Existence theorems for a nonlinear partial differential equation of viscous, incompressible flow

ItIXTHEMhTICAL

263

BIOSCIENCES

Existence Theorems for a Nonlinear Partial Equation of Viscous, Incompressible Flow A.

R.

Differential

ELCRAT

Mathematics

Department,

Communicated

by

Indiana

Cniuersity,

Richard

Bellman

of a solution

is shown

Bloomington,

Indiana+

ABSTRACT Existence partial

differential

incompressible in large adheres

fluid

vessels

[2].

a Banach

The problem

Solutions

a generalization sequences problems

1.

of the tube,

stating

that

which

is intended

boundary

there

may

any two

condition

be many

regular

by

Altman

elements

are

applying ;9].

solutions

through

is guaranteed

by

the

These

imposed

operator

Frkhet

Kantorovich-Newton are defined

Friedman

a theorem

asymptotic

continuously

flow

the fluid

in the equation

boundary-value of

blood

is that

however,

in L” maps

differen-

theorem

or

by convergent

problems

second order are uniformly

theorems

of a viscous,

to represent

are time

solutions

of

for a nonlinear

flow

solutions;

solutions

space Y and is twice

are obtained

and their derivatives

existence

tube

problem

in time-dependent

is set so that the differential

of it by

whose

velocity

the only

space X into a Banach

tiable.

Their

Since

for a boundary-value

for the

in an elastic

[l].

to the wall

has been proven norm

equation

for

linear

HZjlder continuous.

[4].

INTRODUCTION

The goal of this paper is to prove nonlinear

partial

differential

equation

work in [l] and [2] on mathematical Only “classical” derivatives values

solutions

*

for a certain

comes

out of the

that is, ones which have continuous

in the equation

and take on boundary

continuously.

In the past it has been found relatively classical

theorems which

models for blood flow in large vessels.

are sought,

of all orders appearing

existence problem

solutions

Present

address:

of nonlinear Wichita

partial

State

0

1965 by

equations,

but

of

often

University. Mathematical

Copyright

easy to prove uniqueness

differential

American

Biosciences

Elsevier

2, 263-292

Publishing

Company,

(1965) Inc.

very difficult

to show existence.

have weakened theorems.

As a consequence,

the sense of existence

Unfortunately,

uniqueness

theorems

have not always been possible to obtain, about the relationships there

reasons

about

problems

the physical to weaken

and, perhaps,

of a smooth

thus make existence

asymptotic

uniqueness”

incompressible section

11 i has

squared

conditions

tion that the fluid adheres (linear equation)

explicit

series

integro-differential even though that

this approach

operator. solution

square

Thus, the complete

Existence ential

with

and applying

theorems

initial-boundary

These

solutions

are twice

to this one.

which lies behind

an

are proven here.

with in 11:

However,

it must

described

above.

is that

value problem,

It is

the work in

is dealt

that is, the one which

unique (in the usual sense),

up the problems continuously

equations

so that the differ-

Frechet

of L. V. Kantorovich

are constructive,

are constructed,

considered

he obtains

to an unbounded

sense, since he knows

uniqueness”

to make a solution

of operator

* The suggestion problem

In the case of a rigid

for nonlinear

differentiable,

or one of 31. Altman operators

space into another by means of a Newton-type

theorems

condi-

here.

involved

on the solution

dictated

gradient

iteration

the sense of uniqueness

either a theorem

one real Ranach

he avoids imposing

asyvmptotic”

problems

will be shown by setting

operators

pressure

square

asymptotic

has enough such conditions

over a fixed tube

goes to zero as time goes

form of uniqueness

Picard

square

tube flow of a viscous,

in the conventional

to fluid dynamic “Mean

in mind that

is not dealt

of “mean

He is then able to use this special solution,

is “mean

and 121, and only existence be kept

easier to prove.

idea

to the wall of the tube.

it is not unique

any other

this paper.

solution

the

other than the physically

by adapting

in such

to be dropped,

of pulsatile

with an analytic

solution

For these

conditions

of two solutions

By using this weakened

any initial-boundary tube

of these solutions below.

fluid, this being simply that the integral

of the difference

to infinity.

introduced

in his study

In addition,

the sense of uniqueness

and initial

M. Lieberstein

solutions.

significance

a way as to allow some boundary H.

to go with these results

such as the one considered

it seems worthwhile

in the field

and there has been some confusion

between various generalized

is some question

in fluid dynamic

workers

in order to be able to prove existence

in that

the elements

of trying subsequently

sequences

converging

of these sequences

some Newton-type was made in [l,

procedure footnote

mapping

procedure.* to the

being solutions

on the fluid dynamic 1.7, p. 751.

EXISTENCE

of linear equations. of nonlinear

This represents

equations

of its variants an element

where

is employed,

for finding a solution

the

and no method here.

a marked Schauder

of approximate

theorem

with this matter,

existence but they

partial

the solution

are not pursued

which will be expected

THE

PHYSICAL

Suppose tube

made

straight blood

FIG.

will show that

is

for dealing In addition,

the physical

problem

in view of the minimal

PRORLEM

the flow of a viscous,

of compliant

through

this problem

here

equation,

imposed.

we observe

section

is provided

differential

in this paper.

may have many solutions,

”

On the other hand

possibilities

theorem

conditions

or one

a method

of this linear equation

will suggest

the proof using Altman’s boundary

theorem

solutions

for a linear

proofs

over treatments

does not provide

once it is known that one exists.

for approximating

The

advantage fixed-point

since that theory

of the sequence

only by an existence given

265

THEOREMS

material,

restricting

a large blood vessel, is referred

fluid in a

our attention

This is intended

of this tube.

the reader

incompressible to represent

to a fairly the flow of

but for the physiological to

setting

of

[I-31.

1.

The tube section in E3.

and z axis are taken perpendicular denoted

at time t is thought

The axis of the tube section as rectangular

to the x axis.

Cartesian

The lateral

by S(t), and the cross section

the x axis at x by A(x, t).

Various

have to be made subsequently,

of as a domain

is taken

R(t) contained

as the x axis and the y axis coordinate

surface

axes in a plane

of the tube

section

cut by the plane perpendicular

assumptions

but it is assumed

on the regions immediately,

is to

R(t) will for clarity

266

A.

of exposition,

that

R(t)

which the divergence surface

is a simply

theorem

connected

holds),

that

S(t) from the x axis is given by a function

differentiable

on its domain,

dimensional

Green’s

imaginable

region bounded

a smooth

on S(t), rubber

As in [lj driving

[2],

the pressure

Hence,

and U, v, and w denote equation

gradient

if fi denotes

the differential

the function

u’w, +

z’wuy +

(constant)

of viscosity.

density

provides

components

of momentum,

in with

(1)

v>,y+ v,,) = - Py

f-qw,, + w>,,,+

of the fluid and ,U is the

[8] f or a derivation equations.)

the boundary

of these

(constant) equations,

To go with these equations condition

the fluid and the wall.

surface of the tube section has negligible

(2)

WI,)= - p,,

that the fluid be at

to the wall at the wall at all times,

be “no slip” between

values

of motion,

ww*)-

(See

of as the force

w, = 0,

8” +

called the Navier-Stokes

viscous flow theory rest with respect

of

of mass),

d(v, + uvx + vvy + WV,)- p(vxx +

usually

assumption

if he thinks

giving the pressure

the velocity

conservation

(conservation

equations

is thought

giving

respectively,

26, +

d is the

two-

for all x and t.

to be given by the functional

the functions

the x, y, and z directions,

coefficient

on the

tube.

of a given function.

where

curve,

nearly any smoothness

and the reader will not be led astray

and

d(w, +

(one for

which is continuously

by a Jordan

the fluid, and it is assumed

the continuity

region of points

and that A(x, t) is a simply connected

Of course, the physical problem justifies

yields

Green’s

the distance

R. ELCRAT

that is, that

In particular,

there

if the lateral

motion, this gives us the boundary

conditions u=v=w=C) We might However,

use Eqs.

(l),

on S(t)

(2), and

a mathematically

(3) as a model

simpler system

a model for blood flow that is subject assuming

the pressure

to be nearly

that p, and p, are negligible, of velocity,

for all t.

is needed in [l] to formulate

and assuming

2. 263-

292

(1968)

verification.

in cross sections that

v and W, are such that terms involving

~Wathematical Biosciemxx

for our flow problem.

to experimental constant

(3)

the radial

Thus,

A(%, t), so components

them and their deriv-

EXISTENCE

atives

267

THEOREMS

can be neglected,

conservation

of momentum

implies

the single

equation +,

+ SbU,) - /-+,,

for the axial component The

“no slip”

+ GYY+ u,,) = -

of velocity

condition

P,,

(4)

where p, is a function

of x and t only.

now becomes on S(t) for all t.

‘U = 0 (For further

discussion

of the physical

components

of velocity

and pressure gradient

and [2].)

Proving

existence

main

of this

paper.

task

Although

in Eq.

of a solution

velocity

term

W,

appears

axis

is thought

behind neglecting

radial

the reader is referred

to [l]

of Eqs.

(4) the cross components

just as they are in Poisseuille the axial

reasoning

varies

(4) and (5) will be the

of velocity

flow and in pulsatile

with the axial

in the equation. of as arising

(W

distance

so that

This variation

from

the

are neglected

flow in a rigid tube of velocity

pressure

relief

[l],

the nonlinear along the

due to the wall

compliance. Comparing problem

Eqs.

(4) and

for a parabolic

hope for a solution boundary

values

the solution

(5) with

equation,

of Eqs.

the

initial-boundary-value that

(4) and (5) to be unique.

at the ends of the tube

function

usual

we see immediately sections

there

In fact,

and initial

would have to be prescribed

to obtain

is no

additional values

for

uniqueness.

This is just the sort of thing which will not be done.

There are two reasons

for this.

values other than the

First,

physically

imposition

dictated

is not desirable

of initial

condition

and boundary

that the fluid adhere to the wall of the tube

in the model

of physiological

phenomena

the work in this paper because such data are virtually experimentally. boundary-value

Second, problem

in the Introduction; to treat Thus, Eqs.

existence

(4) and (5).

existence

for

impossible the

that is, for the complete to attack

in the

to obtain initialdiscussed

problem it may be necessary

sense or by a nonconstructive the incomplete

problem

The price paid is that the weakened

was discussed

inspired

complete

for Eq. (4) may lead to the difficulties

in a generalized

it seems prudent

which

proving

that

Introduction

must

method.

consisting

of

sense of uniqueness

be adopted.

3. PREREQUISITES We

now introduce

the subsequent

notations

and give results

that

are prerequisite

to

work. Mathewzatical

Hiosciences

2, 263-292

(1968)

A function

p defined

Holder continuous one such

on a region W in (x, y, z, t) space is uniformly

with exponent

SCif u is a real number between zero and

that

b(P) -

1< const. [W, Q)l", P,QE~,

v(Q)

where

d(P,Q) = [(x - x’)~ + (v - Y’)~ + (2 - x’)~ + jt - t’ll’P if P = (x, y, For

such functions

2,

q,

It can be shown

(see [4, Chapter

denoted

is a Banach

C,(W)*,

We denote

Q = (x’, y’, z’, t’).

and

t)

31) that

space

the set of all such functions,

under

the norm

th e set of all functions

by CzTn(@

q defined

on L@ such

that

It can be shown (see [4, Chapter

31) that C, t,(9?)

is a Banach

space under

the norm

The following proposition in later + with the

sections:

Tills is slightly two different

notation

a somewhat

on extension

A function different types

C,(a) different

from

of Banach

f uniformly the notation

will be used free15

Hiilder continuous of Friedman

spaces of HGlder

for the space described space.

of functions

above

on a set in

in [4], where

continuous

functions.

and the notation

he deals He uses

used above

for

EXISTENCE

269

THEOREMS

(x, y, z, t) space can be extended to the whole @ace in such a *way that it is still uniformly A proof

Hiilder continuous

can be found

in

and its norm

[7, Chapter

The regions 3 which are considered and are contained simply

connected,

and t = t,, respectively, for t, < f < t,.

will be bounded regions

2 depicts

2. Composition

such

n {(x,

2,

a region

of the

where h is a function

.Y

We see that

tz).

g.

space-time

region

H.

E if there is a four-dimensional

?’ = h(.r, 2, t),

all of whose derivatives

and are Holder continuous

(exponent

a).

by one of

x = h(y, z, t), through

second order exist

9 has property

E if every point

has it.

The nornzal boundary of a region 3’ such as the one described is the

of

on t = t,

manifold

V of the point such that V fl 9’ can be represented z = 12(x, ‘J’, t),

of 9’

R(t,)

regions as well.

z) E aR(t), t, < t <

t)I(x,y,

A point of .Y is said to have property

-

by the closures and

y, z, I) 11= t>

three-dimensional

9 = {(x, y,

neighborhood

K(t,)

t = constant,

and by an open three-dimensional

will be simply connected,

FIG.

two hyperplanes

The sets

R(t) = .&

Figure

They

three-dimensional

not been increased.

in this paper are simply connected

in the region between

say t = t, and t = t, (tl < tz).

If 1, has

71.

set

R(t,) U .‘9’ (the base

and side of the

“cylinder”

above

depicted

in

270

A. Ii.

Fig. 2). As a consequence [4, Chapter

should be prescribed For

of the maximum princi$le for parabolic

21 it is this manifold

example,

the maximum

principle

y=o

zero in 9?‘, so that

$ defined

C, +a if there

a solution

functions,

on the normal

is a function FE

such that

function unique.

if

boundary

on the normal

and g are continuous

A function class

that,

equation

of 9,

of

in 99,

P=g

f

implies

on the normal

VAT-vp,=f

where

of a parabolic

in :‘R,

vdrp-q+=O

then p is identically

equations

on which values of a solution

to make the solution

ELCKAT

IJ = !P on the normal

boundary

of 3,

is unique. boundary

of 9

is said to be of

!P, C,+.(@, boundary

of W.

For such functions

VW

define

I*1

2+a

inf{lYl,+,

--

j YEC2+.(B)

and Z+!I = Y-’ on the normal The following theorem For

two theorems,

for the heat equation,

proofs, THEOREM

see

[4, Chapter

1. Suppose

boundary

an a priori inequality are fundamental

of W}. and an existence

to the work in this paper.

41.

that 9! has property

is of class C, f 1 on the normal boundary of W.

E, that

f E C,(S),

a?zd $

Then, if y is a soktion

of

in M,

v&--t=f

on the normal boundary

P=#

of &‘,

and y E C, +.(9?), there is a constant B depending only on v, u, and 3? such that

bl Mathewmtical

Hzoscienc~s

2+a

2, 263-292

G BW,. (196X)

+

IfI,).

EXISTENCE

THEOREMS

THEOREM

2.

271 W has property

Suppose

E, f E C,(W),

and f is zero on

Then there is a (uniqzhe) solution of the problem

ai?(

in 2,

yAP,-vi=f

on the normal boundary

cp=o

of 99,

und this solution is an element of C, ,.(92). In later sections we will investigate from one Banach

space to another

them and linear operators precisely,

suppose

continuously Then,

of nonlinear

operators

connections

between

which are local approximations

P maps

an open

into the Banach

subset

space

if there is a linear operator

linear operators

the properties by establishing

A of the

Y, and that

L in B(X;

to them. Banach

More

space

x is an element

X

of A.

Y), the space of continuous

from X into Y, such that P(x + 1%)= P(x)

+ L(h) + N(h),

(6)

where

we say that

P is Fre’chet differentiable

x, and we call L the F-differential F-differential denoted

is unique if it exists

as P’(x).

Then

the Eq.

that

satisfies

Eq.

In a normed the set of points

an operator

F-differentiable

at

It can be shown that the

(see [S, Chapter

81).

In this case L is

(6) can be written

J’(x + h) = P(x) where o(h) denotes

or, more simply,

of P at x.

+ P’(x)(h)

defined

+ o(h),

in a neighborhood

of zero in X

(7). space the “segment” a + E(b -

I\ proof of the following

a) where

important

connecting 8 varies

proposition

two points

a and b is

over the interval

[0, 11.

can be found in [6, Chapter

Xl. THEOREM

3.

Suppose

that P maps an open neighborhood

A of the

segment S connecting the points a and b of the Banach space X’ into the Banach space Y, and that P is F-differentiable

at each point of ,4.

Then, for each

3’ in A IP(b) -

p(a) -

P’(y)(b

-

a)\\< [lb ,~Iathematical

all 2~

IlP’(x)

Biosciemxs

-

P’(y)Il.

2, 263-292

(1968)

Suppose

now that

the mapping

1’ is F-differentiable

x + P’(x)

is F-differentiable

of A into B(X;

at a point

While this mapping

x we denote

is an element

with an element

of 23(X, X;

from XxX

into

Y, since H(X;

isomorphic

(see 16, Chapter

Fdifferentiable into

B(X,

X;

at each Y)

differentiable,

at each point of A and that

Y) is continuous.

of B(X;

13(X;

point

is continuous,

H(X;

Y))

as P”(x).

Y)), it can be identified

Y), the space of continuous 51).

If this mapping

its F-differential

and H(X, X;

bilinear

operators

Y) are naturally

Thus, it can bc shown that if I-’ is twice of A and the mapping that

is, if 1’ is twice

x -

P”(x)

of A

continuously

F-

then

P(x + h) = P(x) \vhere the last neighborhood

term

+ I”(x)(h)

on the right

of zero in X that

+ JP”(x)(h, represents

vanishes

h) + O(ljhJz),

an operator

more rapidly

than

defined

in a

I~Jz]~~as lIh!l

goes to zero. \Ve are now ready to state the following found

in [5, Chapter

‘THEOREM 4 (Kantorovich). the s@ere

A = (x 1 Ijx -

Sf&bose

The proof is to be

P is a continuous

ma$fiizg

v/

xOjj < I?> in the Banuck @ace X &to the Banach

s$ace Y, and that P is twice continwusly A, = (x ! 11%-

theorem.

17j.

x0:] < 71, where Y < R.

(1)

[P’(xa)J-i

exists,

(4)

l[P”(X)lj


.2:

in

diJferentiable 012the closed sfihere

F?lrtllrr,

slcfiposc that

a-1.

‘I‘lzela, if h = h’B2q < 4 and Y,, = (1 I’(x)

111 -

2/a)Bq/h < Y, t/w equation

= 0

has a solzltion x* z~t the sphere {x ) l/x0 ~ x;j < Yo} to which the seqccence

I -‘cn; 1 = %, converges.

fn

addition,

the sepence

P’(~“)l-lwc,)

EXISTESCE

is

273

THEOKEMS

well defined

an.d converges

The

to x*.

following

ewor

hold:

bounds

II%* - x,,jl < (Zh)2”?gz2”, and

the solution

x*

is unique

in

A,.

The process of forming the sequence the Newton Newtolz

space that

be a one-to-one Y.

above

{x,}

of forming theorem

mapping

in the above theorem the sequence

requires

that

of the Banach

We shall give a theorem

P’(x,,)

{xl]

is called

the modified

the linear

operator

space X onto the Banach

below that

removes

the restriction

be one-to-one.

Supposing space

The

method.

P’(xJ

and that

method,

X, to be a closed linear set in a normed space X, the vector

of cosets

X/X,={x/x=x+X,,xEX} is a normed

space

with

the norm

defined

and it can be shown that X/X,, is complete suppose

that L is a bounded

X onto (not necessarily

b>

if X is [5, Chapter

linear operator

in a one-to-one

mapping

manner)

121.

Thus,

the Banach

space

the Banach

space Y, and

let

x, = Then, let 9

be the linear operator

L-l(O).

from X/X,

onto Y induced by L, that

is,

q-q 9

is a bounded

in a one-to-one

= L(x)

linear operator manner,

where

mapping

so the bounded

XEX

one Banach inverse

;IlatWematical

space onto another

theorem

Nioscie~zces

implies 1, 263-

that

the

292 (1968)

_\. Ii. ELCRAT

274 inverse

mapping,

is a continuous

Z-l,

P to be a continuous space

X into

(possibly

the Banach

linear

nonlinear)

space

Y,

operator.

operator

that

P

Now, suppose

mapping

the Banach

is F-differentiable

in the

sphere S(X,, 7) = {x j 11~~- ~11< Y>, and that L = P’(x,J

maps X onto

mapping

Y induced

Let

X/X,

onto

E be an arbitrary

positive x1 =

There

is an element

-

there

in this

number.

be the operator as follows:

Set

Set

/ 1x1:i < j IX1i 1 + F.

x-lP(x,

+ Xl).

ljxz -

x1/( < jlX_salready

Zr(j(l + E).

If

been defined

let Xn=

The method

2

we proceed

X1, . . . , X,,_ 1 and xi, . . . , x,, _ 1 have

manner,

and choose

letting

9-lP(x,).

is an x2 in X2 such that

the elements

Then,

x1 in Xi such that X2 = x1 -

Then

Y.

by L as above,

Xn_l

-

Z-lP(x, + x,,_ 1)’

x, in %+,such that

used in forming

the sequence

{?c,, + xx} which is defined

in

this way will be called the underdetermined modified Newton method. Theorem 5, which gives conditions from

for the convergence

of this sequence,

[9].

THEOREM

5 (A&man).

Szt$$ose that I’ is as above, that

and that

IIP’(x) -

P'(x,)//
for

XES(X,,

Then, if BC
Bioscimces

and

Lzrj< r(1 -

2, 263- 292 (196X)

K),

Y).

is taken

EXISTENCE

275

THEOREMS

the sequence {x,, + x,} to a solution

defined above is contained in S(x,, Y) and converges

x* of the equation P(x) = 0.

This solution

x* is an element of the closed sphere S(x,, Y).

\Ve shall prove

the following

our later needs using essentially Suppose

6.

THEOREM

simpler

theorem

Altman’s

proof

P is a continuously

which is tailored from

to

[9].

F-differentiable

ma$$Gag

of a Banach space X into a Banach space Y, that P’(0) maps X onto Y, and that IlP’(x) -

P’(y)11 < IIx -

Let dp be the mapping

of X/X,

y1i

for all x and y i~z X.

onto Y induced by L = P’(O), and assume II-Lp-ljI < B.

Then, if r is chosen strictly less that l/B

and if

IlP(O)//< “17< ~(1 the sequence {xn}

defined previously

Br)IB,

(x0 = 0 here) is contained in S(0, Y)

and converges to an element x* of S(0, V) which is a solution of the equation P(x) = 0. Proof:

Since 1

Br< there is a positive

number

Hr(l + E) < 1 We use this

E

and

to define

Brj < r(l -

Br),

E such that and

By + e < ~(1 -

the underdetermined

Br(1 + F)).

Newton

sequence

{x~>.

We have

jlXllid Ij%ll+

E<

Ij~-lll /lp(o)/I+

E<

+

+

E<

r(l - BQ + E)) < y

(8) so that x1 E S(0, Y).

Recalling

the definition Mathematical

of z,,, we have Biosciences

2, 263-

292

(1968)

270

a\. Ii.

x2 -

$1 = 21 -

and,

applying

9-l~P(x1) Theorem

ljxg- x11/d 11x2 - q(l

- P(0): == - z-‘[P(x,)

-

P(0) -

BLCR.1-I’

P’(O)(x,)],

3,

+ &I < (1 + 4B

IIWx,) - ~‘(O)I~ * +&

SUP lJ:.:bI

-wl

<

Let

cc denote

Kr(l

jjxzii < 11x2-

+ 8).

Then,

+ ~)llXll,.

using Eqs.

xl/j +- (1x1/1< (1 + 4 = (1 -

so that x, -

so that

xz t S(0, r). x,* __1 = -

For

n larger

22-1 [I+,‘

J -

than I’(.x,,

(8) and

(0) (!I),

lxllj < (1 + a)(1 -- 4~

r-f)y < Y, two the definitions .1) -

I”(O)(X,~~ 1 -

imply

N,,_:!)l,

EXISTENCE

so that limit,

THEOIIEBIS

{xPL} has a limit

0 =

and

x* in S(0,

we

-

lim Z-~P(.X~ 71+ rzI

Remark.

By

see that

11P(O);l).

maximizing

the

choice

Y =

th is case

In

l/(SS) the

Eq.

(12)

gives

approximate

4. SETTING

theorem

us the

_FIP(x*),

earlier,

precisely,

K(t)

solutions

our main

P(X*).

YE (0, l/W, restrictive

on q (and,

hence,

on

l/(4.32). bound

x,Il <

jb’Y(1 + F)]‘lY

{ x~}. PROBLEM goal in this paper

of the boundary-value

problem

is to prove

consisting

the existence

of Eqs.

(4) and (5).

if g

where

-

requires

error

THE MATHEMATICAL

As stated of a solution More

{A?~} also has a

function

is least

i/x* the

=

LZY)/B,

7j <

for

_ I) =

9(O)

f(Y) = Y(l -

Blso,

the sequence

have 0 =

we

Then

Y).

so

= {(x,

y>z, t) 1(% y>2) E n(t), t, < t e tz},

is the tube section

described

in Section

Mathematical

2, we wish to prove

Biosciences

4,

263 - 292

the

(1968)

278 existence

of a solution*

(4

ZLt +

(b)

U= 0

ZLU,

of Y(U,, +

~

on S(t)

IL,.).

(13)

for all t,

where / = $,/d and S(t) is the lateral surface of R(t). this by proving space-time

existence

region,

\+‘e suppose

and

that

of a function which

in .9,

UJ + i = 0

+

\Ve shall accomplish

which has as its domain

solves

Eqs.

(13) when

a larger

restricted

to 2.

the manifold

.Y’, = {(X, y, 2, t) ~(x, y, z) E S(t), t, < t < tZ} has Property

E at each of its points,

We make two successive in a “spacelike” extension,

manner,

which

is assumed

ends” to the three-dimensional

region,

to be such

which we shall denote that

9, = ((x, y, has Property

2,

E.

A&, is accomplished

between

region 9, the first manner.

The first

by adding,

region R(t),

as R,(t).

t) I (x,

3’,

at

to form

This extension

z) E aR,(t), t, < t < &} that is, so that the space-time

The second extension, by adding

a “finite

from W, to a space-time history”

region region

to .%?I. More precisely,

92 = {(x, J’> 2, t) 1(x, y, z) E R,(t), t, < t < tz}, where

of C,(W).

the manifold

E at each of its points,

B1 has Property

of the space-time in a “timelike”

as -tiI, is accomplished

WC shall denote

each time t, “smooth an “ellipsoidal”

extensions the second

and that f is an element

to -c t,,

R,(t) is a simply connected three-dimensional region for each t t, and t,, K,(t) = K,(t) for t, < t < t,, 2, has Property E, and

the set

* See the

first

paragraph

of the

Introduction

EXISTESCE

THEOREMS

279

where

is a cylinder with respect to the time coordinate. of W are depicted

in Fig.

The successive

extensions

4.

FIG. 4.

The above extensions theorems linear

of &? are made in order to be able to apply the

given in Section

equations

3 on solutions

in our later

ends” to the tube sections that

have Property

preventing

work.

of boundary-value

In particular,

problems

addition

for

of “smooth

is done in order to work in space-time

regions

E, the sharp edges at the ends of the tube sections

this condition

from being fulfilled

by 92.

The addition

of the

part of giz between t, and t, is made so that we may make a (mathematically) necessary

assumption

about

f

is an element

of C,(&)

and that theorem,

It is clear now that

a function

IfI,

(a)

L’t+ vu, -

(b)

v=o

a function

f

f to

imposing

this restric-

92s in such a way that

is zero on aR,(t,,) without

Section

u which is a solution

Y(v,,X + zj,, t

increasing

3).

a,,) +

f= 0

of the problem in W,, (14)

on 9,

is such that its restriction of such

without

In fact, we extend

(extension

its norm,

f

the function

tion on the physical problem.

to W is a solution

v is what

we shall

of Eqs.

(13).

subsequently

Mathematical

Biosciences

The existence

establish. 2, 263-

292

(1968)

280 5.

.A. R. ELCRAT

PRELIRIINARY

REStlLTS

For ZI an element

of C, , .(A$) we define

P(u) = PR0~0sIT10ii

1.

7ft

+

P(26)

IUf,

iS

-

Y(?f,, + uv,. + u,,) + f.

UTZ &W6&

Of

c&).

PYOO!: It suffices to show that gh E C,(.Z,) if g and b do. We note that

so that

where

and

the result PROPOSITION

Proof:

is proven. 2.

P maps C,.,_,(9,,)

into C,(W,)

continuously.

IA “6°ECl!+1(‘%3,

and let t: be a positive

number.

121- ZZO;Z+n< 6 Since

me have

We seek 6 = a(.~, u”) such that

implies

Iw

-

P(u”) lx < E.

EXISTENCE

281.

THEORERIS

TO deaI with the second term on the right we note that (fqx so that,

(ZP),

= (U2 -

zql

= (U,+-

z&26 + 240) -t (24 -

ZP)(ZL,+ U(1).

using the fact that Ikl, G IflA&

we have 1(2b2)xFinally,

(ldo2)xi2 <

(‘t41z +

using the fact JR -

it follows

4,

1U”/z)114,-

24:1, +

(jU&

+

~Z~~~J~~t”-

%I,.

that

< c

implies

that

:gj, < IhI, + c>

that 1P(UO) -

P(U) ja < 62 + S(1 -+ 3V + IZLOlq + lzqa)

when ever 1~4~ - 7ii2 +_r< 6. The

proof

is completed.

PROPOSITION

Proof:

3.

P is F-differentiable

at each point of C,l .(a,),

with

By definition P(u + 9) = W

+ L(q) + h(p),

where

We must first show that L is an element linearity

and that L(q) E C,(W,)

to show that

L is bounded.

of B(C,+z(W,);

if goE C,_,x(ZJ

are immediate,

Let r+~be an element Mathenmtical

C,(&?a)).

it suffices

of C,+,(9$\,).

Bioscietices

Since Then

2, 263 - 292 (196X)

A. R. ELCRAT

282

I-WI, g IPtln+ l4l%l~ + I%l,lP;,+ ~$P,zI,+ byvIa+ Id) < max (1, v, [u(,,[~~,(a)(p[2~_n, so that

L is indeed

bounded,

and

IWl d max (1, v, I& Now, noting

that

h(v) is an element

Iu,J,).

of C,(W,),

we have

1~1~ or /qlija is zero, h(q) is zero, so, after

If either

PROPOSITION

4.

The

noting

that

ma$@g 21 + P’(u)

o/ C, tu(Si?P) into R(C,

La(J?2) ; C,(&))

sa tis~f ies

the

uniform

Liflschitz

tion

Proof:

IYe set M = P’(d)

so that

M is the operator M(p) = (Zllpl), -

in R(CzI (&p),

-

,(a,);

= (26; -

Thus,

Mathematical

Riosciences

2,

263-

292

P’(d),

(1968)

C,(%,)) u;,p

given

+ (u1 -

bl

212)p*

condi-

EXISTENCE

283

THEOREMS

PROPOSITION

The ma$#Gg

6.

of C, +a(92) into B(C,+,(W,); F-differential

C,(W,))

at u corresponds

F(h, k) = (hk) ,, Referring

Proof:

is F-differentiable

at each u, and its

to the bilinear operator

to Proposition

h, k E C,(%). 3, we have

J”(tt + h)(f) = P’(u)(k) + (hk),, so that P’(u + h) = P’(u) + T(h), where

T maps

Jz into

the linear

operator

T,

defined

by

k E C, + .(9q.

T,(k) = (hk),, Since

The

mapping

T is linear,

and

liTI = so

that

that

T is an element

sup

of B(C,+,(XJ;

T is the F-differential

llTbll G I B(C,,

m(W2); C,(S?,))).

It follows

of the mapping u -+ P’(U)

at each

zc.

Using

the identification P”(h, k) = T,(k),

which

was mentioned

in Section

3. the result Mathematical

follows.

Biosciences

2, 263-292

(1968)

lie~~zark 1. is twice

It now follows

continuously

continuous),

immediately

differentiable

(a constant

Kemnark 2.

mapping

P

is certainly

< 1.

If X and Y are closed linear manifolds

respectively,

such that

hold with

Y

into Y for each ZL,all the above results

C, Lr*(L%Z)and C,(.@,)

Actually,

in C, i_,(@,,) and

P maps X into Y, P’(U) maps X into

for each 21, and P”(U) maps XxX

section

5 that

and that

p”(z,)II

C,(%,),

from Proposition

replaced

by X

and Y, respectivel!-.

this is the form in which we shall use the material

of this

in \chat follows.

fi. .4 SOI.LU’IOS

U’e shall Section

LISINC; THE KAXTOROVICH

now give

a proof

THEOREM

of the existence

theorem

promised

in

4.

Let X be the subset of C,+,(d,) identically consisting

consisting

zero on the normal boundary of those functions

identically

of those functions

zero on the surface

the sum of zeros is zero and the limit of a sequence

aA’,(t,).

and Y are closed linear manifolds

in C,,,(W,)

5 maps X into Y because

to be zero on aA’,(t,),

and, moreover,

and all of its derivatives 8R,(t,).

Similar

and that

reasoning

P”(U)

The problem a solution

that

maps

into

respectively.

f

is defined

in X is such that

appear in the definition

shows that

XxX

and C,(W,),

any function

Since

of zeros is zer-o, X

The operator

P defined in Section

which are

of B2 and Y the subset of C,(&)

it

of P are zero on

P’(U) maps X into

Y for each 21,

Y for each 1~.

which we shall now solve is that of proving

existence

of

ZI in X of the equation P(M) = 0,

where P is thought

of as an operator mapping X into Y. Since the manifold

yJJILis a subset of the normal boundary of Eqs.

(l-4)) and its restriction

\Ve ha1.c seen that that

of Si?,, such a function

to .oRis therefore

P is twice continuously

u is a solution

a solution F-differentiable

of Eqs.

(13).

on X and

EXISTENCE

285

THEOREMS

for u in X in Section

5. Theorems

1 and 2 enable us to prove the following

lemma. 1.

L~~li.4

Tlze

bounded

linear

operator

L, = P’(0)

E R(X;

Y)

is

aj+earing

in the a jwiori inequalit_y given by Theorenz 1.

inxtertible, and

7elhere B is the co&ant Proo,/:

1Ve have

seen that

L,(q) = cp*- vhp. If wt. show that

has a unique

the equation

solution

in X for each h in Y, or, equivalently,

initial-boundary-value

YAP; = h

yr -

theorem

solution

will imply

boundary

of BZ

f or each h in Y, the bounded

in C, +,(W,)

X).

Since W, has Property

E and an It in Y is zero on aR,(t,)

Theorem

the existence

Theorem

2 guarantees principle

1, with

inverse

that LO’ E R(Y;

The maximum

the

in ‘g2, on the normal

q=o has a unique

that

problem

of a solution

implies that this solution

by definition,

of the above equation. is unique.

In addition,

4 = 0, yields

so that

For clarity immediate

of exposition

corollary

we state

of Theorem

the following

lemma

which

is an

4. Mnthematicnl

Rioscienc~s

2.

263-292

(1968)

286

A. R.

LEMMA 2. mapping

Suppose

that P is a twice continuously

ETLKAT

IT-differentiable

of a Banach space X7 into a Balzach space Y such that P’(0)

invertible,

is

and that

(1) II[P’(W1/I < Br (2)

lip”(u) 11< 1

for each u i% X.

Then, if

where ljl <

the Newton

1/(2fq,

sequence 21, +, =

u, -

[p'(%)l-lp(a%J, uug= 0,

is well defined and converges to a solution u* of the equation P(u) = 0, where

u*

is an element of the closed sphere S(u, Y,, =

(1 -

‘Jr-

Y,,),

B2$/B.

The erYoY bougtd j (u* holds for the sequence {un}.

u,,/( < (2B2~)2”/2”B

Further, the solution u* is unique in the sfihere

S(u, YJ where y1 = (1 + vl The

requirements

of Lemma

B2r)/B.

2 are satisfied

P(0) = f satisfies ;f InG 171 where

Mathematical

Biosciences

2, 263 - 292

(1968)

by our operator

P if

EXISTENCE

287

THEOREMS

and B is the constant this condition

referred

to in Lemma

1.

We are forced to fulfill

by fiat. The

i%SUMPTION.

pressure-gradient

function

p, satisfies

IPxlx< WB2) in the region B (and, hence, its extension

to W, satisfies the same inequality

there). Summarizing

we have

If the $ressure-gradient

THEOREM.

the problem

our results,

(13) has a solution

Eqs.

proven

the following

function satisfies the above condition

u. Further,

u is aw element

of C, +,(.9Q,

and 2 +a < l/B. IUUI 1.

Remark

Our application

not only yields existence solution can be generated. {u,},

CT1+ (WP) -

y&

zero in .cZ~, and u,,+r is the solution

on the normal

The solution that

is the unique F1 -

is, generating solution

is contained the sequence

(Theorem

Thus,

where 2,; = uO, and I&, , problem

in W,,

be included

by applying

in the results

Altman’s

each

only by means 2).

as such for

using the modified Newton

{u:}

on the normal

well we will “obtain” solutions

and uniqueness

of the initial-boundary-value

Ydp, = P(&

will, in fact,

are obtained

of 9,.

in the proof of the Kantorovich

could also be generated

$V=o This

existence

boundary

and the proof of the fact that it has a unique solution

at each stage in the iteration method,

of the

in B?s,

= JY%)

we have not investigated

this linear problem, theorem.

the above theorem

problem

$9=0 However,

2 to obtain

but points out a way in which this

In fact, the solution u is the limit of a sequence

where u,, is identically

initial-boundary-value

of Lemma

of a solution

element

in the

of 9s

of the next

theorem.

of an existence

we have

boundary

However, sequence

theorem

section

which

in this case as of aphroximate

for a linear

problem

the error bound Ttiatlzematical

Biosciences

2, 263-292

(1968)

288

guaranteed modified

by Lemma 2, and we could give other, less desirable ones for the Newton

do not really leaves

method

have

open the question

of the above

(one is given in the next

the elements

linear

of how one might

problems

with such approximations. IZew~~r/z2. solution space X.

and combine

This matter

1Ve have included

of the operator Actually,

equation

Ilt + 2111 .\-

is “globally

VA I6 + f = 0

this

the results

in Lemma

the solution

on the normal

problem

Eqs.

(15).

with arbitrary

ever, this approach

does not appear

problem

in 9

A

Thus,

in,

SOLUTIOK

We shall

USING

again

to the problem A function

problem

being

for the “complete”

existence

technique

of a solution

of

values of class C, 1x. How-

to be applicable estimate

to such an initial-

edges at the ends of

such as that in Theorem

the sense of uniqueness

the problem

theorem

of Eqs.

for the problem

will be somewhat

has helped in

different

(14) in order

of Eqs. here,

on the normal

class C, + 1 there will be called an mdisturbed if

(15)

of S2,

THEOREM

# which is defined

to be ,mdisturbed,

of the

solution.

ALTMAN’S

attack

the desired existence

this

theorem

owing to the sharp

weakening

for a smooth

boundary

In fact, the general

prescribed

the tube sections which prevent an a$riori

5.

obtained

to be unique.

the similar problem

the quest

the

for that one, and a solution

which we have used could be used for proving

1 from holding.

that

it is the problem of Eqs. (13)

we are interested

\C’e have proven an existence

initial-boundary-value

boundary-value

above

in d2,

[4, Chapter 2 1. However,

one which

(13) is not expected

Kenzurk 3.

obtained

2 the statement

merely a device with which to prove existence of Eqs.

This

the solutions

problem

unique”

than

approximate

is unique in some sphere in the Banach

U= 0

rather

but we

sequences.

is not dealt with in this paper.

it could be shown that

initial-boundary-value

paragraph),

of the approximating

(13).

to obtain

The approach

however. boundary

boundary

of W, and of

functiolz,

or said

EXISTENCE

THEOREMS

2x9

(1)

$J = 0 on ,Y’,_, and

(2)

+, = #, = #,, = $J,,,, = lClfZ = 0 on aW,).

We define

X’ = (111zl

E

ZI has undisturbed

C,+&%,),

and let Y be the Banach

space so designated

1’ are closed linear manifolds the operator and P”(u)

map X’ and X’xX’, the existence

in Section

in C,_+ .(.%?,) and C,@,),

P defined in Section

shall prove

boundary

respectively,

6.

Then X’ and

respectively,

5 maps X’ into Y.

of a solution

values}

and

In addition

P’(U)

into Y for each 21 in X’.

zc in X’

\Vr

of the equation

P(u) = 0, where P is thought

of as an operator

this will yield the desired will be accomplished the results X’

existence

by applying

of Section

5 are that

mapping

X’ into Y, and, as before,

theorem

for Eqs.

Theorem

6.

In fact,

P is continuously

(13).

The proof

included

and that

its F-differentials

l,E.\IMA.

The lineur operator L, = P’(0) E B(X ; Y) ma/w X’

If X;, = L;l(O)

SF1 is invertible,

where B is the constant a$pearin, Proof:

Since a function boundary

u in

Lemma

in CztU (9,)

where X is the space introduced Lemma

maps X’ onto 1 of Section

1 of the previous

which is identically initial-boundary

of g2

section

zero on the values,

X’

in the previous

paragraph.

Y since it maps the subset

Thus,

X of X’ onto

L,

1’ 1)~

6.

\I’e shall denote by @ the class of all functions boundary

onto Y.

and

of A?, has undisturbed

xc certainly

on

satisfy

and 5ifl is the one-to-o+ae operator nzajq%zg X1/X:, onto

1’ that L, induces,

normal

among

F-differentiable

which

defined on the normal

al-e undisturbed. Mnthematical

Biosciences

2, 263-292

(19&S)

Suppose k is an element initial-boundary-value -k(P)

of Y, and let ~,4be an element

solution

on the normal

of a solution

of d,

of X’.

The existence

4 in X of the initial-boundary-

problem, in &,

h

L,@) = f/6=0 where

k = F, -

on the normal

L,(Y),

Y

being

with #on the normal boundary The

boundary

cp, and this p is an element

follows from the existence value

Then the

in 9&.,

= h

cp=# has a unique

of 4!.

problem

existence

implies that maximum

of Q follows L,(Y)

an element

of 9s.

Hence,

of d,,

of C,, %(&?a) which

agrees

In fact, we need only set v = Q!?+ If/.

from Theorem

is zero on 3R,(t,).

principle.

only one element

boundary

2 since

Uniqueness

to each $ in %! there

$ being

undisturbed

of cp follows from the corresponds

one, and

of the set L, ‘(h) c X’.

Since

we have

where

B is the constant

Theorem

1.

Therefore,

appearing

jj9”;‘jj and the lemma

Mathematical

inequality

given

by

< B,

is proven.

Now, all the requirements will be fulfilled

in the a priori

we have

if we assume

Biosciences

2, 263-

of Theorem that

292

P(0)

(1968)

6, with Y chosen to be 1/(2B), =

f

satisfies

EXISTESCE

291

THEOKEMS

where 7 < 1/(4P). We make

this assumption.

sequence

{zt*},

ah in X’

Then,

as defined

an underdetermined

prior to Theorem

modified

6, converges

Newton

to a solution

of the equation P(U) = 0.

In addition

the error

bounds

hold where E is the positive We could

Remark.

of the previous

number

equally

paragraph.

used in defining

well apply

Then

Theorem

the operator

one to one, and the underdetermined

the sequence

P’(0)

6 to the in that

{.u,$}.

problem

theorem

is

Newton process reduces to the mod-

ified Newton method. The number E used in defining the sequence of approximate

solutions

can then

be taken

to be zero, and the error

holds (if we choose Y to be 1/(2B)).

This error bound is markedly

than the one provided by Kantorovich’s method

as applied to our problem.

of Altman’s

theorem

SUMMARY

OF

partial

in flow of a viscous,

This solution

that the velocity

bound

satisfies

differential

on the C,-norm

of

stating

VLlZd+ p, = 0 incompressible

the physically

of a solution

in a larger,

set in such a way that

that there exists

equation

fluid in a compliant

dictated

be zero at the wall of the tube.

existence

for this equation being

as we have seen, application

a smaller

result of this paper is a theorem

of the nonlinear

for the velocity

by proving

better Newton

function.

d(Zl, + UU,) -

tube.

for the modified

RESULTS

The central a solution

theorem

However,

here requires

the pressure-gradient

bound

boundary

of an initial-boundary-value

“nonphysical” a solution

region,

this second

of it restricted

Mathematical

condition

The result is obtained

Biosciences

problem problem

to the original 2, 263-

292 (196s)

region

is a solution

problem equation showing

for an operator that,

problem.

The assumptions

Existence

of finding

one Ranach

assumptions,

of Kantorovich’s

space

pressure-gradient

function

a certain

among these assumptions.

proportional

an a jwiori

inequality

and

r-t, Chapter

another

involved

or a generalization

and

satisfies

of it due to

be uniformly

bound.

of the bound on the C, norm of the pr(xure-gradient is inversely

into

the operator

theorem

with its C, norm satisfying

the most restrictive

for the second

a zero of an operator

involved are that a certain manifold be “smooth”

and that the (prescribed) continuous

mapping

under certain

the hypotheses Altman.

of the original

is shown by posing a problem

to the square derived

function

is by far

In fact, the required

of a constant

by Friedman

Hijlder

The assumption

which

(see Theorem

4_!), and we gi1.e no upper bound

bound

appears

in

1, Section

3,

for the value of this

constant. It should be pointed problem adhere obtained

only

the

out that we ha\.e imposed

above-mentioned

to the wall of the tube. to be unique,

of in light uniqueness” HEFEl
of those has been

boundary

Thus,

and tile results

in

[l]

and

demonstrated

[a]

on the fluid dJ.namic

condition

that

the

fluid

there is no hope for the solution of this paper where

“mean

for this problem.

are to be thought squarcl

asymptotic