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NEW YORK UNIVERSITY INSTITUTE OF MATHEMATICAL SCIENCES, LIBRARY 4 Washington Place, New York 3, N. Y

NEW YORK UNIVERSITY INSTITUTE OF MATHEMATICAL SCIENCES

IMM-NYU 258 MAY 1959

Existence and Uniqueness for a Third Order Non-Linear Partial Differential Equation

CHESTER B. SENSENIG

A3

IT)

M

PREPARED UNDER CONTRACT NO. Nonr-285(46) WITH THE OFFICE OF NAVAL RESEARCH, UNITED STATES NAVY AND CONTRACT NO. NSF-G6331 WITH THE

NATIONAL SCIENCE FOUNDATION

REPRODUCTION IN WHOLE OR IN PART.

IS PERMITTED FOR ANY PURPOSE OF THE UNITED STATES GOVERNMENT.

NEW YORK UNIVERSITY JF"?^258

INSTITUTE OF MATHEMATICAL SCIENCES May 1959

LIBRARY A Washington Place, New York 3, N, Yf

New York University Institute of Mathematical Sciences

EXISTENCE AND UNIQUENESS FOR A THIRD ORDER NON-LINEAR PARTIAL DIFFERENTIAL EQUATION

This report represents results obtained at the Institute of Mathematical Sciences, New York University, sponsored by the Office of Naval Research, United States Navy, Contract No. Nonr-285(^6) , and the- National Science Foundation, Contract No. NSF-G6331.

Introduction The puroose of this paper is to investigate the existence and uniqueness of a solution of the equation

(1)

(^"uy^ + ux^HAu-A) = o

where u is a real valued function of the real variables x, y, and

2 2

tj A = -£-13 + -^-5 ; and X is a positive constant. 9x <9y2

Equation (1) has arisen as an elementary mathematical model in meteorology [1], In this model x and y are position variables in two dimensional Euclidean space, and t is the time. We may think of u as the effective depth of the atmosphere, of (-u ,u ) as the velocity vector of the air particles, and of /\\x as the vertical component of vorticity. VJe will thus speak of the

solutions of the ordinary differential equations 4£ = -u (x,y,t)

dv ^

and g^r = u (x,y,t) as parametric representations for the curves

followed by air particles in the xy - plane. It is then clear

from (1) that the Helmholtzian, /yi - X2u, is constant along the

air particle paths.

For convenience we will restrict ourselves to the consideration of existence and uniqueness of a solution of (1) in fy = )(x,y,t)| -co <x<oo, y>0, 0 < t < cv where c is a positive constant.

Let /\u -X u = h where u is a solution of (1) in # . If h is smooth enough, it is well known that when y > 0 then

u(x,y,t) = i ^ gUjy^W^tJdgdf

'/>0 (2) '- co

- ^ j tt(g,0,t) i K'Uv)d£

-©

-

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.

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T " *p 2

where v = /(£-x) + y and where we have used the appropriate

Green's function g(x,y,*£, *|) and Bessel function K(x). That is,

we let p =/(€ -x)2 + (A/-y)"2, p = /(7-x)2+(>?+y)2, and

g(x,y;g,7) = K(Xp)-K(Xp) where K(x) is the modified Bessel

p function of the second kind of order zero. Then g + g -X g =

p SEE +s*n " * s = °' S(x,0;5,^) = 0, and g behaves like log p for

(£,>?) near (x,y).

We will use the above physical terminology in the following heuristic derivation of the appropriate initial and boundary conditions for equation (1).

The right side of (2) depends on h and u(x,0,t). Since h is constant along the air particle paths, vie see that h can be given everywhere in w in terms of its values at points where air particle paths enter Xy (i.e. at points where u (x,0,t) > 0 or t = 0), In particular u can be expressed in terms of h at points where the air particle paths enter 0 and in terms of u(x,0,t). It therefore seems natural to prescribe the values of u on the xt - plane, to prescribe the values of /\u. - X u on the half plane t = 0 and y > 0, and to prescribe /_yu - X u at points on the xt - plane where u > 0. That this prescription of initial and boundary values constituted a well posed problem was suggested by E. Isaacson; earlier workers in meteorology learned this from numerical experiments.

For convenience we will assume that air particle paths leave S (i.e. u < 0) at points in a simply connected open set of the xt - plane, and air particle paths enter ^(i.e. u > 0) at points of the xt - plane exterior to the above mentioned simply connected open set.

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We will find it convenient to consider a solution which at infinity does not deviate "too much" from a uniform flow parallel to the y - axis. Such a uniform flow, u" = ax+b where a and b are constants and a > 0, satisfies (2) and hence any function u which satisfies (2) will also satisfy (3).

u(x,y,t) =^\ J g(x,y;?,t)[h(5l'llt) + A"(a£+b)] d£d^

h>0 (3) cp

- ^ I M€,0,t)-a£-b]i K'(^v)dC + ax + b

when y > 0. We will choose to work with (3) rather than (2) since we will be placing certain restrictions at infinity on u - ax - b and h+ A (ax + b).

Next we define what we mean by a weak solution of (1) in £J, In Part I we will show that a weak solution satisfjring certain initial and boundary conditions exists with relatively weak restrictions placed on the prescribed initial and boundary con- ditions. In Part II x^e will show that as we gradually strengthen the restrictions placed on the initial and boundary conditions the solution is also gradually strengthened until we have existence of an ordinary solution of (1) satisfying the prescribed initial and boundary conditions. In Part III we prove a uniqueness theorem,

Let U be any real valued function with domain XV such that U and U are continuous. We require that the solutions to the

x y

ordinary differential equations -?£■ = -U (x,y,t) and -*£ = U (x,y,t) exist in the large in &J and are unique. The curves in /£) described by the vector [x(t ),y(t ),t] will be called the air

I

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particle paths of U. Let H be any real valued function with domain /j) such that along each air particle path of U, H is constant (excepting possibly at points where the air particle path is tangent to the xt - plane). We will call H a pseudo- He lmho It zi an of U. We also require that \\ g(x,y;£, \) [H(£, *] ,t ) + X2(a?+b)]d£d^ exists for (x,y,t) in A . \f (3) is valid for u replaced by U and h replaced by some such H, then we call U a weak solution of (1) in /V.

We note that for U to be a weak solution of (1) in /y , U and U are the only derivatives whose existence we are assuming. In the remainder of this paper we will use the notation u (and h) for genuine and weak solutions (for Helmholtzians and pseudo- He lmholtzians ) and the reader should be forewarned.

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Part I

Existence of a Weak Solution"

We will let cf> and \J/, denote the prescribed values of u and h

respectively on the plane y = 0, and \[/p will denote the prescribed

values of h on the plane t = 0. We will prove the existence of a

weak solution in Theorem 1 below for 0 < t < c, (where c, > 0 is

introduced in the statement of the theorem). Heuristically the

proof is based on the following construction. For each n=l,2,3,»..

we define functions h , u , x , and y inductively in the strips

kc, (k+l)c, n n n n

— i < t < - for k=0,l, . . . ,n-l. u may be thought of as an

approximate weak solution, x and y describe the air particle

paths of u , and h may be thought of as an approximate pseudo-

Helmholtzian of u . We show that a subsequence of j x_ !- and y y^ I

converges to limit functions x and y respectively. We use the

functions x and y to define functions h and u. We then show that

the curves described by x and y are the air particle paths of u,

that h is a pseudo-Helmholtzian of u, that u is a weak solution of

(1), and that u and h satisfy the prescribed initial and boundary

conditions.

Theorem 1. Let c|> be a real valued function whose domain is

j(x,t) | - co < x < co and 0 < t < c where c is a positive constants,

Let 4 also satisfy (1a), (1r)> an^ (!c)»

(1.) 4, (Ji - and A are continuous. Also for some constants L and i such that L > 0 and 0 < i < 1 we have 14 v(x,t)-c|> (x,t)| <

2CX. 2wC ■"

Llx-xl1 for all (x,t) and (x,t) in the domain of 4>.

Certain syrabols are used throughout a large part of this report, and a glossary of such syrabols is contained at the end of this report.

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(1R) <J>(x,t ) - ax - b, $ (x,t)-a, and <j> (x,t) are bounded where a and b are real constants with a > 0.

(1„) Let the boundary of the region of outgoing particles on the (x,t) plane be given by x,(t) and Xp(t). That is, let x, and x0 satisfy a uniform Lipschitz condition with x, (t) < Xp(t) for 0 < t < c, call C, the curve consisting of the points tx-,(t)}0,t] for 0 < t < c, and call C? the curve consisting of the points [x?(t),0,t] for 0 < t < c. Let <{> (x,t) = 0 for (x,0,t) on C-, or Cp» let cj> (x>t) < 0 for x, (t) < x < Xp(t), and let <{> (x,t) > 0 for x < x, (t) or x > Xp(t).

Let \|/, be a real valued function whose domain is j (x, t ) | (x,t ) is in the domain of 4 ar*d c|> (x,t) > ol. Let \K also satisfy (2.).

(2.) i|r, is continuous and \|/, (x,t) + A (ax + b) is bounded where X is a positive constant.

Let \J/„ be a real valued function whose domain is

f V

l(x>y) I -°° < x < co and y > o(. Let \J/? also satisfy (3A) and (3r)<

2

(3A) ^p ^s continuous and i|/p(x,y) + \ (ax + b) is bounded.

(3B) \|fp(x,0) = i|/,(x, 0) for (x,0) in the domain of both *ff-_ and \J/p.

Then for all small enough positive c, there exists a real valued function u with domain /y , = j(x,y,t)|-co < x < co , y > 0, 0 < t < c;m such that u satisfies (Jj-a), (^-r)» and (Uc ) •

(kA) u(x,0,t) = cj>(x,t).

ik-a) There exists a pseudo-Helmholtzian h of u such that h(x,0,t) = ij/-(x,t) when (x,t) is in the domain of \|/, , h(x,y,0) = ^o(x»y)» and (3) Is valid for u and this h.

(4C) u(x,y,t ) - ax -b, ux(x,y,t)-a, u (x,y,t), and

p h(x,y,t) + A (ax+b) are all bounded.

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We will start the proof of Theorem 1 by examining the second integral in (3).

Lemma (1.1). Let w be the function with domain Ay defined by

co

w(x,y,t) = &E f [<K^,t) - a£ -b] i K'(Av)d£ when y > 0 where

_ — _-do

v =s/(?-x) +y and K(x) is the modified Bessel function of the second kind of order zero, and by w(x,0,t) = - cj>(x,t ) + ax + b for y = 0. Then w and its first and second derivatives with respect to x and y are continuous in (x,y,t) and are bounded. Also Aw - A w = 0.

Proof of Lemma (1.1). Since 4(£»t ) - a£ - b is bounded and continu- ous, we could easily show that w is continuous for y > 0.

Nov/ consider a fixed ooint (x ,0,t ). For y > 0 we have

CO o o

w(x,y,t) - w(x,0,t) = 12L ( [4(5, t) -<Kx,t) + a(x-£)] i K' (Xv)d? +

00 -co

[4(x,t) -ax -b][-»I \ i K' (Av)d£ + 1]. He observe that ~ -ob TC

M f I k1 (Av)d£ = ^ i Ay sec 9 E* (Ay sec 9)d9. Since -00 -&

Ay sec 9 K (Ay sec 6) is a measurable function of 9 for each y > 0,

since | Ay sec 9 K (Ay sec 9)| < M, and since lim Ay sec 9

y->0+

K (Ay sec 9) — -1 for almost all 9, then by the Lebesgue convergence

oo

theorem we have 4? 1 y K' (Xv)dE, — > -1 as y — > 0+. Furthermore

-oo

the convergence is uniform with respect to (x,t) since

1 2 i

- j Ay sec 9 K (Ay sec 9)d9 does not depend on (x,t). Now given

"2

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8

e > 0 we can choose 5 > 0 so that 0 < y < 6 imolies |w(x,y,t) -

op

w(x,0,t)| < Me + -^ I |<K^,t) - d>(x,t) + a(x-£)| ^ |Kf(Xv)|d§ for

-CD

all (x,t) inhere M" is chosen so that |cj»(x,t) - ax - b | < M. Now let R be any positive number such that R > 2|x |. Since cj>(£,t) - a£ is uniformly continuous in (£,t) for |C I < R and 0 < t < c, then we can choose 6"> 0 so that |cj>(5, t ) - (j>(x, t ) + a(x-5 ) I < e for |£-x| < 6""' < | , |x| < | , and 0 < t < c. Then for |x| < | , 0 < y < 5, and 0 < t < c we have |w(x,y,t)-w(x,0,t)| < M e +

CD / c """ \

it j v lv b it I J

in J • j ,i irt;^ < Me +Me +

-co V-co x+6'V

CD

kzM j — ££ < (M+M)e + te!£j . Now choose o > 0 so that 71 x+6" (£-xr *" 7t6"

6 < 6 and ^^ < e. Then for |x| < £ , 0 < y < 6, and 0 < t < c

its'" " * "p "

we have |w(x,y,t) -w(x,0,t)| < (M+M+l)e. Now for |x| < | ,

enough to (xQ,0,to). Thus w is continuous at (x ,0,t ). This

completes the proof that w is continuous.

(1), To see that w is bounded x-je observe that |w(x,y,t)| <

cp

M

—* d£ = MM for y > 0 and |w(x,0,t)| < M.

-oo Av

CO

We observe that the only hypothesis used, to show that w is continuous and bounded, was c|>(x, t ) - ax - b is continuous and bounded.

Next we will show that w (x,y,t) exists and is continuous and bounded. Since cj>(x,t) - ax - b is continuous and bounded, we could

1 See page 13 for the choice of M.

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show that differentiation under the integral sign with respect to

x is raermitted for y > 0. For y > 0 we have w (x,y,t) = oo cox

M j [<K€,t)-a€-b] JL [1 Kf(Xv)]d€ - - ■& J [<KS,t)-a£-b] ^

-oo -oo

CO

[i K* (Xv)]d£ = ■& ( Ux(£,t)-a] i KT (Xv)d£. Since w(x,0,t) =

-00

- cj>(x,t) + ax + b, we have w (x,0,t) = ~cj> (x,t) + a. We list this

as

CO

& j [4x(?,t)-a] i K'(Xv)d£ f or y > 0

-00

(1.1.1) wx(x,v,t)=

- 6 (x,t) + a for y = 0.

rx

Since <j> (£,t)-a is continuous and bounded, the continuity and

boundedness of w follows exactly as it did for w.

Next we will show that w exists and is continuous and

bounded. Again we could show that differentiation under the

integral sign with respect to y is permitted for y > 0 since

4(x,t) - ax - b is continuous and bounded. For y > 0 we have oo

wy(x,y,t) =| | [<K5,t)-a£-b] £ [& K'(Xv)] d£ =

-co

CO

1

it -co

2 2

[4(5,t)-a^-b][i^|i- Xk'(Xv) +^ X2K"(Xv)]d£. Next we

v3 v

2 _2 . ,„ _v2

observe that -2— K(Xv ) = 2_ \k' (Xv) + ^"^ ■ X2k"(Xv). Hence for

^ ™ v v

oo -

y > 0, wy(x,y,t) =| J [<KC,t)-a?-b][^ k'(Xv) + X2k" ( Xv)--^ K(Xvfl

GO

d? = | \ [<K^,t) -a£- b] [X2 K(Xv) - -^ K(Xv)] d? =

-co

.

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0D

= | \ U24(S,t) - X2(a£+b) - 43QC(5#t)3K(Xv)d€. We observe

-co

that this last integral exists for y > 0. Since X cJ>(x,t)-X (ax+b)< 4 v(x>t) is continuous and bounded, we could show that this last

XX

integral is continuous in (x,y,t) for -co < x < co , y > 0, and 0 < t < c. Hence w exists for y > 0, and w coincides for y > 0

- - y y

with a function which is continuous for y > 0. It follows that w exists and is continuous for y > 0, and we have

co

r

(1.1.2) wy(x,y,t) =1 j [X2cJ>(£,t)-X2(a£+b)-cJ>xx(5,t)]KUv)d£ .

-co

To see that w is bounded we choose M so that

y cp

op _ co _0°

CO 7-3 ?j CO

Iwy(x,y,t)| <f JMe"^z+y dz < 2f je'^dz. If 0 < y < |,

o £- 2 o

then |wy(x,y,t)| < ^ (-M log X Jz2^j Xz + ^ J

-\42+y2 ,_ _ afa i -,_ .w„w A 21 f -xz . 5

Me

bounded.

? -X ^

dz < - ~- 2 log Xzdz + ^-^ \ ft" dz. Hence w is

— y

Using (1.1.1) and the fact that cj> (x,t) is continuous and bounded we can show that w is continuous and bounded in the

XX

same way we showed w was continuous and bounded. Also we obtain

X

GO

1 r,'

4xx(5,t) i K (Xv)d£ for y > 0 (1.1.3) w (x,y,t) = {

XX

'-(txx(x,t) for y = 0 .

11

We could show that w Is continuous for y > 0 and

°P 2

w (x,y,t) = | j [cJ)x(C,t)-a][A2K(Xv) - ^-_ K(\v)]d£ f or y > 0 In

-co the same way we obtained the similar result for w . Hence ^oo co y

wyx(x,y,t) = ^ j" [*x(€,t)-a]K(Xv)de+i j ♦„(«•*) $=r Xk' (Xv)dg

-CO -CO

CO

for y > 0. For y > 0, | j ^(^t) ^~ \K!(Xv)d£ =

-GO

CO c°

i J ^(x+z.t) ~=r KuJ^+p)dz = | j [4xy(x+z,t)-4xx(x-Z,t)] -co /z +y 6

. Xz K?(X /z2+y )dz. Thus vie have

wyx(x,y,t) = \- J [4x(?,t)-a]K(Xv)d5 (1.1.4)

+

71

00 1

Lv(x+z,t)-<Ly (x-z,t)] -~z- k' (x7?+y^)dz •

yz2+y2

So far we have claimed that (1.1. ij.) is valid for y > 0, Now

we notice that the integrals in (1.1. \\) converge for y > 0, and we

could show that they are continuous for y > 0. Hence w __(x,y,t)

— yx

coincides when y > 0 with a function which is continuous for y > 0. Hence w exists for y > 0, and (1.1. 1+) is valid for

— yx

y > 0.

We can show that the first integral in (1.1.1+) is bounded in the same way we showed w is bounded. For the second integral in

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(1.1.1*) we have || \ [<L _(x+z,t )-(L„ (x-z,t )] Xz K (x/z2+y2)dz |

1 TC I kTxx' ■ XX ^ y

o 7z +y

00 ^

<i ( L 21 z1 X|KIiX>/*24yZ)ida

In ^ ^ co -rs — »

A"' Mz1 dz + ^ ffei e-A/z +y dZ for 0 < y < 1

— it

/op r^ — 2

I \ Mz1 e~X^Z +y dz for y > 1

loJ

<-y 1 00

< i-k / i" _** dz + Mz1 e~Xz dz J . This completes the proof

\o z -1 i /

that w is bounded and continuous.

Since w , w , and w are continuous, then w„„ exists and is x* y* yx xy

continuous and w = w .

Me could shox^ that (1.1.2) can be differentiated under the

integral sign with respect to y for y > 0. Hence we obtain

op

w (x,y,t) = *Z ( [X2cKS,t)-X2(aS+b)-<|> (£,t)] i K* (Xv)d£ for

-co

y > 0.

The function defined by the last integral for y > 0 and by

p p -X 4(x,t)+A (ax+b)+cj> (x,t) for y = 0 is continuous and bounded

for y > 0. The proof of this is the same as the proof that w is

continuous since X cj>(£;,t)-X (a£+b)-c|> (£,t) is continuous and

rxx

bounded. Hence w is continuous and bounded for y > 0 and

yy -

f °° (1.1.5) \lF ) ^24(^,t)-X2(a^+b)-(|xx(^,t)]iK,(Xv)d? for y > 0

/ i. \ i -co w (x.y.t) =■{

yy ,J"

(- X24(x,t)+X2(ax+b)+4VT(x,t) for y = 0.

'XX

From (1.1.3) and (1.1.5) we easily obtain /\w - xS; = 0. This completes the proof of Lemma (1.1).

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13

Next we will choose several constants which we will be using. Using the properties of K, t-i » gnd typ we see that there is a real constant M such that |K(Xx)| < M| log x| for 0 < x < | , |K(Xx) | < M e"7^ f or x > | , |~ K(Xx) | = X|K* (Ax) | < | f or x > 0 , |^ K(Xx)| = X|k' (Xx)| < M e"^ for x > 1, | -^ K(Xx) | =

X2|k"(Xx)| : il for x > 0, \-^> K(Xx) | = X2|k"(Xx)| < M e-Xx for

x > 1, |\J/;L(x,t)+X2(ax+b) | < M, and k2(x,y )+X2(ax+b) | < M.

Let W be an upper bound of the absolute values of the first and second derivatives of w with respect to x and y. Let D-, = k M2(l +-i>) +W +a and Dp = 5 2M2 + ^|- + 2W . Let c, be any

X \i

positive number such that c, < c, aP^D-jC, < M, and 2 exp(-2DpC^)>l,

We are now ready to construct the functions h , u , x , and

V

For each positive integer n let h (xQ,y ,tQ) = ^^o'^o^ for

cl -co < x < co , y > 0, and 0 < t < — .

Lemma (1.2). h is a continuous function of (x0»y0»tQ) at almost all points on each plane t = constant. Also l^n^o^o*"^) + X2(axQ+b) | < 2M.

The proof of Lemma (1.2) follows immediately from the definition of h . Clearly we could omit the word "almost", and we could replace 2M by M. We have stated the lemma as we did so that it remains valid when we get to larger values of t which will be shown as we extend the construction to later time intervals

Let vn(x,y,t) = -^ \ } gU,y;?/<) [hn(E,/ctt )+X2(a£+b) Jd£df for

-co < x < co , y > 0, and 0 < t < -=• , where g(x,y;£/>) = K(Xp)-K(Xp) and p and p are defined as p = /(^-x)2*^/ -y ) , p = /(£-x) +(';+y) .

.

Ik

Lemma (1.3). vn, v , and v are continuous. Ivnxl < 1|M (H-=w)

^ X

|v I < li^ll+i), When 0 < s a ./(x-xj^+fy-y)2 < i we have

lvnx(^»y't)"vnx{x^r't)| < -(52M2+i^-)a log s and |v (x,y,t)-v (x,y,t)| < -(52M2 +i£§-)s log s.

and

\"

These estimates are weaker than a Lipschltz condition and

stronger than a Holder condition and are used later to establish

the uniqueness of air particle paths.

Proof of Lemma (1.3). We could show that v and v exist and are x y

continuous since h is continuous almost everyv.here on each horizontal plane, but we omit the proof.

For (x,y,t) in the domain of v we have

Iv^U.y.t)! = |^ )§ gx(x,y;C,^)[hn(C,^,t)+ X2(a£ + b)]d?d 1 \

h>0 oo oo

<ijj lsx(x,yj€,^)|d^d>Z <|] \ |gx<x,yj5,^)|dgd'2

1 >0 -oo -co

00 CO

MV i j2%iiK'(Xp)| + \hi*gl Z* (XP)\l d^

-oo -co v. K y

00 CO CO OD

= f M l^f^K'fxplldedv <f j j |XK'uP>|d^

-00 -CO -00 -co

<f \}|d?d, ♦ f JJ H .-* «d*

P<1 P>1

OM2 2it 1 OT/f2 2n oo -

< 2|- j j dpd9 + 2£ j J p e^P dpdS

0 0 0 0

= UM2 + kW2 (- •§ - i) e"Xp|ro= [iM2(l+i) .

A X^ 0 X^

Similarly |v <x,y,t)| < l|l!2(l+-io . y X

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15

Let (x,y,t) and (x,y,t) be in the domain of v . Let = y(x-x)2+(y-y)2 and P]_ = JiZ-x)2+(1 'J)2 • For 0 < s < |

we have |v (x,yft )-vnx(x,y,t ) |

- 1 1 C (

= l^c J ]Csx(x,y;5,»Z)-gx(x,y;5,>?)] [hn(£//,t ) + \d(aE,+b)]a£>ai |

GO 00

M

<

| ) \ lgx(",y;5//)-gx(x,y,^^)|d4d^

-CD -CO CO CO

-CO -00

<~ J[ X|K,(Ap1)|d^d^ + ~ ^( \|K? Up) Ideal

p<2s p<2s

P>2S^ -L 1 -L

p,<3s p<2s

2s<p<l+s X X -1

+ f H C(^+ix^|i^)Me-^l+X2|p1-p||K,UP^)|]d^ p>l+s -1- 1

(where p" and p* are between p and p, )

< 12M2s + 8M2s + f H t^f + -^JWi

2s<p<l+s pl (p'::")

. 211 ( (' ,2Ms "Xpl „ -Xp*,,,,, + — J ) (-p- e + Mse w )d?d^

p>l+s 1

< sah + aga. \\ i\ + \)&m + Sufi i \ (3e"XPl t.-^aedJ

2s<p<l+s pl P p>l+s

16

< 20M2s

P 2it 3/2 ? 2.71 3/2

71 J ) p, Hl K J J p

0 s x 0 2s

0 2tc oo , rt

6n2s f f „ ~Apl . Aa+l

+ — — J ] Px e dp1d9 +

0 2s 2ti oo

O I [ p o"XP dpdG

0 0

< 20M2s + 12!-T2s log p1

0 0 y2 + !|II2s log p\l(2

2s

+ 12M2S(. £ . ^)8-Apl | » + MI2S(„ £ . ^)e-Xp

X X^

co 0

2

< 20H2s + I6ll2s log 3/2 - I6ll2s log s - i|M2s log 2 + l6M~a

A"1

< (20 + i|)II2s + I6ll2s- 16K2s log s = (36 + i|)Il2s- 16M2s log s

x^ x^

< -(52 + i|)M2s log s. X

16 w2.

Similarly |v (x,y,t )-v (x,y,t ) | < -{52 + ±§)M s log s for

ny

0 < s < j-.

Now let u (x,y,t) = v (x,y,t)-w(x,y,t)+ax+b for -co < x < cd ,

cl y > 0, and 0 < t < — - .

Lemma (1»1±). u , u . and u _. are continuous. |u I < D, and

— — - — — — — — — n, XL*-,. ny ila j.

^ p _ ~ O "I

|u ; < D-, . V.'hen 0 < s = y(x-x) + (y-y)t~ -: tt we have ny j. M-

|u (x,y,t)-u _'x,y,t)| < -D?s log s and |u (x,y,t )-u (x,y,t ) | <

nx -DpS log s.

nx

'2

ny

The proof of Lemma (I.J4.) is obvious using Lemmas (1.2) and

(1.3).

To make it easier to discuss the behavior of the air particle

paths of u at the boundary y = 0 we would like to extend the air

17

particle paths of u into the region where y < 0. To do this we

introduce new functions P , and Pp. Let F ,(x,y,t) = -u (x,y,t)

and F ~(x,y,t) = u (x,y,t) for -co < x < oo , y : 0, and nd nx

0 < t < — . Let Fnl(x,y,t) = -uny(x,-y,t) and Fn2(x,y,t) =

cl u (x,-y,t) for -co < x < co , y < 0, and 0 < t < — -. That is,

F - and P 0 are the even extensions of -u and u „ respectively nl n2 ny nx

across the (x,t) plane.

Lemma (1,5) . P -• and F ~ are continuous. |p , | < D, for i=l,2. When 0 < s =v/(x-x) +(y-y) < j- we have |Fnl(x,y,t )-Pni(x,y,t ) | < -DpS log s for 1=1,2.

The proof of Lemma (1.5) follows trivially from Lemma (l.lj.).

Lemma (1.6). Let (x ,y ,t ) be any point in the domain of F , and

F 0. Then there exist unique functions x (t) and y„(t) defined n2 cx n n d (t)

for 0 < t < -- such that xn(tQ) = xQ, yn(tQ) = yQ, and — ^ — =

Pnlfxn(t), yn(t), t] and — gg - = ^^(t), yn(t), t] for

cl 0 < t < — . Since x (t) and y (t) also depend on (x -y -t ), we

— — n n ''n o'^o' o '

also use the notation x (x -y .t ,t) for x (t) and y„(x .y .t . t )

n o' o' o' n n o'^o* o7

for yn(t).

Proof of Lemma (1.6). The existence of x and y follows since F .(1=1,2) is continuous and bounded [2]. The uniqueness of x and yn follows since |Fnl(x,y,t )-Fni(x,y,t ) | < -D2s log s (1=1,2)

for 0 < s = 7(x-x)2+(y-yP < \ [3] .

Lemma (1.7). Let (x ,y ,t ,t) and (x0>yo»t ,t) be any points in the domain of xn end yn< Let s =J(^0~^0) + (y0~yo^2+^o~to ' and

'

•

'

18

let sn(t) =

ext>(-2Dpc, ) , 1 exp(2D2c)

Then Sn(t) < [2(D1+l)s] d x when s<sq = g(D +17^)

exp[2Dp(t -t)3 Proof of Lemma (1.7). Let z(t) = [2(D1+l)s] for

c

t < t < -* and 0 < s < s . Then z(t ) = 2(Dn+l)s , o — — n o o 1

exp(-2D?c ) 2 (t) = -2D2z(t) log z(t), and z(t) < [2(D]L+l)s] c x since

, exp(2D?c) 2(D,+l)s < (jj) < 1 and exp [2D2(tQ-t)] > exp(-2D2c1).

We will show that S (t) < z(t) thus establishing the lemma for

cl 0 < s < s and t < t < -— • o o — — n

exp(-2D2c1)

For s < s we have [2(D,+l)s] <

. exp(2D?c) «P(-2D2«i) [(i) * ] < £ . Hence z(t) < J f or s < sQ.

For 0 < s < sQ we have l*n(xo J0, VV'VV^o'VV I = t

|x"o +l°Pnirxn(3Eo^o^o^)' ya(5o»*o-V*,'5ld€ - xQ | <

fco |x0-xQ| +D1|to-to|(note |Fnll < Dx) < (D1+l)s. Similarly

'yn(xo'^o'*ojto)""3rn(xo'yo,to»to) ' < (Di+1)s» and hence Sn(tQ) < 2(D1+l)s = z(tQ) for 0 < s < s0.

Suppose S (t") > z(t") for some s and t " such that 0 < s < s

and t" > t . Since S (t ) < z(t ), S (t"") > z(t*"), and S and 2

are continuous in t, then there is a t, such that t, > t ,

' 1 1 o*

S (t) < z(t) for t < t < t,, and S (t, ) = z(t1). For t < t < t,, we have Sn(t) < z(t) < j- and hence Un(xo,yo,to, t )-xn(xQ,yo,to,t ) |

= |x (x ,y ,t ,t )+\ F , [x (x ,y ,t ,5),y (x ,y ,t f£).£]d£ ' n o,Jo* o* o ) nl n o,Jo' o,,Jn o^o* o' '

t o

.

■

19 t

" xh^o'yo»to'to)-JPnlCxii(xo^o'to^)' 3TnU0,3T0,t0,S),S3dS|

t t

< (D^Ds -D2 ] Sn(C) log Sn(C)dC. Similarly

\

l^o^o^o^^yn^o^o^o'^l < (V1)s-D2l Sn(^ log Sn(^

t *o

and hence Sn(t) < 2(D1+l)s - 2D2 \ Sn(£) log Sn(£)d£ for tQ < t < tj.

t o

d£

For tQ < £ < tx we have Sn(£) < z(€) < |, -Sn(€) log Sn(€)

fcr1

< -z(£) log z(£), and Sn(t1) < 2(3^+1)8-202 \ Sn(€) log Sn(£)

fci to h

< 2(D1+l)s-2D2 \ z(5) log z(5)d? = 2(D1+l)s +\ z (g)dg = z(tx).

o o

Since this contradicts S (t, ) = z(t, ), we have S_(t) < z(t) exp(-2D?c,)n X X n

< [2(D1+l)s] d x for t > tQ and 0 < s < sQ.

Similarly the lemma can be proved when t < t and 0 < s < s .

Lemma (1.8). x and y are uniformly continuous functions of ■ n " n

(x ,y ,t ,t) in their domain. o o o

The proof of Lemma (1.8) follows easily from Lemma (1.7) and the fact that |x . | < D, and lyntl < Dt»

Let (x ,y ,t ) be in the domain of u . We wish to define o o o n

functions an, Pn, Yn so that [an(x0,y0,t0),Pn(x0,y0,t0),

Y (x ,y .t )] is the most recent point where the air particle path 'n o,Jo' o

of u through (x ,y . t ) "enters" the domain of u (either p is n "ooo n n

zero or y is zero depending on whether the particle path hits the (x,t) plane of the (x,y) plane).

.

20

Cl

For -oo < xQ < oo , yQ > 0, and 0 < tQ < — let Yn0 be the

largest number such that yno < tQ and 7n^0»J0> t0»Yno^ = °* If

no such rno exists, let yn0 = 0*

When -oo < x0 < oo , y0 = 0, and 0 < tQ < — let rn0 = tQ if

<{> (x ,t ) > 0. If 4x(x0,tQ) < 0, let Yno be the largest number

such that rno < t0 and yn(x0,Y0,tQ, rn0 ) = 0. If no such rn0

exists let y^« = °» no

When -oo < xq < oo , yQ > 0, and tQ = 0, let Yn0 = 0.

We have now associated a number Yn0 with each point (x0,y0,to)

cl such that -co < xQ < oo , yQ > 0, and 0 < tQ < — . Let an, Pn,

and yn be the functions defined by an(xQ,y0,t0) = xn(x0»yo»t0»Yno),

^V^'V = yn(xo'5ro'to'Yno)' and ^WV = Yno for

cl -co < xQ < oo , yQ > 0, and 0 < tQ < — . Then (an>Pn>Yn) is a

point where the curve, generated by [xn(xo,y0,tQ,t ),yn(xQ,y0, tQ, t),t] enters the domain of u as t increases (except possibly when Pn = 0 and 4xUn,Yn) = 0).

From here on we let anQ = a(x0,yo,tQ), anQ = an(xo,yo,to ) ,

Pno = fV^o'V^no = ^o^o'V^no = ^o'^o'V > and

Yno = Yn(^o»^oJ^o) where ^o'V^ and ^o^o,S) are any Points

in the domain of u .

n

Lemma (1,9)* an»Pn> an^ Yn are continuous at points (x ,y , t )

for which y ~ = 0 or y >0 with (a ^.0,y „) not on C, or C0. no no no7 " no l d

It is clear from the definitions of a . S . and r that a

n* Kn' 'n n

and 6^, are continuous at those ooints where y~ Is continuous. It n ' n

is easy to show that the statement of Lemma (1.9) about Yn is true using the uniform continuity x and y and the definition of Yn»

'

21

We new wish to extend h in the t direction so that its

C . 2cl_>

domain is }(x ,y ,t )|-co < XQ < co , yQ > 0, 0 < tQ < — J . Let

cl 2cl (x ,j ,t ) be a point in the new domain such that —- < t < -— — .

cl Go straight down to the point (x ,y ,t - — -) which is in the

region where u and its air particle paths are defined. Follow

n cl

the air particle path of u from (x ,y , t - —-) down to the

nearest boundary point of the domain of u . We define h at

" r n n

(x ,y ,t ) to be the value of \|/, or ty~ at this boundary point,

cl 2cl More precisely, when -co < x < co , y > 0, and -— < t < — — ,

we extend the definition of hn by letting hn(x0»yo»t0 ) =

c c c

^l[an(xo^o'to-lT)'Yn(xo'yo»to--ir)] lf Yn(x ,y0,t0 - -i) > 0, and by letting hn(xo,yo,to) = ♦2tan(xo'yo'to"^;)»Pn(xo'yo'V"^)]

if rB(xo.y0.to-^) = o.

We will now show that Lemma (1.2) remains valid for the extended h ,

Proof of Lemma (1.2). Since we have already observed that the

-::- -::- cl

lemma is true for planes t = c where 0 < c < — ,we will prove

• cl * 2cin the lemma for planes t = c where — — < c < — — . A similar

argument can then be used to extend the proof to planes t = c" for

larger c" as the definition of h is extended further.

-::- cl * 2cl

Consider a fixed plane t = c where — < c < . Let

n — — n

xni(^ = xn[xi(r)'°>^>c'::'-ir] and yni(r) = yn[xim'°'r'c" ~ !T]

for 1=1,2 (see glossary for x,^) and Xp(f))» Let R. =-(x ,y ,c"")|

cl 1 x = x . (t) and y = y^-jft) for some t such that 0 < t < ~ [• for

1=1,2. Then clearly the set of points on the plane t = c", at

which h is discontinuous in (x ,y ,t ), is a subset of R,(J R?.

Vie will show that R. and Rp have measure zero.

•

- '

■

22

Choose / so that |x]L(r)-x1(t') I < /Itr-'cl for 0 < £, £< cr s (1) c.

Then for It - t| < T+f and ° 1 ^> *" 1 " we have

c c

|xnl(^)-xnlC2-)| = |xn[x1(^),o,r,c":-Ti]-xn[x1(r),o/^c^1i]|

r ^exp(-2Dpc- )

< J2(D1+l)[|x1(^)-x1(£-)l+ lr-^3| (see Lemma 1.7)

L

exp(-2Dpc, )

< [2(D1+l)(/(/+l)|;?-d] • Let

exp(-2D?c, ) H = [2^+1 )(/+!)] d L . Then |xnl(S)-xnl(fc) |

exo(-2DpC, ) c^ s

< H|r -rl f or 0 < r, £< —■ and |£ -tj < -^ .

Let k be a oositive integer and choose k so that k > k

• ^ o o

C, S 1C-,

implies £± < j£ . Fix k > kQ. Let f± = -^± for i=0,l,2, . . . ,k.

Let S be the set of points (x .y .c") within and on the circle in [i, ' o' ° o'

the plane t = c with center at [x , (-£■ )»5rni(^ )jOl and radius

c exp(-2D2c1)

1) for u.=1.2 k-1. For-?

2H(t^) for u=l,2, ...,k-l. For * , : ^ < ?- - we have

lr->' * E < FT and hence /[x^W-x^^Jl^Cy^fe)-^^)]^

c exp(-2D2c1)

s KaW^V1 + ^m^-ymV1 - 2H(^) so that

[xnl(r),ynl(r),c'"'] is in S^ for t , < r < tr+1 . Clearly

2 exp(-2D2c , ) RjC S1U 32U . .. JSk_1 and m(S ) = }4^(t~) for

u=l,2, . . . ,k-l where m(S ) is the plane Lebesgue measure of S .

k-1 . c. 2 exP(-2D2Cl) lm2 exp(.2DpCl)

Hence m(R, ) < > m(S ) < i|.itH ( — ) k ^ x for

U-l

See glossary for s •

'

I

23

each positive integer k > k where m(R,) is the plane exterior

measure of R,. Since 1-2 exp(-2D2c1) < 0 by the choice of c1,

1-2 exp(-2DpC, ) then k — > 0 as k -^ co . Therefore mjR,) = 0 and

R.. has measure zero. Similarly R~ has measure zero. This

completes the oroof that hn is a continuous in (xo,y0,tQ) at

almost all points on each plane t = constant.

o

We have yet to show that |hn(£,^ ,t )+A2(a£+b ) | < 2M for

IT < fc < -IT ' If Yn(5^'t"TT:) = °» then lhn(^^'t)+A2(a^+b)l

< \*2[^A,t--}),Vn(£,,'l,t-1±)] + A2[aan(5,^,t-1i)+b]|

+ a\2|5-an(5,/?,t-1i)|

O G C C

< M-«-aA2|xn(?,^,t-1i,t-1i)-xn[5,^,t-1i,vn(^,^t--i)]|

0 c

< M+ a\2D, It- -i - v U,^*t--i)| < M+aXD,c, < 2M where we have — 1 n 'n l n — i l —

used the fact that |i|/p(x,y ) + A. (ax+b) | < M and aX D-^ < M.

Similarly we obtain |h (g,^,t)+X (a£+b)| < 2M when Yn(^,7,t--^) > 0.

From Lemma (1.2) we see that ^ J J g(x,y;^,^) [h (€,'j,t )

V >°

2c

P X

+\ (a^+bHd^d't exists for -co < x < co , y > 0, and 0 < t < — - .

We extend the definition of v by letting

v„U,y,t) = h \\ 6U.yf€.tHM5.*ft) + \2(a£+b)]d£d4

2c for -co <x< oo, y > 0, and 0 < t < — -— .

Lemma (1,3) remains valid for the extended v .

Next we extend the definition of u by letting u (x,y,t)

n n 2c,'

= v (x,y,t )-w(x,y,t )+ax+b for -co < x < oo , y > 0, and 0 < t < -=-*<

/j >0

•

.' '

•

2k

Lemma (l.Ij.) remains valid for the extended u .

n

Next we extend the definition of P , end P ~ by replacing

cl 2cl

_±, by in the orevious definition of P , and P 0. Then

n " n ■ n± nc

Lemma (1.5) remains valid for the extended P ■, and F «,

c 2c nl n2

Now replace -~ by — — - in Lemma (1.6). The lemma remains

valid and it extends the domain of x and y to 3 (x ,y ,t ,t)|

n 2c n L 2c "^° -co < xQ < 00 , -co < yQ < co , 0 < tQ < -^i , 0 < t < -^i { .

Lemmas (1.7) and (1.8) remain valid for the extended x and y .

Next we extend the definitions of a , S , and v by replacing p n' rn' 'n '

cl 1

— =• by in their previous definition. Then Lemma (1.9) remains

n * n

valid for the extended a , S , and y •

n* n n

We can thus extend the functions h , v . u,F,,FOJ x,

n n' n' n± iad' n

y , a , B . and v stepwise in time until 0 < t < cn and ■'n* n' ^n.' 'n — o — 1

0 < t < c, • That is, to define h at a point P in a new time

cl strip we ao back a distance — in time to a point P . We define

n o

hn at P to be Vz(a-n»Pn) at Po if Yn = ° at Po» and v,e define hn at P to be ^i (an>Yn) at PQ if Pn = 0 at Pq. We then define the remaining functions at P as previously. Lemmas (1.2) through (1.9) remain valid for these extended functions.

We will show that a subsequence of ■> u V converges to a weak solution of (1) in *v, which has the properties mentioned in the theorem. Lemma (1.10). There is a subsequence, -fn, 7, of the positive

integers such that j"xn (xo,yo,tQ,t ) j and f yn Uo,y0,tQ,t A

converge for all (x ,y ,t ,t) in the domain of x and y am

o o o nk nk

such that the convergence is uniform in every bounded subset.

:

.

2$

Proof of Lemma (I. 10). Since |xn(x0,y0,tQ,t )-xQ |

= ll ^l^n^o^o^o^^n^o^V^^^3^! < V^o' ± Dlcl>

t o

then the sequence {xn(xo,yo,tQ,t )-xQj is bounded uniformly with

respect to (x0,y0,t0,t ) and n'

For any (x ,y ,T ,t) and (xQ,yo,t0,t) in the domain of xn

; p _ p — 2

let s =n/(xo-xo) +(y0-y0) +^t0~to) . For s < sQ (see glossary) we have |xn(xQ J0,tQ, t )-xQ-xn(x0,y0,to,t )+xQ |

< Ix^o^^n^o^o^o^^^^o^o^o^) l+|xn(x0,y0,tQ,t)

exp(-2DpC, ) - xn(xQ,y0,to,t)| < s+ [2(1^+1)8] +D1|t-t|. It

follows that the sequence j 'Xn(xQ,yo, tQ,t )-XQt is uniformly

equicontinuous in (x ,y ,t ,t).

Similarly the sequence ]yn(x0,y0,tQ,t )-yJ is uniformly bounded and uniformly equicontinuous.

It follows from well known arguments that a subsequence, (n,^, of the positive integers exists having the properties listed in Lemma (1,10).

Let x(x ,y ,tQ,t) = lim xn (x0,y0,tQ,t) and y (x0,y0,tQ,t )

k—5-co k

= lim yn (xQ,y0,t ,t) for -co < xQ < oo , -co < yQ < co , k— >co k

0 < t < c, , and 0 < t < c, . — o — . J. - —x

Lemma (1.11). Let (xQ,y0,to,t) and (x0,yQ,t0,t) be any points in the domain of x and y. Let S(t) =y[x(xo>y0,to,t)-x(xoSyo,to,t)]H+[y(xo,yo,to,t)-y(x0,y0,to,t)]^

and let s = /(xQ-xo)2+(y0-yo)2+(t0-t0)2. Then

26

exp(-2Dpc, ) , , exp(2Dpc)

S(t) < [2(D1+l)s] c ± when s < sQ = ^g +± ] (jt) * .

Also |x(x0,yQ,t0,t)-x(x0,y0,t0,t)| < D± | t-t | and

ly(xo,yo,to,t)-y(xo,yo,to,t)| < D1|t-t| for 0 < t, t < c^.

Lemma (1.11) follows easily from Lemma (1.7) and the fact

that (F , ) < D, for 1=1,2. ni i '

Lemma (1.12). x(x ,y ,t ,t) and y (x ,y , t ,t ) are uniformly con- tinuous functions of (x .y -t , t) in their domain

oo o

Lemma (1.12) follows easily from Lemma (1.11).

For (x ,y ,t ) in A, with y > 0 and t > 0 let yo be the

largest number such that y < t and y(x ,y ,t ,y ) - 0. If no

such Y exists, let v =0. o ' o

For (xQ,0,to) in >(S1 with t > 0 let yQ = t if

4 (x ,tQ) > 0. If 4x(x ,t ) < 0, let y0 be the largest number

such that yo < t and y(x ,y ,t .y ) = 0. If no such yo exists,

let y = 0, o

For (xQ,yo,0) in Q 2 let Yq = 0.

We have associated a number y~ with each (x ,y ,t ) in £* , ,

o oJoo 1

We define functions a, p, and y with domain a) , by a(x ,y ,t ) = x(xo,yo,tQ,Y0), P(x0,y0,tQ) = y(x0,yQ,to,Y0), and Y^xo'^o,-to^ = Yo* Tben (a,p,y) is the most recent point before time t = t where the curve [x(x ,y ,t , t ),y (x ,y ,t , t ) ,t ] enters /l , as t increases excent possibly when p = 0 and (a,0,y) is on CL or C2„

-

27

In the following we will let a = a(x ,y ,t ), a = a(x ,y",t ), ° o o,Jo' o ' o o'Jo* o '

Po = PtVJTo'V.' Fo = PC3S)*y0^o>- Yo = ^VVV' and

"o = Y(x0,y0^0).

Lemma (1,13). Let (x0iy0»t0) ^e any point in Xy 1 such that

YQ = 0 or y0 > 0 with (ao»0>Yo) not on C, or Cp. Then

lira a = a lira 8 = p , and lira y = Yn.

k-^oo nko ° k->oo nko ° k->oo nko °

Lemma (1.13) follows from the fact that for each (x ,y ,t )

0 0 o

we have X^U0,J0^t) — > xU0,y0,tplt) and 7^0a0^0*t) ~>

y(x ,yo,tQ,t) uniformly in t as k — > co .

Lemma (1,11;). a, (3, and y are continuous at points (x ,y ,t )

for which y = 0 or y^ > 0 with (a ,0,y ) not on C. or C0. o o o'o 1 2

Lemma (l.ll|) follows from the fact that x(x ,y ,t , t) and y(x ,y , t ,t) are uniformly continuous.

Let h be the function with domain /d*, defined by

^V^o'V = VVV when Yo > ° and ^WV = ^VfV

when y =0. o

Lemma (1.1$). Let (xo,yo>tQ) be any point in A such that yQ * o or yQ > 0 with (ao,0,Yo) not on C, or Cp. Then h is con- tinuous at (xo,yQ,to) and j^^n^o^o' to) = ^Wo'V'

Lemma (1.1$) follows easily using Lemraas (1.13) and (l.lij.)

and the definition of h.

Lemma (1.16). h is a continuous function of (x ,y ,t ) almost 1 0 0 o

everywhere on each plane t = constant. Also |h(x ,y ,t ) +

o o,Jo* o

X2(ax +b)| < 2M.

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The proof of Lemma (1.16) is similar to the proof of Lemma (1.2) for extended h .

From Lemma (1.16) we see that

■^ ]} gU,y;5/I)[h(4,'?,t)+\2(a^+b)]d5d'Z exists for each (x,y,t)

in /§ , . We define v to be the function with domain A? , whose values are given by

v(x,y,t) =^ \ g(x,y;£,'?)[hUAt)+\2(a£+b)]d4d>? .

Lemma (1.17 ) • v, v , and v are continuous. |v | < I4.ll (1+ —5)

X y * *

and |v I < !j.M2(l+-^). For 0 < s = 7(x-x)2+(y-y j* < r ,

X 2

|vx(x,y,t)-vx(x,y,t)| < -(52M2 + i^-)s log s and

^ / 2 |vny(x,y,t)-vny(x,y,t)| < -(52M2 + i£§-)s log s.

The proof of Lemma (1.17) is the same as that of Lemma (1.3).

Let u(x,y,t) = v( x,y, t )-w(x,y,t )+ax+b for (x,y,t) in/:),.

Lemma (1.18). u, u , end u. are continuous. |u I < D, and 1 x y xx

|u I < D1. When 0 < s = y^-xp+ty-y)2 < i , then kx(x,y,t)-ux(x,y,t ) I < -D2s log s and |u (x,y,t )-u (x,y,t ) | < -DpS log s.

Lemma (1.18) follows from the definition of u.

Let (x,y,t) be any point in Aj t It is then clear that

lim g(x,y;£,t)[h U,*l,t) + X2(a£+b)] k-^-co k

= gU,y;5/<'Hh(£,'?,t) + ^2(a^+b)] for almost all (£,'?) with ^ > 0,

|g(x,y;S/})[h (?,n,t) + X2(a£+b)]I < 2M|g(x,y,-£f >/ ) | for all (£,£) nk

.

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29

with 1 > 0 and for all k, and g(x,y;€,4) [h,. (?,^,t )+\2(a£+b )] is

nk a measurable function of (£,4) for all k. Hence by the Lebesgue

convergence theorem we have lim u (x,y,t) = u(x,y,t). Similarly

k->co nk lim u„ (x,y,t) = u(x,y,t) and lim u (x,y,t) = u (x,y,t). k^-oo nkx X k->co nky y

Let F1(x,y,t) = -u (x,y,t) and P2(x,y,t) = ux(x,y,t) for

(x,y,t) in/51 and let P1(x,y,t) = -u (x,-y,t) and P2(x,y,t) =

u (x,-y,t) for (x,-y,t) in & . Then lim P (x,y,t) =P1(x,y,t) x x k->co kl

and lira F (x,y,t) = Fp(x,y,t).

k-^oo V /

Lemma (1.19). Let (x >yo,tQ) be in XV ^ and choose t > tQ so that

y(x »y »^ »*) > ° f°r *-Q < t < t. Then the curve described by

[x(xo,yo,to,t),y(xo,yo,to,t),t] for yq < t < t is the unique air

particle path of u through (x ,y , t ) •

Proof of Lemma (1.19 )* Por fixed UoJyo,tQ) in M , let Zfc(£) = ^[xn, (xo'yo'to^)"x(xo^o'to^)]2+Cyn1 ^o^o'V5 ^WV^ )]

k

for 0 < £ < c, . Then Z,(£,) ->• 0 as k -> co for each £. Given £

o o

choose k„ so that k > k^ implies Zk(£) < j- . Then for k > kQ

we have

Kjxn (V5ro'to^)^VUo)yo,to^)^]"Fl[x(Xo'yo^to^)' lc k k

y(x^»y^»tn>^)^i I

oJ,/o' O'

" Pn1/X(XoJo'to'?)j(Xo,y°,to'"U1 ' + |Fnkl[x(xo,yo,t0,?),y(x0,y0,to,5),^]

" piCx(xo»yo,to'^)'3r(xo»3ro'to,^,^:i '

< - D2 Zk(£) log Zk(C)

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+ |Fn tx(x0,yo,t0,€),y(xo,y0,t0,§),€3

k1 - P1[x(x0,y0,t0,5),y(x0,y0,t0,C),^]| — > 0 as k -^ oo

Thus lira Pn Cx (xo,y0,to,$),y (xo,yo,to,?),a

K— >-CO , 1 K K

k = P1[x(x0,y0,t0,5),y(x0,y0,t0,4)JC] for each?, and

Pn txn ^xo,yo»to'^)»yn {xo,yo' to»5 } »^ is a measurable function

k1 k k

of ? and its absolute value is less than D, for each k and £.

Therefore by the Lebesgue convergence theorem

3c<xo'yo'to't) = }±rn xn,(xo'yo'Vt) . k-^co k

=x +lim \ Pn [x (x0,y ,tQ,?),y (xQ,y0,t 4),C3d5

k->-co £ ,1k k

t ° = xo + J pitxtxo»yo*'to^^y(-xo»yofto'5,»5^de •

t

Similarly we obtain

t

y(xo'yo'to't) =yo+l p2Cx<xo'yo'to'','y(xo'yo'to'£,'53d* •

t o

Thus x. and y. exist and

xt(xo'yo,to't) = pi[x^x0»yo,to,t)'y^xo,yo'to,t^t^ and yt(xo'yo'to't) = I?2U(xo,yo,to,t),y(xo,yo,to,t),t].

When (x , y ,t ) is in /u-, and y < t < t, we have

O O 0 JL o — —

xt(xQ,yo,to,t) = -uy[x(x0,y0,t0,t),y(x0,y0,t0,t),t] and

yt(xo'yo,to't) = uxCx(xo'yo'to't,'y<xo»yo'to't)»t3, Prom the inequalities in Lemma (1.18) it is clear that [x(x ,y ,t ,t),

y(x ,y ,t ,t),t] represents the unique air particle path of u

through (x0,yo,tQ).

31

Proof of Theorem I: On each air particle path of u, h by defini- tion is constant except possibly at points where the air particle path meets C, or Cp. Hence h is a pseudo-Helmholtzirn of u. Since u and h satisfy (3) by the definition of u, then u is a weak solution of (1). We now observe that u(x,0,t) = <|>(x,t), h(x,0,t) = \Jr_j (x,t) when (x,t) is in the domain of ty,, h(x,y,0) = ^2(x,y), |ux(x,y,t )-a | < D]_+a, |u (x,y,t)| < D^ and |h(x,y,t)+X2(ax+b) | < 2M.

To complete the proof of Theorem I we have yet to show that u(x,y,t )-ax-b is bounded. We have |u(x,y,t) - ax - b| = |v(x,y,t) - w(x,y,t)|

< -k- \\ |g(x,y;£,^ ) |2r!d5d^ + W (where W is an upper bound of |w|)

<j*>0

00* CO

< \ \ \ lg(x,y;?/<)|d£d^ + W

-co -co

00 CO 271 CO

< ~ \ ( |K(Xp)]dgdf + W = f~ j j p|K(Xp)|dpd9 + W which is a

-co -co 0 0

constant.

Thus for all small enough c-, > 0 there is a weak solution

with domain <&. satisfying the conditions of Theorem I.

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32

Part II Getting Stronger Solutions by Strengthening Hypotheses For the rest of this report we let u be a weak solution constructed as in the proof of Theorem I, and we let M, W, D, , Dp, and c, be fixed numbers choosen as in the proof of Theorem I. We also let v, P1, P2, 3c(xQ,y0,t0,t ), y(xQ,yo, tQ,t ), a, p, y, and h denote the same functions as in the proof of Theorem I. Theorem II, Let c|>, ty. , and tp satisfy the hypothesis of Theorem I

with the exception that (2.) and (3a ) are replaced by (2.) and

A, „« ^k, -^ x^^™ UJ ,cA

(3;).

(2») ty, is uniformly Holder continuous and \J/,(x,t)+X (ax+b) is bounded.

(3A) ^p is uniformly Holder continuous and \|/p(x,y)+X (ax+b) is bounded.

Let (x >y ,t ) be any point in -$, such that p > 0 or p = 0

with (a ,0,y ) not on C, or Cp. Then the second derivatives of u

with respect to x and y exist and are continuous at (x , y ,t ) and they satisfy /\u - X u = h. Thus h is the true Helmholtzian of u at U0,y0,to).

Theorem II is proved with the aid of several lemmas which follow.

For (x »y ft ) In A)- and 6 > 0 let R6 ~ !(x,y,t)|(x,y,t) ei^, U-xQ| < 6, |y-yQ| < 6, and |t-tQ|<6-.

Let aQ = a(x0,y0,t*), p* = P(x*,y*,t*), and Y* = Y(x*,y*,t*).

J

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Lemma

(2.1). Let (x ,y0,tQ) be any point in^1 such that pQ > 0,

or 0 =0 with (a ,0,y ) not on C, or Cp. Then there are constants

H > 0 and 6 > 0 such that (x*,y*,t*) and (xQ,y0,t0) in R6 implies

exp(-2D~c, ) .. exp(-2D?cn) Ur-aJ < H s 2 X , IPq-PqI < H s * L , and

.<. exp(-2Dnc,; , — - j S~"Z — ? *TZ — 2

hr*-7J < H s ^ x where s = /U^) +(yQ-y0) +(t0-to) '

Proof of Lemma (2.1).

Case I (p > 0). Since p is continuous at (x0,y0,tQ), we can

choose 6 small enough so that (x ,y0,TT0) in R5 implies pQ > 0.

Then (x0,yQ,^0) in Rg implies y0 = 0, aQ = x(xo,yo,?o,0) , and

3 = v(x ,t ,t ,0). If we also choose 6 small enough so that the o o o o

diameter of R5 is less than s , then Lemma (2.1) follows easily from Lemma (1.7 )•

Case II [p = 0, y > 0* and (a ,0,y«) is not on C, or C,l. r o o OO ± d.

Since yfc is continuous and yt(xo,yo,tQ,Y0) = <l>x(a0»Y0) > 0 (note

(a ,0,y ) is not on Cn or C0), we can choose positive constants v o' ' 'o 1 2 '

6, tlf and e2 so that yt (x0,y0,£0,t) > e1 > 0 for (xQ,yo,^0) in R5 and |t-YQl < Bg,

Since v is continuous at (x ,y . t ), we can choose 6 smaller

ooo

if necessary so that (x >y_,t ) in Rg implies Y0 > 0 and

'V^o1 ^ e2'

Wow let (x ,y ,t ) and (x0,yo,to') be in Rg. Assume without

loss of generality that yo < Y0« Tnen since y(x0,yQ,to,Yo) = Pq ■

o(Y*>o), we have y(x*,y*,t*,Y0) = yU^y*,t*,Y0)-y(x*,y*,t* Y*) ■

(y -Y )yj.(x ,y ,y »t ) where t" is between xl and 7 . Since 'o o Jt o,Jo' o* o o

I .

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3U

Iv^-y I < £o and Iy ~Y~1 < ^> it follows that |t"-Y^I < e0 ar*d 00 — 2 ' ' o O — 2 O — 2

hence y(x*,y*,t*,Y0) = (V^t^o^c'V**1 > ^(y^Y*) =

ei'~o"Yol* Usins y^0»y0»^o»^o^ = ^o = 0("o>0^ we now have

'Yo""Yo' - e" l^xo'yo'to»^o^'y^o*'?o»^o*^o^* NoW choose 6 smaller if necessary so that the diameter of RA is less than s . Then

,_ -•■-. 1 exp(-2DpCT J

from Lemma (1.7) we have Iy«-Y„I < t~ [2(D,+l)s] x . The

results for a and p follow in an obvious manner.

Case III (B = v =0 and (a . 0,y,J is not on C, or C0). As o o o' ' o 1 2

in Case II we can choose positive constants 6, e, , and 6p so that

yt(^o,yo,^o,t) - el > ° for **o'yo'^ ln R6 and ^"Yo' - e2» so that I Y-"Y~ I < £o for (x" ,y\,^L) in RR) and so that the diameter

00 — 2 O'^O'O 0'

of RR is less than s . o o

Now let (xo,yo/to) and (x*,y*,t*) be in Rg. If y0 = Yq = 0,

our conclusion follows as in Case I. If yq > 0 and yo > 0> our conclusion follows as in Case II. If y0 = ° a^d y'Q > 0, the con- tinuity of y in Rr can be used to conclude that there is a (x ,y ,% ) on the straight line segment from (x »y ,F ) to

»*- <*

(x ,y ,t*) such that Y(x >yQ,t ) = 0 but Y(x,y,t) > 0 (hence

B(x,y,t) = 0) on the straight line segment between (x »y /EQ) and

(x ,y ,t ). Since B is continuous in Rg, it follows that

B(x ,y ."£ ) = 0. The methods used in Case II can be used to show r o* J o' o

that |y(x ,y j£ )~Y I o,Jo* o ' o

, exp(-2D?c, ) „ P ,, P ^ ,, P •«exp(-2DpC1 )

<i[2<D1+u] - 21 t(i0-*o> +(?o-yo» +(V*o» '

, exp(-2DpC, ) „ ^

< i [2(D1+l)s] . But |y0-Y0I = Y0 = lY(X0,y0,t0)-Y0l

and our lemma follows.

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35

Lemma (2.2). Let (x ,y ,t ) be in ^ 1 such that PQ > 0 or PQ = 0 with (a ,0,yo) not on C, or C?. Then h is uniformly Holder con- tinuous in a neighborhood of (x ,y ,t ).

Proof of Lemma (2.2). Using Lemma (2.1) and the fact that v|/. and \|/p are uniformly Holder continuous we can show that there are

-,k it it v

constants H > 0. 6 > 0, and 0 < e < 1 such that (x .y -t ) and

' » ooo

(xo,yo,tQ) in R5 implies |h(x^,y^, t^)-h(xQ,yo, tQ) | < Hse where

s =\/(xo"x0^ +^n-5ro^ +^to~to^ * Jt is then clear that there are

positive constants 6, < 6 and 60 such that (x .y .t ) in R» and

l ^ * o o* o o.

s < 60 implies that (x",y" t'r) is in RR. Hence for (x ,y ,t ) in

d. 0 0 O 0 0 0 0

R5 we have |h(x^,yo', t'0)-h(xQ,yo, tQ) \ < Hse for s < 6^. That is,

h is uniformly Holder continuous in Re .

°1

Proof of Theorem II. Let (x »y,t ) be in/', such that p > 0 or P = 0 with (a ,0,yo) not on C, or C~. For arbitrary (x,y,t ) in /' , we have

(11. 1) vx(x,y,tQ) =-^]| gx(x,y^/n[h(4/(,to)+\2(a4+b)]d^dl = ^ )( gx(x,y^,v)[h(^,V,to)-h(xo,yo,to)+aA2(^-x0)]d4d^

(11. 2) vy(x,y,tQ) = ^ 'jj ( gy(x,y^,7)[h(?,1,to)+X2(a^+b)]d4d^

■ Si J) gy(x,y^/0[h(C,/(,to)-h(xo,yo,to)+aX2(C-xo)]dCd'i

?.>0

co

- [h(xQ,yo,to)+X2(axo+b)] | j K(\v)d£ .

-co

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To get the second representation of v we added and subtracted rhUo,yo,to)+X2(axo+b)] ^ \ j gx(x,y;£,>* )d£d1 to the first integral

of (II. 1), and then we observed that \ \ gx(x,y;£,'i )d£,d>i

ti>0

g„(x,y;^, ^)d^dV = 0. Vie obtained the second representa- >l >0 tion of v in a similar manner by observing that

CO

^J jgy(x,y;?,1)dCdl = ^JJ^[K(Xp)+K(Xp)]d?d«z = - | J K(Xv)d£ . >i>0 *£>0 -co

Since h is Holder continuous at (xQ>y0>tQ) , we could show

that differentiation with respect to x and y at (x,y,t )

= (x ,y ,t ) is permitted under the integral sign in the second

representations of (II. 1) and (II. 2). The resulting expressions

are also valid for (x,y,t) in some neighborhood of (x0»y0»t0)

since h is also Holder continuous in some neighborhood of

(x ,y ,t ). Hence for (x,y,t) in some neighborhood of (x0»yo>*0)

we have

:n.3) v^u^t) =^-[

gxx(^y;S,Wh(£,^t)..h(x,y,t)+a\2(£-x)]d£d4 ,

^>0 :iI.I0 vxy(x,y,t) = ^ ^(gxy(x,y;?,^)[h(?/(.,t)-h(x,y,t)+aX2(?-x)]d?d^ ,

;il.5) vyy(x,y,t) = ■^Jjgyy(x,y|C,/7)[h(5,^,t)-l:(x,y>t)+a\2(4-s)]d?d^

•— s CO

=$ \ -ft' (Xv)dE, if y > 0

P J -co + [h(x,y,t)+X^(ax+b)] <

1 if y * 0

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Since the Holder continuity of h is uniform in some neighbor- hood of (x ,y ,t ), we can show with the aid of (II. 3), (II. k) » 00 o

and (II. 5) that vxx, v , and vyy are continuous at (xo,y0,tQ).

2 2

Next we wish to show that Av - X v = h(x,y,t)+X ( ax+b ) at

points (x,y,t) where (II. 3), (II. I;), and (II. 5) are valid. To aid

co

f"¥ \v K'(Xv)d^ if y > °

us here we introduce w(x,y,t) = <j and

1 if y = 0

v(x,y,t) = ^L U g(x,y;C/i)d?d4 . We will first show that

w(x,y,t) = e"Xy and v(x,y,t) = l(e"Xy-l) .

X^

Replacing 4(5, t )-(a^+b ) by 1 in Lemma (1.1), we see w has continuous bounded first and second derivatives with respect to x

_ o

and y for y > 0 and Aw - X w = 0. Since w(x,y,t)

co

= M 1 ■ -1- — k' (\>/z +y2)d£, we have w = 0 and hence it J /g ? X

-co yz +y

w - X w = 0. Therefore w = c, e~ ' + c0 e y. Since w is yy 12

bounded, then c? = 0. Since w(x,0,t) =1, c, = 1 and w = e" y.

Replacing h(£/t,t )+X2(a£+b) by 1 in (II. 1) and (II. 5) we see that v = 0 and v (x,y,t) = w(x,y,t) = e y. Hence v = — k e" * + v-.y + Vp. Since v is bounded, then v, = 0. Since

v(x,0,t) = 0, then v? = -| and v = i (e~Ay-l).

X X

p

We now have Av - X v 88 "Si U Ag(x»y;^,/?)[h(?,'?,t)-h(x,y,t)+aX2(C-x)] d^dl

+ w(x,y,t)[h(x,y,t)+X2(ax+b)]

g(x,y;^,^)[h(5,/?,t)+X^(a^+b)]d?d^. Using the fact that

-sj

^>0

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38

As(x>y;£/<) = x &(x>y;ZA) v© have /\v- \ v

= [h(x,y,t)+X2(ax+b)] [w(x,y, t )-\2v(x,y, t ) ] = h(x,y,t )+X2(ax+b).

Since u = v-w+ax+b, then u has continuous second derivatives

with respect to x and y at (x_,y .t ), and Au(x_,y_,t ) -

o o o ■ o o o

x2u(xo'yo'to) = h(x0'yo'to)+x2(axo+b)"°" x2(axo+b) = h(xo'yo'to)* This completes the proof of Theorem II.

In our next theorem we assume that the prescribed values of

the Helmholtzian are constant in a strip along the curves C, and

C0. We can then show that u , u , and u exist and are con- d xx _».y yy

tinuous for all small enough t.

Theorem III. Let 4, ^-i a^d \|/, satisfy the hypothesis of Theorem

II. Let c|> satisfy a uniform Lipschitz condition with respect to

t. For the functions x,(t) and Xp(t) of (lc) in Theorem I let

d, = min x, (t), d? = max x, (t), d-, = min x_(t), d, = max xp(t), 1 0<t<c x 0<t<c x J 0<t<c d * 0<t<c eL

and assume dp < d,. Let (3r) and (2R) also be satisfied.

(3r) For some positive number f and real numbers p, and Pp we have dp+^d^-^, \|/p(x,y) = p, for d, -"^ < x < dp+2" and 0 < y < f, and tp(x,y) = p? for d,-'<r < x < d. +t and 0 < y <'£-.

(2„) There is a positive number o* such that a~ < t and such that (x ,0,t ) on C. implies i!/-(x,t) = p. when (x,t) is in the domain of ty, and both |x-x | < cr and |t-t | < <r (i=l,2).

Then there is a cp such that 0 < Cp < c, , u has continuous second derivatives with respect to x and y in k)~ = j(x,y,t ) | -co <x<co, y > 0, 0 < t < c2f , and /\u-X u = h in $? so that h is the true Helmholtzian of u in ^L.

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39

Theorem III is proved using the lemmas vhich follovj. Choose IL so that |i (x, t ) -cL (x,t ) | : M, | t-t | for all (x,t) and (x,t) in the domain of cj>.

Let O = gib 4 (x,t) where the greatest lower bound is taken

x

over all (x,t) such that <!> (x,t) > 0, lx_x0l > <J" or |t-tQ| > <j-for each (x ,0,t ) on C1 or Cg, and d1-( 2D1c+sq+ K) < x < d.+^c+s^fc Then u> > 0.

Choose Co so that 0 < Cp < c, , D,Cp < r-, D-iCp < ?r, and c2(WD1+M1-D1D2 log Dj^Cp) < ^ .

Lemma (3.1 )» h is uniformly HBlder continuous in some neighborhood of each point in £'p.

Proof of Lemma (3*1 )» Let (x ,y , t ) be any point in>£p. If

B > 0 or 3 =0 with (a , O.y ) not on C, or COJ then h is uniformly

0 O 0*0 L d

Holder continuous in some neighborhood of (x0>70tt ) ^>y Lem^a (2.2). The only remaining case is the one in which 3=0 and (ao»0>Yo) is on C. or Cp.

Case I [3 = 0, y0 > 0» an<3 (ao»0»Yo) is on C,]. Suppose there is a point (x ,y ,t ) in /J ~ such that

s =7(x^-xo)2+(yo-y0)2+(t0-t0)^ < sQ, y0 > 0, and either |aQ-x| ><r or | y -t | > <r for each (x,0,t) on C,. Then 3=0 and

'y(VVVz)l = l3r(x0,yo,¥o,z)-y(x0,yo,¥o,Y0)| < dJz-yJ

< D,Cp < t- for 0 < z < Cp. Thus for 0 < z < c? we have

= |P2[x(xo,yo,?o,z), y(x0,yo,t0,z),z] -F2(%'°^o)'

< |P2[x(xo,yo,to,z), y(x0,yo,tQ,z),z] - P2[x(xo,yo,tQ,z ),0,z] | + |<L[x(x .y .T .z),z]-<L (aft,Yn)l

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k-o

< -D2DlC2 log D1c2+W|x(x0,y0J0,2)-x(x0,y0,f0,Y0)l+Ml|a-Y0l (since |y(xo,y0,tQ, z ) | < D^g)

< "D2D1c2 log D1c2+VJD1|z-Y0l + \ |z-Y0 I

< c2(WD1+M1-D2D1 log Dxc2) < ^ .

Also |a0-o0l = l(a0-X0)+(x0-x0)+(xo-%)l < V2+so+Dlc2 S° that a > a -(2D, c0+s J > d, -(2D, c+s + fr) . Similarly

O — O 1 d O — L ± O

a < d, +2D, c+s +C and hence from the definition of cj we have o - 14. 1 0

4x(vV ± w-

Since lyt(xo,yo,t0,z)-yt(xo,yo,^0,Y0)l < $ for 0 < z < c2 and since <l>x(ao, YQ) > tJ , then yt(x0,y0»^0»z ) > yt (x0»y0>^0»Y0)- ^ = 4x(aQ,"Y0)- I > ^- f = % for 0 < z < Og.

Since (a ,0,y ) is on C, we have either Iy0"YqI > ^~

a -a I > - by the_choice of (x >yo,TL). Thus either

Y ~

-2 (°^ <£l«° yt^o^o^o*z)dzl

or

<T < Y -Y "o ' o

d£

Y,

■ I ly(x0,y0,^0.Y0)-y(x0,y0J0,Y0)l - a ly(x0,y0,^0,Y0)l = £ |y^o,yo,^0,Y0)-y(x0,y0,to,Yo)l

p exp(-2D?c, )

^D [2(Dl+1)s3 ■ or <r < |a0-aQ|

= lx(x0,y0,T0,Y0)-x(x0,y0,t0,Y0) I

< U(xo»5F0^o^o)-x(xo'yo'to^0)l + \xl*0,70.*0.r0)-*l*0,70,t0.r0)\

exp(-2D0c.) 2D, exp(-2D?c, )■

< [2(D1+l)s] d X +D1lY0-Yol 1 (l^i)[2(D1+l)s]

From this we see that all small enough neighborhoods of (x0>yo»^0) do not contain any such points (x J J ). Therefore for all small enough neighborhoods of (x ,y , t ) we have (xQ>J0>t'0) in the neighborhood and Yq > 0 implies h(x^,y^,to) = ^ ( a^ , y^ ) = P^«

'

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in

Now suppose there is a point (xQ,y0,t0) in 4^ such that y = 0 and s < s . Then 0 < pQ = F0"P0

< ly( V^o'V V^o^o^o'V l + l3r(5c0,y0,Fo,Y0)-y(x0,y0,t0,Y0) I

exp(-2Dpc, )

< D1Iy0-Y0I+[2(D1+1)s] * X . Since ^ Iyo"Y0I < D.^ < ^ >

then 0 < p < f for all such (xo,yo,tQ) near enough to UQ,yo,to).

Also aQ = x(x0,yo,70,Yo)-x(xo,yo,^0,Y0)+x(x0,yo,^0,Y )

. , exp(-2D0cn )

- x(xo,yo,to,Y0)+ao < D1Iy0-V0I+C2(D1+1)s] 2 x + aQ. Again

D-, I Y -Y I < 2^ • Also a < dp since (ao*0>Yo) is on C,. Hence

a < d~+t and similarly aQ > d1-f for all such (xo,y0,tQ) near

enough to (x ,y„,t ). It follows that for all such (x .y ."E ) ooo ooo

near enough to (x-0tJ0»b0) we have d-^-t < aQ < d2+t , 0 < P0 <t, and hence h(5cQ,yo,to) = *2(aQ,P0) = p^.

Therefore in Case I h(x0,y0,tQ) = p1 for all (xQ,y0,^0) near enough to (x ,y ,t ). Hence h is uniformly Htilder continuous in a neighborhood of (xo>yo>b ) .

Case II [PQ =0, y0 = 0, and (ao,0>Yo) is on ci^« Since

a, p, and y are continuous at (x ,y ,t ), then for (x .y ,t ) in * • ' ' 0 0 o o o o

A) P near enough to (x >y ,t ) we have h(x ,y ,F ) = p,, and h is

uniformly Htilder continuous in some neighborhood of (x ,y »t ).

Similarly h is uniformly Httlder continuous in a neighborhood of (x-Q»yQ>t ) when p = 0 and (ao>0>Yo) is on Cp. This completes the proof of Lemma (3.1).

Theorem III follows from Lemma (3.1) as Theorem II followed

from Lemma ( 2.2) .

Next we want u , u , and u to be bounded at infinity. xx' xy' yy J

This is accomplished in Theorem IV.

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1+2

Theorem IV. Let 4, <lh, and \J/p satisfy the hypothesis of Theor

III. Let 4 also satisfy (1D).

(ln) gib 4 (x,t) > 0 where the greatest lower bound is taken

over the set of all (x,t) such that 4 (xft) > 0 and either |x-x \xr

or |t-t > cr for each (x ,0,t ) on C, or C0. o — o' * o 1 2

Then there is a c, such that 0 < c, < c~ and u has bounded second derivatives with respect to x and y in £)•> = j (x,y,t ) | -co <x<co, y > 0, 0 < t < c^ i .

Let oj = gib (p (x,t) xvhere the greatest lower bound is taken over the set specified in ln of Theorem IV. Then 0 < ^ < uj.

»

Let Co satisfy 0 < c, < c^ and c, (WD, +M.j-D,Dplog 0,0^) < % Again we prove several lemmas to aid us with the proof of the theorem.

Lemma (Lj..l). The HBlder continuity of h in (x,y) is uniform in £3 = j(x,y,t)|{x,y,t) e & with x < d-L-2ir-D1c3-l or x > di +2<r +D1Co+l or y > 2D,c-+lj- with respect to (x,y) and t.

Proof of Lemma (lj.1). Let (xQ,y ,t ) be any point in ^ ^ and suppose p = 0. We will show that then y.(x ,y ,t ,t) > % for 0 < t < Cy We have yQ = y(xo,yo,tQ, tQ )-y(xQ,yo, tQ, Yq)

< DtI^q'YqI < Dic"3« Therefore since (x ,y ,t ) is in $ , we have

x^ < d, -2cr -D, c-,-1 or x > d, +2o_ +D, c-,+1. Hence o—l 13 0 — q. 13

ao = x(xo^o»to'Yo)-x(xo»y0'to»to)+xo ^ Dlc3 + xo ± V2"--1 aQ > d, +2S"+1, and thus $x(a0>ro) > <£>• Then for 0 < t < c*

have lyt(xo,yo,to,t)-yt(x0,yo,t0,Y0)l

< |F2[x(xo,yo,to,t), y(x0,yo,to,t),t]- F2[x(xo,yo,to,t),0,t]|

+ 14 [x(x ,y ,t ,t),t]-4 (a ,y )| X o,Jo* o* ' Tx o''o '

or

we

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>

k3

+ M1|t-rol < -D2D,c^ log D^c^+WD^Co+MnC^ < £y and hence

yt^o'V^^' ^ 3Tt(x0,y0,t0,Y0)- 2 = *x(ao'Yo)" 2 ^ 2 for

0 < t < c-.

Now

let (x0,y0,tQ) and (xQ,yo,^0) be in $3 with

s =V(x -x ) +(y -y ) +(t -t ) < s . Consider the case where vv o o wo Jo o o o v —

p = 0. When yo > Y0, then "S = 0 and 0 < yq-Y0 = | |° 2 d^

W Y

Y0 To

< i f yt< WV*^ = - ytx0,y0,t0,Y0)

° _ exp(-2Dpc, )

■ z ry(»0.y0'to'Y0)-y(x0,y0,t0,Y0)] < i [2<d1+i)s]

Y When y0 < Y0» then 0 < YQ-Y0 = Z ) % d^

Yo Y

i = ) yt(w*o*€)a6 = " - ^wW

L> - O

3 ° P exp(-2Dpc,)

< z Ey(vyo'fo'Y0>-y(x0,y0,t0,Y0)] <r [2(d1+ds]

Co LJ

P exp( -2DpC, )

Hence |y -Y I < - [2(Dn+l)s] when P = 0 and s < s .

oo—— 1 *o o

Now consider the case where P > 0. If 6 =0 for some

o o

UQ,yo,to) in ^ ^ with s < sQ, we obtain I Y0"Y0 I

2 exp(-2D2c1)

< — [2(D, +l)s] as in the previous case. If jJ > 0, we

U) _ 1 have Y0 = Y0 = 0.

p exp(-2DpC )

Thus |Y -Y I < Z [2(D1+l)s] for s < s . Since

to

ao ~ x(xo»yo,to,Yo) and Po = y^xo,yo,toYo^' a similar result follows for a and p. Since i|/, and typ are uniformly Holder con- tinuous, we can easily obtain the conclusion for Lemma (lj.,1).

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Proof of Theorem IV. Using the integral representations given in

(II. 3), (II.lj.), and (II. 5) of the oroof of Theorem II; the fact

that h(£,>£, t)+A (a£+b) is bounded; and the result of Lemma (lj..l),

we could show that v , v , and v are bounded in jj\ -,. It fol-

xx xy yj 2>

lows that u , u , and u are bounded in ~jq ~. Since u , u ,

XX siy Jtf J A A A j

and u are continuous in *\ by Theorem III, then u , u , and yy j xx xy

u are bounded in the closure of /^ - $ ,. Hence u , u , and u are bounded in fcK .

yy 3

We now come to our final existence theorem. Theorem V. Let 4» t-j, and to satisfy the hypotheses of Theorem IV. Let (}>, \{/, , and \|/p also satisfy the following assumptions some of which are repetitions.

(1.) 4> <t i and (}> are continuous and have continuous

JrL X AA

bounded first derivatives with respect to x and t. Also

^xxx^^xxx**'10' - Li^-xl1 and Uxxt(x,t Mxxt(x, t ) | < Llx-xl1

for all (x,t) and (x,t) in the domain of (J>.

n (2.) \J/, , f, , and \|/,. are continuous. \j/, and ty,t are

bounded and uniformly Htilder continuous.

(3A) $2} ^2x* £nd ^2v are continuous« ^px and ^2v are bounded and uniformly Holder continuous.

(3g) ^(XjO) = ^2(x,0) and

*lt(x>0) = 2£ ^2x(x»0) J J Sy(x,0;£,4)*2(€,*)d€d4

co

- \ t2x(x,0) i [X2c|,(C,0)-4xx(5,0)]K(Xk-x|)d4-ci)x(xJ0)t2y(x,0)

-00

for (x,0) in the domain of both ij/, and \|/p.

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J

US

Then u satisfies (lfA), ikn) $ (^q)i and ikj)) in/v,.

(if..) u and its first and second partial derivatives with respect to x and y are continuous, and they all have continuous first partial derivatives with respect to x, y, and t.

(h!) (|r - u 4- * U #-)(Au -X2u) = 0.

(i^) u(x,0,t) = 4(x,t),/\u(x,0,t)-X2u(x,0,t) = ^(x^) when (x,t) is in the domain of \|/,, and /\u(x,y,0)-\ u(x,y,0) = *2(x,y).

(if.D) u(x,y,t )-ax-b and its first and second partial derivatives with respect to x and y are bounded.

Again we break up the proof of the theorem into several lemmas.

Let Jr and ^ be defined by -^(x,y,t) = -u (x,y,t) and ^2(x,y,t) = ux(x,y,t) for (x,y,t) in^, and J^(x,y,t ) = u (x,-y,t)-2u (x,0,t) and c?p(x,y,t) = 2ct> (x,t )-ux(x,-y,t ) when (x,-y,t) is in/C*o» Then 3~-. and *-^ are continuous and have con- tinuous first derivatives with resoect to x and y. For (x .y -t )

a 000

in the domain of J?, and 3"~ let X(t) and Y(t) be functions such that

X(tQ) = xQ, Y(tQ) = yQ, and £j£l = 5\[X(t ),Y(t),t] and $^±

= X[X(t),Y(t),t] for 0 < t < Oy X(t) and Y(t) exist for

0 < t < c, since j?-\ and t? are continuous and bounded. X and Y

are unique since t-% and j~ have continuous bounded first derivatives

with respect to x and y. Since X(t) and Y(t) also depend on

(x ,y ,t ), we use the notation X(x ,y ,t _,t) for X(t) and

Y(x ,y ,t ,t) for Y(t). We also observe that X(x .y^.t .t) and 0 0 o o o o

Y(x ,y ,t ,t) have continuous bounded first derivatives since 7,

0O0 J-

and -?p have continuous bounded first derivatives with respect to x and y.

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^6

Let (x ,v ,t ) be in ^L and let t vary in an interval con- o '* o7 o 5

taining t such that [X(t ) ,Y(t ), t ] is in A,. For such t we have J^[X(t),Y(t),t] = Pi[X(t),Y(t),t] for 1=1,2. Hence

x(xo'yo'to't) = x<'Wto»t) and Y(xo^o'to't) = y(xo,yo,tQ,t) for such (x0,y0,tQ,t). Therefore aQ = X(xQ,y0,t0,Y0) and

p0 = Y(xo*y0' W for (x0'yo'to) in/S3-

Lemma (5.1). a, P, and y have continuous first derivatives at (xo,yo,tQ) in/6^ if pQ > 0 or y0 > 0 with (aQ,0,Yo) not on (^ or

c2.

Proof of Lemma (5.1 ). We will show that y has continuous deriva- tives at the points mentioned. Since a = X(x ,y ,t ,y_) and P = Y(x , y ,t ,y ), the conclusion regarding a and p follows from the fact that X and Y have continuous first derivatives.

Case I (p > 0). Since P is continuous at (x ,y ,t ) , we can choose a neighborhood Rfi of (x ,y ,t ) such that (x ,y ,t ) in RQ implies p > 0. In such a neighborhood we have yq = 0 so that y has continuous first derivatives at (x ,y ,t ).

Case II (y > 0 with (a _,0,Yn) not on C, or Cp). Since a and y are continuous at (x ,y ,t ), we can choose a neighborhood Rg of (x , y ,t ) such that (x ,y ,t ) in R5 implies yo > 0 and (aQ,0,Yo) is not on C1 or C2» Hence FQ = Y(xo,yo,tQ, yq) = 0 for

(VVV in R6« Since VV^o'W = *x( W > ° for

(x ,y ,r ) in Re. we conclude from the implicit function theorem ooo o

that y has continuous first derivatives at (x ,y ,t ).

Lemma (5*2). h has continuous first derivatives in/6L.

Proof of Lemma (5.2). The proof follows from Lemma (5.1) if Po>0

or y >0 with (a .O.y^) not on Cn or C0 since then h(x .y^,t )

' O 00 X d. OOO

- t1(aQ,P0) or *2(a0,Y0).

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If 6 =0 and (a ,0,y ) is on C, or COJ then h is a constant

"O O' ' O 1 d

in some neighborhood of (x0>yo> *-<-,)•

The remaining case is where P0 = YQ = 0 and (ao>°>0) is not on C, or Cp. We note that a, p, and y are continuous at (x0>y0>t ) and <}» (a ,0) f 0. Suppose there is a sequence A ( x , yo* t ) | of points in -$o such that Y(xn,y0,tQ) = 0, xn~x0 f 0, and xn -> xQ as n — > co . Then

°<V W-% . »(«n.V*o.O)-X(Vy0.to.O) (x ,,.».,„) as

x -x„ *„-x« x ' o,Jo* o*

no no o

P(x ,y ,t )-(3

n -> co , " — ° Y° -> Yv (x ,y ,t ,0) as n -> co , and hence

xn"xo xQ o o o

h(xn,yo>to)-h(xQ,yo>to) _ »2[a(xn,yo,to)>p(xn>y0>to)]-»2(ao>0)

xn~xo xn"xo

~ > Xx txo#yo.tOf0,*2x(ao*0)+Tat (xo^o'to'0),"2y(ao'0) as n ~> °° *

o

2y'

Suppose there is a sequence \(xn>Y0ttQ) t of points in /j~

such that y(x ,y„,t ) > 0, x -x„ 5^ 0, and x_ -> x^ as n -> co . Then noo 'no' ' n o

^o'Vy^yyo'V3 = Y^o^0>to^(xn,yo,to)]-Y(xo,y0,to,yo) xn"xo " xn"xo

y(x ,y ,t )-y

= W^o'W where Yn 1« between Y(xn,y0,tQ)

n o

and yQ.

Also Y[xn,yo,to,y(xn,yo,to)] = P(xn,yo,tQ) = 0 and

Y[xo,yo,to,y(xn,yo,to)3 = Y[vyo^o,Y(xn,yo,t0)]-Y&cn,y0>to,y(xn,y0,t0)]

X>-,~X« X«~X«

no no

= - Yx ^n^o^o'^n^o^o^ where xn is between xn and xQ. o

Y(x_,y0,t )-y0

Therefore \ ° ° W^o'V^

n o

= " Yx tVy0'to'Y(xn'yo'to,]' As n "* °° VwW o

— > Y. (x ,y ,t ,y ) r 0« Therefore for n large enough we have t o o o o

.

Yt(xo'yo'W ^ 0 and —

Yx tvyo^o'^^yo^o" n VXo,y°'to,0)

= ^=—7 3: — m — t — > -t — r— — pn as n — *• 00 •

An a(xn^o*to)"ao X[xn>yo^o^(xn^o^to)3-X(xo^o*to>Yo) Also _^ = __

no no

Y| y "IT "t / " Y*

= V^,y°'to'Y(Xn'y°'to)]* "'v^o ° VVWV

where x is between x and x and yn is between y0 an^ "V(xn»y0»t0)»

/ 4. \ Y (x ,y ,t ,0)

a(x ,y ,t )-a x x owo» o*

HenCe W ° - Xx (xo^o'V0) + V(an,0l V V°>°>

no o x o

as n -3>- 00 .

Let \j7.,(x,t) = \J/,(x,t) when (x,t) is in the domain of \Jr, and, when (x,t) is not in the domain of ij/-,, define \L so that ij7, is continuous and has continuous derivatives everywhere. Then h(xn»yo*to)"h(xo^o^o) = t1[a(xn,yo,to),Y(xn,yo,to)]-?1(ao>Y0) xn"xo " Xn"xo

= ^V^V^o +r(yy0.t0)-Y0

x -x YlxL n' ' v n,Jo» o/J x -x yltv o* 'n

no no

(where a is between a(x ,y„.t^) and a and 7 is between n n'^o'o o n

Y<WV and Yo}

r* Yx ^xo,yo,t ,0^ "^

~* jXx (V^o'V0^ % (a ,0)° Vao'°>0) *lx(V0)

Y (x ,y ,t ,0) y

x o,Jo* o*

° $ (a ,0) *lt(ao>0) ^ n -^ co.

Tx os

Prom (3td) of the theorem we obtain

h(x ,y ,t )-h(x ,y ,t )

2_2 — 2 2__2 — 2 > x (x ,y ,t ,0\|/o (a ,0)

x-x xvo*Jo,o,y2xvo'

no o

+ Yx (xo,yo,to,0)\|f2 (aQf0) as n -> 00.

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We may now conclude that h (x ,y ,t ) exists and

xo'^o'o

hx (xo'^o'to) =Xx <xo»yo'to'0)*2x(ao'0,+Yx (xo>W0)*2y(V0)

o o o "

(we use h , h , and h. to denote the derivatives of h since h

xo yo o was defined as a function of (x ,y ,t Q)).

The continuity of h in //, follows easily using (3R) of the

xo -5 theorem.

Similarly we can show that h and h. exist and are con-

yo to tinuous

Lemma (5.3 )» The first partial derivatives of h are bounded in

Proof of Lemma (5.3). By examining the expressions for the first

derivatives of h we can easily show that the first derivatives

are bounded in any set such that if (x ,y ,t ) is the set and

000

PQ = 0, then 4x(<*0,y0) > ^' Since the set of points (x ,y ,t ) for which p = 0 and <f> (a ,y ) < £3 is a bounded set, and since the first derivatives of h are continuous everywhere, it follows that the first derivatives of h are bounded.

Lemma {$.}+). (hp.) is valid in /a ?,

Proof of Lemma (5.'-j-)» We have already shown that u, u , u , u , " x y j&yi

u , and u are continuous in /S' ~, We have yet to show that

u., u. , and u. exist and are continuous in /v, and that u , t tx &y j xx

u , and u have continuous first derivatives with respect to x, y, and t in S -.

We could show that w and its first and second derivatives with respect to x and y have continuous bounded first derivatives with respect to x, y, and t using the same methods used to orove Lemma (1.1),

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In a straight forward manner we can show that v. , v. , and

v. exist and are continuous since h. is continuous and bounded, ty t

Hence we may conclude that u. , u. , and u. exist and are continuous.

Since h has bounded first derivatives, h is Httlder continuous in (x,y) where the Holder continuity is uniform with respect to (x,y) and t. Hence, using (II. 3), (II. k) t and (II. 5) of the proof of Theorem II, we can show that v , v , and v are Holder continuous in (x,y) where the Holder continuity is uniform with respect to both (x,y) and t. This can be shown with arguments similar to those used in proving Lemma (1.3) for all the integrals excepting the last. We can show that the last integral has continuous bounded first derivatives with respect to x and y so the result follows for the last integral also.

Since w , w , and w have bounded first derivatives with

respect to x and y in A* -,, then w , w. , and w are Holder con-

s> xx xy yy

tinuous in (x,y) where the Holder continuity is uniform with

respect to both (x,y) and t.

Since u = v-w+ax+b, it follows in A/, that u , u , and u

' 3 xx' xy' yy

are Holder continuous in (x,y) and that the Holder continuity is uniform with respect to both (x,y) and t.

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Next we will show that the first derivatives of X(x ,y ,t ,t) and Y(x ,y ,t ,t) with respect to x , y , and t are HBlder con- tinuous in (x ,y ) and that the Holder continuity is uniform o' o

with respect to (xQ,yQ), tQ, and t. Let (x0,y0,tQ) and (x0,y0,tQ)

be any points in //', with s = J (x -x ) +(y0*yo) • Let

Zl(t) = |Xx (xQ,y0,t0,t)-Xx (x0,y0,t0,t)| and Zg(t)

= 'Yx IVVV'^ (xo,y°,t0,t)|. Then Xx (xo,yo,tQ,t) o o o

= 1 ♦ 5 Xx U0)y0,V5)>lx[X(x0,y0,t0,5),Y(x0,y0,t0,5),5]d§ t ° t °

+ \ Yx ^o'yo'to'5)>lyCX(xo^o'to'5)'Y(xo'yo»to^,'«]d«- Then

t ° o

there are constants M and e (0 < e < 1) such that for s small

enough we have z, (t) t x

l([Xx (x0,y0,to,^Xx(x0,yo,to,a]^lx[X(xo,yo,t0^),Y(x0^o,t0^U]c? A o o

r

+ j ^0(x0.yo.t0.5)|>l3t[X(3c0,y0,t0,e)1Y(xo,y0,to>5),S]

o

-/lx[X(xo,yo,to,^),Y(x0,y0,t0^)^3j d? + etc. |

t t

t_ t

< | J [MZl(^)+Mse+Mz2(^)+Mse]d^| <H|j [z1(^)+z2(^)]d^+2Mc

0 t

Similarly z2(£) < m|( [z^S )+z2(£ )]d£ |+2Mc3se. Let

t o

(t) = 1 1 [z1(^)+z2(^)d^|. For t > tQ we have

R .

i p a -2H(t-fr_J _ -2M(t-t )

R (t) < UMc3sE+ 2MR(t), ^ R(t) e ° < i|Mc3se e

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-2E(t-t ) . -2M(t-t ) 2M(t-t )

R(t) e ° < -2c3se(e ° -1) , R(t) < 2c3se(e ° -1)

< 2c3se(e2^c-l). Thus z-^t) < "E (t )+2Mc3se < 2Mc3see2Mc. We obtain the same result when t < t . In a similar way we can show that the other first derivatives of X and Y are Holder continuous in (x ,j ) uniformly with respect to (x ,y ), t , and t.

Now we could show that in some neighborhood of a point (xo,yo,ro) the first derivatives of a(x0,y0,to),p(x0,y0,t0), and y(x ,y ,t ) with resnect to x -y^. and t are Htilder continuous in

OOO * O ' O ' o

(x ,y ) where the HBlder continuity is uniform with respect to (xQ,yo) and tQ provided that p(xo,y0,?o) > 0 or P(x0,yo,tQ) ■ 0 with [a(xQ,yo,to),0,Y(xo,yo,^o)] not on C1 or C2.

Next we could show that in some neighborhood of each point in A'~ the first derivatives of h(g,^,t) are Htflder continuous in (^j'V) where the HBlder continuity is uniform with respect to (C,1 ) and t.

For an arbitrary point (x ,y ,t) we have

(5.^.1) v^U^t) = ^ }) Sx(x,y;^,'?)[h^(?/(,t) + aX2]d?dn

t

>0

= h. lUx(x»y^^/Hhc(^/(,t)-.hc(xo,yo,t)]dgd^,

^>0 (5.IJ..2) vxy(x,y,t) = it [\ Sy(x»y^/0[hc(C/(,t) + aX2]d?da

^>0 ^>0

co

" h^(xo'yo't) 5 J K^v^d^ * and

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(5.i+.6) vyy(x,y,t) =-^ jj ^(xfyj5f1)^(Wft)«d*

/»>0

oo - & | ^K1 Uv)[h(£,0,t)+X2(a£+b)]d£d>7 .

-oo

Since bu. and h. are Htflder continuous in (x,y) uniformly with respect to (x,y) and t for (x,y,t) in some neighborhood of (x »y >t), we can differentiate under the integral signs with respect to x and y at (x ,y ,t), and we can show that the resulting derivatives are continuous at (x0>yo»t).

We could show that we can differentiate under the integral sign with respect to t in (II. 3), (II. ij.), and (II. 5) (contained in the proof of Theorem II), and from the resulting expressions

we could show that v. , vtxv» an<* vtvv are continuous.

Pinallv it follows that u , u , and u have continuous

xx' xy' yy

first derivatives with resoect to x, y, and t. We remark that it

would also be possible to show that the first derivatives of u , ^ xx*

u , and u with respect to x, y, and t are Holder continuous in (x,y).

Lemma (5«5). (ij-g) is valid in/a-,.

Lemma (5»5) is obvious since h is constant along the air particle paths of u.

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Part III Uniqueness Uniqueness Theorem. Let cj>, t-i and typ satisfy the hypothesis of Theorem V. Let u be any real valued function with domain /y o

such that (UA)» (^r)> ^C ^ ' and ^D^ are valid with u replaced by u. Then u = u in A) ?»

Proof. Let h = Au ■ ^ u. Prom (luj we see that u(x,y,t )-ax-b,

u„(x,y,t)-a, u (x,y,t), and h(x,y,t)+\ (ax+b) are bounded in /Q~t x y j

and hence we can show that (3) is valid with u and h replaced by u and h respectively. This result follows from

1 J [A5 ~ X2u + X2(a^+b)] g(x,y;€,1)d£<tt

-J [hU^,t)+X2(a£+b)]g(x,y;£,>2)d£d>, =|[gd('I^"bi - (u-ag-b)§§]ds

where the double integration is over the region defined by

2 2 2 ^ > 0, £ + /^ < R , and p < e, and the single integral is taken

along the boundary of the above region in the positive sense.

Letting R -> co and then e -a- 0 we obtain (3) with u and h replaced

by u and H respectively.

Obtain functions vlr J^V, X, ¥, a, ]?, and y from u as Jr,, ^,

X, Y, a, p, and f respectively were obtained from u. Methods

similar to those used previously can be used to show that j-~ , ^o>

X, and Y have bounded first derivatives with respect to all their

variables. Choose D to be an upper bound in /j * of the absolute

values of/., 7^,, /., j~~ and the first partial derivatives of -A,,

3 2, ?"1$ ^2, X, X, Y, ?, and h.

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Uniqueness Uniqueness Theorer.. Let i, '.. ana v~ satisfy -he hvpcthesis of Theorem 7. Let u be an:* res! value:. :" an crier, --irh rcr.air. / - such that (ii.^), (!i3), (k-c), and (— ) are valla with Q re; L::ei by u. Then u s a in /)..

Proof. Let E = /^u - X2u. Prom (LI) we see that u(x,y,t )-ar.-t , ux(x,y,t)-a, u (x,y, t ), and h(x,y,t)+X (ax-*fc) are bcunaea in /? ., , and hence *..:e can shov.- that (3) is valaa with a ana h realarea by u and h respectively. This resale follows f r : i

1 l[/\u-X2u + X2(a^+o)3 g<x,y;€,-l)d£d*

[K(5,^t)+X2(a4+b)]g(x,y;£,^;dLa; = )Cg1(u":-'b } - (~-ag-b |# ! as v?here the double integration is ever the region defined by

/?> C, 5

2 . .2

T1

< R , and p < e, ana a he single integral is t alien

along the boundary of the above region in the positive sense. Letting ?. — s>» cc =na aher. e — > I we obtain (3) with u ana h replaced by u and h respectively.

Obtain functions^, J^V, X, 7, o, 3", ana ~ fror. u as >/,, -T- , X, Y, c, ,3, and y respectively *:ere obtained from a. Methods similar to those used previously can be used a: shev: that j% , ^-, X, and Y have bounded first derivatives with res;e:: ;: all their variables. Choose D to be an upper bound in,/. - :f the absolute values of _71 , r^^x* ^~o ~-~ -r-e --— ~ partial derivatives of -a,, ->*?> »7^» ^2J Xf '*' *-' Y» and "*

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Assume u(x,y,t) j~ u(x,y,t) in/>,. Let c - sup t where the sup is taken over all t > 0 such that u(x,y,t) = u(x,y,t) for (x,y,t) in/CL and 0 < t < t. Possibly c =0. If c a 0, then u(x,y,c") = u(x,y,c") follows from (3)» If c > 0, then u(x,y,c") = u(x,y,c") follows from the continuity of u and u and the fact that u(x,y,t) = u(x,y,t) for 0 < t < c".

Assume c" < c,. Then we will arrive at a contradiction by showing that there is an e > 0 such that u(x,y,t) = u(x, y,t) for c" < t < c"+e. It follows then that c" = c^, and Theorem VI is proved.

We have shown that h is identically p. in some neighborhood of each point on C. (i=l,2). Hence we can choose 6, > 0 so that h(x,y,c") = p. for |x-x.(c")| < 6, (1=1,2) and 0 < y < 6,, and also h(x,0,t) = p. when |x-x.(c")| < 5, (i=l,2) and c" < t < c"+6,.

Then we choose 6- > 0 so that 6p < 6.. and |x. ( t )-x. (c ") | < -^

(1=1,2) for c " < t < c"+6p. Then h(x,0,t) = p. when (x,t) is in the domain of \J/, if |x-x.(c")| < 6, (1=1,2) and c" < t < c'+6p.

Since u(x,y,t) = u(x,y,t) for 0 < t < c", then h(x,y,,c") = h(x,y,c"). Also h(x,0,t) = \J/,(x,t) = h(x,0,t) x^hen (x,t) is in the domain of \J/,. Therefore h(x,y,c") = p. for |x-x.(c")| < 6.. (1=1,2) and 0 < y < 6^, and h(:-:,0,t) = p. when (x,t) is in the domain of \J/, if |x-x.(c")| < 5.. (i=l,2) and c < t < c"+6p.

Let w" = gib (J) (x,t) where the greatest lower bound is taken

over all (x,t) such that 4 (x,t) 0, c" < t < c"'"+5„, and either

26 x "26, " ., d

x < x1(c") ji or x > x2(c") + —y- . Then w > 0.

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f 6 Choose e > 0 so that e < 6 , 3D e (2D+1) < ~, 2D e < -y, and

t oT,ffi-,~ / t , 1 ^ / 1 j 2D+1 n 2Dc 1 12MDe(l+— «) (1+ z- )e < ?.

Let N(u-u) = l|ux-ux|| + l|uy-uy|| with l|ux-uxl|

= sup |ux(x,y,t)-ux(x,y,t) | and ||u y-u || = sup |u (x,y,t )-u (x,y,t )

where the sup is taken over all (x,y,t) in *■'■ , such that

it Jt

c ' < t < c "+e.

We now insert several lemmas. Lemma (uT.l). |x(xQ,yo, tQ, t )-X(xo,yo,tQ,t ) | < 3 e e2Dc N(u-u) and

'7(xo'yo'tojt)'Y(xo'yo'to*t)l ^ 3 e e2D° IJ(u-u) for c* < tQ < c'::"+e and c " < t < c "+e.

Proof of Lemma (uT.l). For any fixed (x ,y ,t ) with

< tQ < c*+e let 2l(t) = |x(xo,yo,to,t)-X(xo,y0,to,t)| for

< t < c +8 and z2(t) = |Y(xo,yo,tQ,t )-Y(xo,yo,tQ,t ) | for

c < t < c +e.

Then Zl(t) = |xQ+ j ^-1[X(xo,yo,to,?), Y(xo,yo,tQ,S ),£]d£

- x o

>1[X(xo,yo,to,?), Y(xo,yo,to,C),?]d^|

< \]&mx0,J0,t0,V,*ix0,70,t0$S),&^

+ ij p1tX(xo,yo,t0,?),Y(xoiyo,t0,4)^]

->1[X(xo,yo,to,?), Y(xo,yo,to,4),^]j d£

t J

<D|j [z1(C)+z2(5)3d^| + 3||u -u || |t-tQ|

< 3e llu -u || + D|] [z1(?)+z2(4)]d^|. Similarly

we obtain

57

z2(t) < 3e llux-ux|| + D|\ [z1(?)+z2(?)]d?| so that

0 t Zl(t)+z2(t) < 3sN(u-u)+2D| j [z1(C)+z2(5)3d5|.

t t °

Let R(t) = h [z1(?)+z2(^)]d^| for o* < t < c':+e. For t>t(

to

we have R (t) = z1(t)+z2(t) < 3eN(u-u)+2DR(t ) ,

-2D(t-t ) _ -2D(t-t )

R (t)-2DR(t) < 3eN(u-u), ^[R(t)e ° ] < 3eN(u-u)e ° ,

-2D(t-t ) % _ -2D(t-t )

R(t)e ° -R(tQ) < - |g N(u-u)[e ° -1], and

•5c 2D(t-t_) o_ ?T\n

R(t) < g N(u-u)[e ° -1] < ^ N(u-u)(e^c-l). Similarly we

obtain the same result when t < t . Therefore z.(t) < z(t)+z?(t) < 3eN(u-u)+2DR(t) < 3ee2Dc N(u-u) for i=l,2.

Lemma (Tu.2). |E(x.y.t ) -h(x.y. t) | : 6De(l+-^li)e2Dc N(u-u)

— — — — — — — — — — O O O O O O *~ V) «

for (x ,y ,t ) in w, with c < t < c +e. oo'o 5 — o —

Let (x >yD»t ) be any point in /~J~> with c" < t < c"+e. If y > 0 and t > c " let t, (t sub-boundary) be the largest number such that c" < t, < t and Y{?i ty ,t ,t.) = °» If no such tb exists, let t, = c'.

If yQ = 0, tQ > C*, and 4>x( vV > 0, let tfe = tQ. If 4 (x ,t ) < 0, let t, be the largest number such that c" < t, < t

X OOD — DO

and Y(x ,y ,t t, ) = 0. If no such t, exists let t, = c".

If t = c", let t, = c",

o * b

Let xb = X(x0,y0,t0,tb) and yb = YU0,y0,t0,tb). Then (xb,yb,tfe) is a point where the air particle path of u enters the slab c" < t < c"+e.

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In a similar manner we obtain numbers x\, y, , and "E^ using X and Y.

Consider the case where t, > c " and ^(x^t^) > to* Then for c*< t < c'"'+s we have yfe = 0 and |Yfc(x0,y0,t0,t )-Yt(x0,y0,tQ,tb) |

< |>2[X(xo,yQ,to,t), Y(xo,yo,tQ,t),t]

" ^2»(xofyo,t0,t), Y(xo,yo,to,tb),t] | + l4x[X(xo,y0,t0,t),t]-4x(xb,tb)|

< 3D[|Y(xo,yo,to,t)-Y(xo,yo,to,tb)|+|X(xo,yo,to,t)-xb|+|t-tb|]

< 3D[2D+1] |t-tb| < 3De (2D+1) < ^, and Yt (x0>y0,t0,t )

> Yt(x ,y ,t ,tb)- % - "?• Therefore if *D < tb we have

tb .;. tb

o < VFb a h \ 4 d5 1 4- S YtU0,y0,t0,S)dS

H rb

= - 4 Y(xQ,y0,to,tb) < -% [Y(xo,y0,to,tb)-Y(xo,yo,t0,tb)|

< ~ e N(u-u). When t"b > tb we have y b = 0 and

tb # tb

0 < V*b = 5 1 ^ ** - 7} 1 V^o'V*^ fcb *b

= |: Y(xo,yo,t0,rb) = Jfc tY(xo,yo,to,tb)-Y(xo,yo,to,tb)]

< £i e2Dc N(u-u). Hence |t, -t, | < ~ e2Dc N(U-u). When t, > c* and 4>x(xb,tb) >& we now have |5(xo,yo,t0)-h(x0,y0, tQ) |

= |K(x"b,yb,Fb)-h(xb,yb,tb)| = lh(xbJ^b^b)"h(xb'yb»tb)' <since K(x,y,c") = h(x,y,c") and h(x,0,t) = h(x,0,t) for (x,t) in the domain of \|/, )

< D(|xb-xb| + |yb-yb| + | V*b ' > < D ^ Wo> V V-X(xo^o> VV'

+D[|T(xo,yo,to,tb)-Y(xo,yo,toJtb)|

+ lY(xo^o>to^b)-Y(xo^o^to'tb)l] +D»Vtbl

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59

< 6Dee Dc N(u-u) + 2D2|tb-tb| + D|tb-tb|

< 6Dee2Dc N(u-u) + D(2D+1) ~ e2Dc N(u-u) = 6Dz(l+Q$k)eZDc N(u-u).

Similarly we obtain the same result when tb > c '' and 4x(xb,t"b) > to .

When tb > c , tfe > c*"", 4x(xb,tb) < (J'~ , and <l>x(xb,tb) < (j\ then h(xo,yQ,to) = px or p2 and Mx0,yo,tQ) = px or p2. Suppose E( WV = P]_. Then xb = X(xo,yo,t0,tb)-X(xo,yo,to,to)

+ ^x0.y0,t ,t )-X(x ,y ,t .^J-^ < D|tb-t | + DlVV + xb

-::- 2 l *

< 2De + x,(c ") +— r— < x, (c ") + 8, . Thus we must have h(x ,y ,t )=p,.

Similarly if h(xo,yo,tQ) = p2, then h(xo,yo,tQ) = p2# Hence |H(xo,yo,to)-h(xo,yQ,to) | = 0 in this case.

Next v/e consider the case in which t, = c", t, > c" , and cj>x(xb,tb) < lj\ Then H(xo,yo,tQ) = p1 or p2» Assume h(xQ,y ,tQ) = p-. Then xb < x-,(c")+6, as in the previous case. Also xb = X(xo,yo,to,tb)-X(xo,yo,to,to)+X(xo,yo,to,to)

> x^C*) --J--2D6 > X1(c*)-61, and Yb = Y(xQ,yo, tQ,tb )

-Y(xo^o'to'to)+Y(xo^o'to'to)-Y(xo^o>to>Fb)+yb ^ 2De+yb

= 2De (since Yb = 0). Hence h(x0,yo,tQ) h(xb,yb,tb) = p,, and

|h(xo,yo,to)-h(x0,yo,to) | = 0. We get the same result when

E(xo^o'to) * P2*

Similarly when tfe > c , tb = c , and 4>x(xb,tb) < (j , then

iE<xo»y0'to)-h(xo'yo'to)i = °-

The only remaining case is the one where t, = t, = c • In

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We now continue the proof of our uniqueness theorem. Using (3) with c" < t < c "+e we obtain |u (x,y, t )-u (x,y, t ) |

^ ■■ X X

h \\ SJW,Z,1) [h(€,*,t)-h(S,'<,t)]d£d>L |

2% } J ex Jj>0

<^ (1 + ^r) e2Dc S(u-u) [j |gJC(*.yiM>|d€d*

^ ^>°

used ~ j] |gx(x,y;5,^ ) |d^d1 < I).I12(1+J^) from the proof of Lemma

/£>0 X

(1.3). Therefore ||u -u J| < J N(u-u).

X X ~" J)

Similarly ||u -u || < ^ IT(u-u), and hence N(u-u) < -^ N(u-u). It follows that N(u-u) = 0, u (x,y,t) = u (x,y,t), and

X X

u (x,y,t) = u (x,y,t) for c" < t < c"+e. Hence u(x,y,t)

= u(x,y,t )+z(t ) for c" < t < c"+e and for some function z(t).

Since u(x,0,t) = 4(x,t) = u(x,0,t), then z(t) = 0 and u(x,y,t)

= u(x,y,t) for c" < t < c"+e. But this contradicts the choice of

•::- -:c- _ /

c . Hence c = c, and u = u in ^,,

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GAYLORD		PRINTED IN U.S.A.

NYU IMM-258

c.l

Sensenig

Existence and uniqueness for a third order... n.d.e.

NYU IMM-258

c.l

Sensenig

AUTHOR

Existence and uniquenass_£oit

a third order... p.d.e.

DATE DUE

3h/

BORROWER S NAME

mr-fiTTF-

=&

J m.

fh-u^

N. Y. U. Institute of Mathematical Sciences

25 Waverry Place; New York 3, N. Y.

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