¯ istheunitoutwardvectornormaltoS,τ ,α=1,2,arevectors
α
tangent to S, I is the unit matrix, and D(v) is the dilatation tensor of the
form
1 Introduction 3
D(v)={v +v } .
i,xj j,xi i,j=1,2,3
We set S = S ∪ S , where S is parallel to the x axis and S is
1 2 1 3 2
perpendicular to it. Hence
S ={x∈R3 : ϕ (x ,x )=c , −a<x <a},
1 0 1 2 0 3
S (−a)={x∈R3 : ϕ (x ,x )<c , x =−a}, (1.2)
2 0 1 2 0 3
S (a)={x∈R3 : ϕ (x ,x )<c , x =a},
2 0 1 2 0 3
where a, c are given positive numbers and ϕ (x ,x ) = c describes a
0 0 1 2 0
sufficiently smooth closed curve in the plane x =const.
3
To describe the inflow and outflow we define
d1 =−v·n¯|S2(−a), (1.3)
d =v·n¯| ,
2 S2(a)
with d ≥ 0, i = 1,2. Equation (1.1) implies the following compatibility
i 2
condition
(cid:2) (cid:2)
d dS = d dS . (1.4)
1 2 2 2
S2(−a) S2(a)
The goal of this book is to prove the existence of global regular solutions
to problem (1.1)–(1.4) with arbitrary large flux d. In order to demonstrate
such results we are going to proceed in three main steps: first, we show the
existenceofweaksolution,nextwefindtheconditionsguaranteeingregularity
of weak solution for large time, and finally we achieve the existence of global
regular solutions.
Thisbookconsistsof12chapters.InChap.2notationandauxiliaryresults
are introduced.
In Chap.3 the energy type estimate for weak solutions to problem (1.1)–
(1.4)isderived.ThemainresultisformulatedinProposition3.4.Since(1.1)–
(1.4) is an inflow-outflow problem the normal component of velocity on S
2
does not vanish (see (1.1) ). This makes the direct derivation of energy
5
type estimate for solutions to (1.1)–(1.4) impossible. To make it possible
we introduce the Hopf function (3.2) which help us to transform the
problem (1.1)–(1.4) into the new one, with homogeneous Dirichlet boundary
conditions. Namely, we construct some extension δ such that w = v − δ
(see (3.7)) satisfies that w·n¯| = 0. Since w is a solution to problem (3.8)
S
the energy type inequality (3.10) can be derived by integration by parts.
Therefore, in Lemma 3.2, we are able to obtain the energy type estimate
for solutions of problem (3.8). However, to obtain an energy estimate
we need some estimates in weighted Sobolev spaces for the extension δ
