Galdi G.P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

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III The Function Spaces of Hydrodynamics . . . . . . . . . . . . . . . . . . 139

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

III.1 The Helmholtz–Weyl Decomposition of the Space L

. . . . 141

III.2 Relevant Properties of the Spaces H

and G

. . . . . . . . . . . 155

III.3 The Problem ∇ · v = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

III.4 The Spaces H

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

III.4.1 Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

III.4.2 Exterior Doma ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

III.4.3 Domains with Noncompact Boundary . . . . . . . . . . . . . . . . 1 98

III.5 The Spaces D

1,q

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

III.6 Approximation Problems in Spaces H

and D

1,q

. . . . . . . . 218

III.7 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

IV Steady Stokes Flow in Bounded Domains . . . . . . . . . . . . . . . . . 231

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

IV.1 Generalized Solutions. Existence and Uniqueness . . . . . . . . 23 3

IV.2 Existence, Uniqueness, and L

-Estimates in the

Whole Space. The Stokes Fundamental Solution . . . . . . . . . 238

IV.3 Existence, Uniqueness, and L

-Estimates in a

Half- Space. Evaluation of Green’s Tensor . . . . . . . . . . . . . . . 247

IV.4 Interior L

-Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

IV.5 L

-Estimates Near the Boundary . . . . . . . . . . . . . . . . . . . . . . 271

IV.6 Existence, Uniqueness, and L

-Estimates in a Bounded

Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

IV.7 Existence and Uniqueness in H¨older Spaces. Schauder

Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

IV.8 Green’s Tensor, Green’s Identity and Representation

Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

IV.9 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

V Steady Stokes Flow in Exterior Domains . . . . . . . . . . . . . . . . . 299

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

V.1 Generalized Solutions. Preliminary Considerations and

Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

V.2 Existence and Uniq ueness of Generalized Solutions for

Three-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

V.3 Representation of Solutions. Behavior at

Large Distances and Related Results . . . . . . . . . . . . . . . . . . . 310

V.4 Existence, Uniqueness, and L

-Estimates: Strong

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

V.5 Existence, Uniqueness, and L

-Estimates: q-generalized

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

V.6 Green’s Tensor and Some Related Properties . . . . . . . . . . . 349

V.7 A Characterization of Certain Flows with Nonzero

Boundary Data. Another Form of the Stokes Paradox . . . . 351

x Contents

Contents xi

V.8 Further Existence and Uniqueness Results for

q-generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4

V.9 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

VI Steady Stokes Flow in Domains with Unb ounded

Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

VI.1 Leray’s Problem: Existence, Uniqueness, and Regularity . . 370

VI.2 Decay Estimates f or Flow in a Semi-inﬁnite

Straight Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

VI.3 Flow in Unbounded Channels with Unbounded Cross

Sections. Existence, Uniqueness, and Regularity . . . . . . . . . 387

VI.4 Pointwise Decay of Flows in Channels with Unbounded

Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

VI.5 Existence, Uniqueness, and Asymptotic Behavior

of Flow Through an Aperture . . . . . . . . . . . . . . . . . . . . . . . . . 407

VI.6 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

VII Steady Oseen Flow in Exterior Domains . . . . . . . . . . . . . . . . . . 417

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

VII.1 Generalized Solutions. Regul arity and Uniqueness . . . . . . . 420

VII.2 Existence of Generalized Solutions for Three-Dimensional

Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

VII.3 The Oseen Fundamental Solution and the Associated

Volum e Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

VII.4 Existence, Uniqueness, and L

-Estimates in the Whole

Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

VII.5 Existence of Generalized Solutions for Plane Flows in

Exterior Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

VII.6 Representation of Solutions. Behavior at Large Distances

and Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

VII.7 Existence, Uniqueness, and L

-Estimates in Exterior

Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

VII.8 Limit of Vanishing Reynolds Numb er. Transition to the

Stokes Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

VII.9 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

VIII Steady G eneralized Oseen Flow in Exterior Domains . . . . . 495

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

VIII.1 Generalized Solutions. Regul arity and Existence . . . . . . . . . 499

VIII.2 Generalized Solutions.Uniqueness . . . . . . . . . . . . . . . . . . . . . . 505

VIII .3 The Fundamental Solution to the Time-Dependent Oseen

Problem and Related Properties . . . . . . . . . . . . . . . . . . . . . . . 514

VIII.4 On the Unique Solvability of the Oseen Initial-Value

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

VIII.5 Existence, Uniqueness, and Pointwise Estimates of

Solutions in the Whole Space . . . . . . . . . . . . . . . . . . . . . . . . . 546

VIII.6 On the Pointw ise Asymptotic Behavior of Generalized

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

VIII.7 Existence, Uniqueness, and L

-Estimates. The case R = 0 ... 560

VIII.8 Existence, Uniqueness, and L

-Estimates. The Case R 6= 0 . 572

VIII.9 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

IX Steady Navier–Stokes Flow in Bounded Domains . . . . . . . . . 583

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

IX.1 Generalized Solutions. Prelimi nary Considerations . . . . . . . 586

IX.2 On the Uniqueness of Generalized Solutions . . . . . . . . . . . . 591

IX.3 Existence and Uniqueness with Homogeneous Boundary

Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

IX.4 Existence and Uniqueness with Nonho mogeneous

Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

IX.5 Regularity of Generalized Solutions . . . . . . . . . . . . . . . . . . . . 621

IX.6 Limit of Inﬁnite Viscosity: Transition to the Stokes

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

IX.7 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

X Steady Navier–Sto kes Flow in Three-Dimensional

Exterior Domains . Irrotational Case . . . . . . . . . . . . . . . . . . . . . . 649

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

X.1 Generalized Soluti ons. Preliminary Considerations and

Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

X.2 On the Validity of the Energy Equation for Generalized

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

X.3 Some Uniqueness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

X.4 Existence of Generalized Solutions . . . . . . . . . . . . . . . . . . . . . 676

X.5 On the Asymptotic Behavior of Generalized Solutions:

Preliminary Results and Representation Formulas . . . . . . . 688

X.6 Global Summability of Generalized Solutions

when v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

X.7 The Energy Equation and Uniqueness for Generalized

Solutions when v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

X.8 The Asymptotic Structure of Generalized Solutions

when v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

X.9 On the Asymptotic Structure of Generalized Solutions

when v

∞

= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

X.10 Limit of Vanishing Reynolds Number: Transition to the

Stokes Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

X.11 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742

xii Contents

Contents xiii

XI Steady Navier–S tokes Flow in Three-Dimensional

Exterior Domains . Rotational Case . . . . . . . . . . . . . . . . . . . . . . . 745

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

XI.1 Generalized Solutions. Existence of the Pressure and

Regularity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749

XI.2 On the Energy Equation and the Uniqueness of

Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

XI.3 Existence of Generalized Solutions . . . . . . . . . . . . . . . . . . . . . 762

XI.4 Global Summability of Generalized Solutio ns when

· ω 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770

XI.5 The Energy Equation and Uniqueness for Generalized

Solutions when v

· ω 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772

XI.6 On the Asymptotic Structure of Generalized Solutions

When v

· ω 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

XI.7 On the Asymptotic Structure of Generalized Solutions

When v

· ω = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

XI.8 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

XII Steady Navier–Stokes Flow in Two-Dimensional Exterior

Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

XII.1 Generalized Solutions and D-Solutio ns . . . . . . . . . . . . . . . . . 800

XII.2 On the Uniqueness of Generalized Solutio ns . . . . . . . . . . . . 801

XII.3 On the Asymptotic Behavior of D-Solutions . . . . . . . . . . . . 804

XII.4 Asymptotic Decay of the Vorticity and its Relevant

Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

XII.5 Existence and Uniqueness of Solutions f or Small Data

and v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836

XII.6 A Necessary Condition for Non-Existence with Arbitrary

Large Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

XII.7 Global Summability of Generalized Solutions

when v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

XII.8 The Asymptotic Structure of G eneralized Solutions

when v

∞

6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866

XII.9 Limit of Vanishing Reynolds Number: Transition to the

Stokes Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885

XII.10 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891

XIII Steady Navier–Stokes Flow in Domains with Unbounded

Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

XIII.1 Leray’s Problem: Generalized Solutio ns and Related

Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900

XIII.2 On the Uniqueness of g eneralized Solutions to Leray’s

Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903

xiv Contents

XIII.3 Existence and Uniqueness of Solutions to

Leray’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910

XIII.4 Decay Estimates for Steady Flow in a Semi–Inﬁnite

Straight Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916

XIII.5 Flow in an Aperture Doma in. Generalized Solutions and

Related Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926

XIII.6 Energy Equation and Uniqueness for Flows in an

Aperture Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930

XIII.7 Existence and Uniqueness of Flows in an

Aperture Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935

XIII.8 Global Summability of Generalized Solutions for Flow

in an Aperture Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

XIII.9 Asymptotic Structure of Generalized Sol utions fo r Flow

in an Aperture Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958

XIII.10 Notes for the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1009

Steady-State Solutions of the Navier–Stokes

Equations: Statement of the Problem and

Open Questions

O muse o alto ingegno or m’aiutate.

O mente che scrivesti ci`o ch’io vidi

qui si parr`a la tua nobilitate.

DANTE, Inferno II, vv. 7-9

Introduction

Let us consider a viscous ﬂuid of constant density (in short: a viscous liquid)

L moving within a ﬁxed region Ω of three-dimensional space R

. We shall

assume that the generic motion of L, with respect to an inertial frame of

reference, is described by the following system of equations:



∂v

∂t

+ v · ∇v



= µ∆v − ∇π −ρf (x, t) ,

∇· v = 0 ,

(I.0. 1)

where t is the ti me, x = (x

, x

) is a point of Ω, ρ is the constant density

of L, v = v(x, t) = (v

(x, t), v

(x, t)) and π = π(x, t) are the Eulerian

velocity and pressure ﬁelds, respectively, and the positive constant µ is the

shear viscosity coeﬃcient. Moreover,

v · ∇v ≡

i=1

∂v

∂x

is the convective term, and

∆ ≡

i=1

∂

∂x

1G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations:

Steady-State Problems, Springer Monographs in Mathematics,

DOI 10.1007/978-0-387-09620-9_1, © Springer Science+Business Media, LLC 2011

2 I Steady-State Solutions: Formulation of the Problems and O pen Questions

is the Laplace o perator, while

∇ ≡



∂

∂x

∂

∂x

∂

∂x



is the gradient operator, a nd

∇ · v ≡

1=1

∂v

∂x

is the divergence of v. Finally, −f is the external force per unit ma ss (body

force) acting on L.

Equation (I.0.1)

expresses the balance o f linear momentum (Newton’s

law), while (I.0.1)

ensures that the velocity ﬁeld is solenoidal and represents

the equation of conservation of mass (incompressibility condition). Notice that

in contrast to the compressible scheme, here the pressure π is not a thermo-

dynamic variable; rather it represents the “reaction force” tha t must act on

L in order to leave any material volume unchanged.

System (I.0.1) was proposed for the ﬁrst time by the French engineer

C.L.M.H. Navier in 1822, cf. Navier (1827, p. 4 14), on the basis of a suitable

molecular model.

However, it was only later, by the eﬀorts of Poisson (1831),

de Sa int Venant (184 3), and mainly by the clarifying work of Stokes (1845),

that equations (I. 0.1) found a completely satisfactory justiﬁcation on the ba-

sis of the continuum mechanics approach.

Nowadays, equations (I.0.1) are

usually referred to as Navier–Stokes equations.

In the language of modern

rational mechanics we may say that the underlying constitutive assumption

on the liquid L that leads to (I.0. 1) is that the dynamical part of the Cauchy

stress tensor T is linearly related to the stretching tensor D, namely,

T = −πI + 2µD, (I.0. 2)

In this regard, we wish to quote the following comment of Truesdell (1953, p. 455):

“Such models were not new, having occurred in philosophical or qualita-

tive speculations for millennia past. Navier’s magniﬁcent achievement was

to put these notions into suﬃciently concrete form that he could derive

equations of motion for them.”

A detailed account of the history of the Navier–Stokes equations can be found in

the beautiful paper of Darrigol (2002).

Some authors, showing questionable semantic taste, often speak of “incompress-

ible” and “compressible” Navier-Stokes equations. The latter deﬁnition i s given

to the generalization of (I.0.1), that t akes i nto account the variation of the density

in space and time. Besides the awkward nomenclature, it should be observed, for

the sake of precision, that C.L.M.H. Navier never obtained such a generalization

and that it was derived for the ﬁrst ti me by S.D. Poisson in 1829 (Poisson 1831),

and later clariﬁed on a sound and clear phenomenological basis by G.G. Stokes

(Stokes 1845).

I. Introduction 3

where I is the identity matrix and D = {D

} with



∂v

∂x

∂v

∂x



. (I.0. 3)

Liquids satisfying the constitutive equation (I.0.2 ) are also called Newtonian

liquids.

In several mathematical questions related to the unique solvability of prob-

lem (I.0.1), two-dimensional solutions describing the plane motions of L de-

serve separate attention. For these solutions the ﬁelds v and p depend only on

, x

(say), and t, and, moreover, v

≡ 0. Consequently, the relevant (spatial)

region of ﬂow Ω becomes a subset of the plane R

Till a f ew decades ago (early 1930s), there was unanimous opinion that the

Navier–Stokes equations were useful (in agreement with the experiments, that

is) only at “low”-velocity regimes. It is also thanks to the eﬀorts of outstanding

mathematicians such as Jean L eray, Eb erhard Hopf, Ol ga Ladyzhenskaya, and

Robert Finn that they are nowadays regarded as the universal foundation of

ﬂuid mechanics, no matter how com plicated and unpredictable the behavior of

its solutions may be. In fact, when we struggl e to give an answer to a “simply”

formulated problem, we never know whether it is because of lack of suﬃcient

mathematical knowledge or because of some hidden physical phenomenon.

In any case, there is a ﬁrm belief in the mathematical community that these

equations hide many mysteries and secrets that are still far beyond our current

reach.

When dealing with equations (I.0.1), the primary goal is to study such

signiﬁcant and basic prop erties of solutions v, π as existence, uniqueness, and

regularity, and asymptoti c behavior in space (when Ω is unbounded) and in

time.

Of course, these properties may be rather diﬀerent according to whether

Roughly speaking, the relation (I.0.2) states that the shear stress in a viscous liq-

uid produces a gradient of velocity that is proportional to the stress. In Newton’s

words: “Resistentiam, quæ oritur ex defectu lubricitatis partium Fluidi, cæteris

paribus, proportionalem esse velo citati, qua partes Fluidi separantur ab invicem”

(“The resistance, arising from the want of l ubricity in the parts of a ﬂuid is, cæ-

teris paribus, proportional to the velocity with which the parts of the ﬂuid are

separated from each other”); see N ewton (1686, Book 2, Sect. IX, p. 373).

We wi sh to observe that with a view to solving the above-mentioned questions,

the Navier–Stokes equations can be considered the mathematical prototype of

more complicated models that can be used to take into account other than

purely mechanical phenomena of the liquid, such as thermal conduction in the

Boussinesq approximation or electrical conduction in the nonrelativistic (incom-

pressible) magnetohydrodynamic scheme. In fact, for these more general systems,

one can prove results that are qualitatively analogous to those achieved for the

simpler system (I.0.1) and that present for their proof approximately the same

degree of diﬃculty one encounters f or (I.0.1). However, the situation becomes

completely diﬀerent, and, in principle, more complicated, if the liquid is modeled

4 I Steady-State Solutions: Formulation of the Problems and O pen Questions

we are i nterested in steady or unsteady ﬂows of L, that is, according to whether

the velocity and pressure ﬁelds depend or do not depend explicitly on time.

Throughout this book we shall consider steady motions, for which system

(I.0. 1), with ∂v/∂t ≡ 0 and f = f(x), takes the f orm

ν∆v = v · ∇v + ∇p + f ,

∇ · v = 0 ,

(I.0. 4)

where p = π/ρ, and ν = µ/ρ is the kinema tic viscosity coeﬃcient. We shall

continue to call p “the pressure” (or “the pressure ﬁeld”) o f the liquid L.

In order to perform our study, however, we need to add to (I. 0.4) appropri-

ate supplementary conditions that may depend on the physics of the problem

we want to address, which, in turn, can be broadly characterized in terms of

the type of region Ω where the ﬂow occurs. To this end, we shall distinguish

the following cases:

(i) Ω is bounded;

(ii) Ω is the complement of a bounded region (in short: Ω is an exterior

region).

In both circumstances Ω has a bounded boundary. However, it is also of

great relevance to study ﬂow in regions with an unbounded boundary, such a s

inﬁnite tubes or pipes. Therefore, we shall also consider the following situation:

(iii) Ω has an unbounded boundary.

In all three cases (i), (ii), and (iii), the associated mathematical problems

present several di ﬃculties and basic open questions. The aim of the following

sections is, for each case, to formulate these problems, to outline the related

diﬃculties, and to point out those funda mental questions that still have no

answer.

I.1 Flow in Bounded Regions

Usually, for ﬂow in bounded regions, the driving mechanism is due either to

the movement of part of the bo unda ry, as in the ﬂow between two rotating,

concentric spheres, or to the injection and removal of liquid through the per-

meable part o f the boundary, as in the case of a ﬂow in a region with a ﬁnite

number of “sources” and “sinks.” In such situatio ns, we append to (I.0.4) the

following condition:

v(y) = v

∗

(y), y ∈ ∂Ω, (I.1.1)

by a constitutive equation diﬀerent from (I.0. 2). These latter liquids are called

non-Newtoni an. We refer the reader to the N otes at the end of this chapter, for

the basic mathematical literature related to certain relevant non-Newtonian li quid

models.

I.1 Flow in Bounded Regions 5

that describes the situation in which the velocity ﬁeld is prescribed at the

bounding walls ∂Ω of the region Ω.

It is worth noticing that i n particular, condition (I.1.1), often referred

to as a no-slip boundary condition, requires that the particles o f the l iquid

“adhere” to the boundary ∂Ω in the case that ∂Ω is motionless, rigid, and

impermeable (v

∗

≡ 0). This fact appears to be veriﬁed within a very good

degree of experimental accuracy.

Because of the solenoidal ity condition (I.0.4 )

, and in view of Gauss the-

orem, it turns out that the ﬁeld v

∗

must satisfy the compatibility condition:

∂Ω

∗

· n = 0 (I.1. 2)

with n unit outer normal to ∂Ω. From the physical viewpoint, relati on (I.1.2)

requires that the total mass ﬂux through the boundary must vanish. If ∂Ω is

constituted by the unio n of m ≥ 1 closed non-intersecting surfaces Γ

, . . . , Γ

condition (I.1. 2) becomes

i=1

∗

· n

≡

i=1

= 0 (I.1. 3)

with n

unit outer normal to Γ

The problem of proving or disproving existence of solutions to (I.0.4),

(I.1. 1), a nd (I.1.2) under no restrictions on v

∗

(other than (I.1.2) and, of

course, certain regularity requirements) and on the number m of surfaces Γ

is among the most outstanding and still open questions, not only i n the steady-

state case but in the whole mathematical theory of Navier–Stokes equations.

Instead of (I.1.1), one may consider diﬀerent boundary conditions. A popular

alternative is the so-called stress-free (or pure slip) boundary condi tion, which

consists in prescribing the normal component of the velocity and the tangential

component of the stress vector at the boundary. This type of condition is often

used in stability problems, since it considerably simpliﬁes the calculations (Chan-

drasekhar 1981). Throughout this book we shall always assume condition (I. 1.1)

at th e bounding walls. We refer the reader t o the literature quoted in the Notes

at the end of this chapter for other work related to the Navier-Stokes equations

with boundary conditions diﬀerent from (I.1. 1).

The adherence condition is quite reasonable, at least for liquids ﬁlling rigid ves-

sels under ordinary conditions (Perucca 1963, Vol. I, p. 451). However, such a

condition need not be satisﬁed in diﬀerent situations. For instance, assuming ad-

herence at the contact line Γ of a free surface of a liquid wit h a rigid plane wall

steadily moving along itsel f would l ead to an inﬁnite dissipation rate of the liquid

near Γ (Pukhnacev & Solonnikov 1982, pp. 961–962). Furthermore, the adher-

ence condition is not expected to hold for ﬂuids other than liquids. I n particular,

experimental evidence shows that in high-altitude aerodynamics an adherence

condition is no longer true (Serrin 1959b, §64); see also Bateman, Dryden, &

Murnaghan (1932; §§1.2, 1.7, 3.2), and Truesdell (1952, §79).