Hyperbolic systems of linear equations with discontinuous coefficients

HYPERBOLIC SYSTEMS OF LINEAR Er)UATIONS WITH DISCflNTINUOUSCOEFFICIENTS* N.N. KUZNETSOV (MOSCOW)

(Received

29 March 1962)

1. In this article we shall consider the hyperbolic system of linear equations of the form (1). The coefficients A of this system are discontinuous on certain lines, but possess bounded derivatives in the regions between these lines. We prove the uniqueness of the piece-wise continuous and piece-wise smooth solution of the Cauchy problem for system (1). For the coupling conditions of the solution on the lines of dis~ntinuity of the coefficients we tske conditions (5). A similar result can be obtained for the non-divergent system of equations (13) if its solution is coupled continuously on the lines of discontinuity. The method we have used is based on a well-known idea of Holmgren. The same idea was used by SK. Godunov [13 to study the uniqueness of the solution of Cauchy’s problem for a certain special system of two equations of the form (1). The uniqueness of the solution for a system of two equations of the form (13) is studied by the method of characteristics in the work [21. In the article [31 an energy integral method is suggested to examine the question of the uniqueness of the solution of system (1) of general form. ~nfort~ately, in order to use this method the distance between the lines of discontinuity of the coefficients must not be less than some positive number. Recently Douglis [41 has used this method to prove the uniqueness of the solution of Cauchy’s problem for one system of two equations of special form. 2. Let II be the rectangle 15 1 < a, 0 < t < T; n, = fl\fi, where f: is the union of the lines LK (k = 1, 2, . . . , N) each of which is a curve of the function x = vk(t) continuous and piece-wise differentiable for 0
Zh.

wych.

mat.,

3, No. 2. 299-313, 1963.

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lie

system

of

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ions

395

in any neighbourhood of P) intersect. Suppose that in the region l’l, we are given the following matrices: A(t, x), with continuous and bounded second derivatives, F(t, x), continuously differentiable and bounded, and also the summable column matrix f(t, x). We consider tions

in TI the hyperbolic

(in the narrow sense)

system of

equa-

(1) where u(t, x) is a column matrix, defined in WI. The assumption about the hyperbolicity means that at each point of 1’11the eigenvalues A,, A2’ .1., A,, of the matrix A are real and different (hi - hi+l > p > 0), so that there exists in Tll a non-degenerate matrix C(t, x) (whose rows are left-hand eigenvectors ci(t, X) of the matrix A) such that

CAC-1 = A (*ik

= Eikhk)

and C and A possess

continuous

(4 and bounded second

deriva-

tives. Let N,(P) be an integral line of the equation &/:/dt = h (t, x) passing through the point P E llr. Let the characteristic Nk(p’) lie to the left of the line I!,. and intersect it at the point p. We shall denote such a characteris c*ic by N;(P). We introduce the notation tik((P) similar1~. The characteristics s(P), !$(I’) are said to be inward if & (P) = hk(tP, ZP - O)>Di, hi(P) = hk (tp,zp $0)
We make the following

assumptions.

1) For any point P E LI and any k(l
where 6 does not depend on P or on j. The numbers r, m depend on the point of the line Lj under consideration. We shall assume that any line consists of a finite number of segments on which r, m are constant.

N.N.

396

Kuznetsov

2) At the point P, where N, different above) L, intersect

lines (in the sense indicated

+--a

T<

~n(Nr-2)

nN,-1+b,v2)W~-2)

(4)

’

where W> f ci j f in the neighbourhood of the point in question

tFkhj=

The matrix Xv’ c;(P)

det

?ifj)

det

(i < m + 4,

k

here is ohtained from K, by substituting

for i\
obtained from I!, by the substitution for i\m

for

i>m

the i-th row t a + I, and =I(U) Kk (p) is

of the i-th row by the row - cT(P, + u + 1.

3) We call the function u( t, z), possessing cant inuous and bounded first derivatives in 111, the (generalised) solution of the system of equations (1) if it satisfies it everywhere in II,, and on the lines L, satisfies the condition

tlfB* - AU]

S

(U- -

U*) .& -

(A-U- - A+u+) = 0.

(5)

We put u (0, 3) = lim 24 140 Consider the region C;,, lying in W, and bounded to the left and to the right by the characteristics N, (X,),N,(X,) (tx, = Ix, = 0, XX,< ZX,) respectively. Let If, denote and the straight lines t = 0, t = t
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397

x[wl ; vayq

are constants, and Va depends on the Q rn~~rn~ of the norm of the solution on the top and side boundaries of the region r;oI. where

{Iujlo=

PFoOf. h?t 0 < E ( (fa t ,_,)/2 and let ‘;(: be the sub-region of fTa enclosed by the straight lines t = t, - E, t = t, 1 + E, Let q(t> X) be an arbitrary column matrix, piece-wise continuous-and piece-wise smooth in Cot, and let !dKbe its lines of dfscontinuity. Introducing the operator

(the dash denotes transposition thruugbout), which is conjugate to f,, me obtain for some sub-region Gg+ free of L, and Bfk, the Green formula:

hf -

uL*rp} dtdx =

rphdt

Since c;fi c8n be sub-divided into a finite this kind, suing (6) we obtain

ss OPf -

uL’cp) dtdx -

wdx

+

GE a

-I-

\

number of sub-regions of

(7)

h,cpu) dt -+-

N&G)

(all the integrals increasing t ).

(6)

cpudx =

s It a-l+”

(du--

- cpudx.

\ (qmh, KiX’I)

cpAu) dt

on the right-hand side are taken in the direction

Let r be part of the boundary of Gt consisting and t = t, - E.

of

of the segments h’,, h’,

Lemnrui. In the closed region @a there exists a piece-wise ~utiu~~us and piece-wise smooth function (columu) v’(t, x) which possesses the following properties: L’v’ = 0

631

N.N.

39u

Kuznetsov

at all internal points of smoothness VC1r = C’CU, [ZFlr, =

(9)

o-

(10)

(At a finite numberof points of r and L, respectively (10) need not be satisfied. ) Lenuna2. The family of functions ue(t, n) for is uniformly bounded: pe

(h

4ll\<

equations (9) and

0 < E <

(ta

-

t,_1)/2

vL7,

(11)

where Va is a constant which does not depend on E. 11 uc I(= m;x 1uf 1 . Lemma3. Let z = (C’)-‘vE. .The function vE(t, X) suffers a discontinuity on the characteristics of system (S), and these latter have in the set I: not more than a finite number of commonpoints. On the characteristic N, the function zA is discontinuous, while zi (i f k) are cont inuous. The proofs of Lemmasl-3 are given in Point 5. Let us put the function trE in equation (7). instead of 0. We examine the term in the equation which contains the integrals along the lines M,. On the basis of Lemma3 it can be written in the form

Since uE = C’z at a point of the .line M,(dx/dt = Ah), according to the same lemma, [II”] = [z,lc, and so ([UC], Uhk - Au) = (b&l c/c, Uhk - A,,) = = [Z&l(hk (C#) - (ck, Au)} = 0 and the expression in question is equal to zero. The terms containing integrals along the lines L, are also zero, from (10) and (5). Therefore equation (7) reduces to the form

s

u

Qudx=

It a-c

s +ss

-

v’fdtdx -

vcudx

4 cl++c L(Qu,

CC a

.\

{(Qu, Au) -

N?l

4)dt - \ {(Qu,u) A, - (Qu,Au))dt, NI

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lit

where Q = C’C is a positive region (u, QJ) > q2 11~Ifi.

systems

of

definite

linear

equations

399

matrix, so that everywhere in the

Using the inequalities (Qu, Qu) - h, (Qu, u) = (Cu, CAu) -

h, (Cu, Cu) = (Cu, Mu)

- L (Cu, Cu) = 2 (hl -

(Qu, 4 A, -

(Qu, Au) = 2 (111t

L) (Cl@ 20,

ii) (Cu): > 0,

we obtain q2

\ It

11 u I$dx <

uQudx <

\ II

=-L

a-c

\

< It

and then, using (ll),

1

vCudx+ \\ v'fdtdx\<

be

It a-l+c

0vcIIXII uII& + \\IIvcll4lFIlodtdz9 de a

cz-lSL

we find

Since the function u(t, x) is piece-wise continuous and its lines of discontinuity have not more than a finite number of points in common with any straight line t = const. we can take the limit as E - 0 in this inequality, and as a result we obtain

s It

Finally,

(1 IIu II0&J+ \\IIf II0dtdx I1 a-l

using the Bunyakovskii inequality,

(Jll.L~~~~~(,t\_

a1

l

Ca

we have the inequality

IIU IL& + ~allflIodt~x)

(12)

(t a-1 < t < ta) and this proves the theorem. The theorem shows that the solution of Cauchy’s problem for system (I) is unique. For, putting f = 0 and J IjuIJodx= 0, by successively 1. applying inequalities (12) to the regions G,, G2, . . . t Gp we can prove

400

N.N. Kuznetsov

that

{,” u ~~&c= 0 for any 0
< tp. Then u = 0 as we were required

to prove. We note that the inequality (12) gives more than uniqueness, for it implies the continuous dependence of the solution of Cauchy’s problem on the initial conditions and the right-hand sides in the class of uniformly bounded functions. 4) Let us make two co~ents here. The first concerns the ass~ption of narrow hyperbolicits of system (1). It simplifies the formulation of the assumptions and of the proof, but is not essential. In fact it is quite simple to assume that system (1) is hyperbolic, i. e. there exist real matrices C, A everywhere in fl, which satisfy (2), and they posses continuous and bounded second derivatives. Cur second commentrelates

the solution of which satisfies of dis~ntinuity A.

to the system of equations of the form

the continuity

condition on the lines

For this system we can obtain similarly an inequality of the form (12). The operator conjugate to L, will be analogous to L, and the conditions on the lines of discontinuity A which must be satisfied by the

FIG. 1. conjugate function and which have the sense of conditions (10) have the form (5).

conjugate to

5) At this point we prove Lemmasl-3. We shall consider the region $ and its sub-region ‘;fx_ Suppose that in the region (7’. bounded by the arcs f?1fC2, fC3fC4of the characteristics FJ,(K,), r?~(~~~}and by the segments i$K,, K1K4of the straight lines t = t”, t = t * < t *’ (cf. Fig. I), there is a unique line UT” of the set II. Let t ” - t ’ be so small that at each point of WP’ the inward characteristics are Ni, . . . , N;I

ffyperbol~c

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of

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401

equations

iv+ 9+1, ***’

N”, (m\(r). Put N;(W) = Al!!‘, . . . . N”,(w) = Anfi! In the region G we consider the problem (8)-(1(I). Multiplying (8) on the left by (C’)-’ and using (C’)-IA = A(C’)-’ we obtain

where s =(cBc-~+c~+c~~

A)‘,

2 = (C’)-’ 21.

Let r’ denote the part of the boundary of G’ which consists segments K$,, K&T,, Kkd. Then conditions (9) and (10) give 2

xc; i

of the

P) IPEP = f (P),

(15) (P E PW’)

(P) 2; (P) = xcc:: (P) 2; (P) i

W)

(where “z(P) are given piece-wise smooth functions which are, generally speaking, discontinuous at the point $ condition (16) iskhe continuous condition for Y on. the line Cyy’). Leauna4. There exists ing properties:

in G a function z(t,

r) possessing the follow-

1) it is bounded, coqtinuous and piece-wise continuously differentiable in the region G’ \ R’ where R’ is the union of the lines AIT . ..I A,!!‘, WbE’, and satisfies equation (14) in this region; 2) it

satisfies

condition

3) it has the following

(15) and, for t < t’@, condition

(16);

estimate:

!I z(t, 5)

1< g&e@ (*+ffff*-ff,

(1-7)

where 2, = s;p II2 P) IL

d >/IS Proof.

g=max(1,f),f>nPjI

11iIlG’;

Without loss of generality

onWW’,

IIs II = mgx xIsijl* j we can take ci = r - m so that the

base characteristics on !VY’ are N; , l&‘, . . ., N; , ‘N&, line W" we introducethe vectors

. . ., h$.

on the

402

N.N.

Kuznetsov

2; tt - 0) 6 tt - 0) z:+1 (t - 0)

5 (0 =

2; (t

-

to =

,

*‘1

0)

On the assumption8 we have made, the system of equations (16) is soluble with respect to the ~~nents of c(t) so that 5 0) = FTj (t).

W)

Let I’ be an arbitrary point of the region C’ ,\ R’ and let x = xi( t, P) be the equation of the characteristic N,(P). On the line !f’V we put

W)

z:(Q)=

ii

-

W

2: tQ)= i(Q)

where (! E h’8’, Cj; is the point of intersection

of the ~aracteristic

N,(Q) and a line of r’ and 2i are arbitrary functions possessing continuous bounded derivatives for ’ Q t < t” and satisfying the conditions t

z; (t”-

0) = i: SUP

t’
ma=

mci(r

(W) 1 ii

tm < f G a (t)

Formulae (19) define the vector(:)(P).

1~

2,.

Putting

we define the k-tb approximation at the point P by the formula

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system

of

linear

403

equations

tPi W)

(k-1)

(‘0

Zi(P) =

Zi(Pi)

Sij

-

Zj (5,

X=Xi

where Pi is the point of intersection

P)

(21)

“’

of the characteristic

N,(P) either

(k) (k) if Pi E WI’J’; Zi (Pi) = & (Pi), with the line WW'or with a line of I-‘; (k) Zi (Pi) ‘= ii (Pi), if P, E I-‘. The null approximation is obtained from (19)-(20)

k = 0

where

z(-l)

z

5 0.

(k) We note that the successive approximations z (t, x) are defined, continuous and piece-wise continuously differentiable in the region G’\R: On the lines Ai!Y, VW they are discontinuous, but have limiting values to the left and the right. (k) Let us prove the uniform convergence of .z (t, x) in the region (k-0 G’ \ R ‘. We put o(k) = Tz . Then ‘Pi (Oi

(lib(P)

=

(JJ;k) (pi)

s(2,

s..())!k-1)

-

tp

We note that titk) 1 (iP ) is different

13

3

j

1 “‘Xi

(?a P)

dz.

from zero only when Pi E

WW' and so

Since, from (19) and (20)

,,(&P’) --k$)(P’) (I <

fa [ I(dk-1) II d-z <

[ 11 dk--1) /Id-c, tP

LPI we

fa

have (Iw(k) (P) 11\< (1 +

f) (J s’ \]6+--1) 11CEZ. tP

It clearly follows that I/ dk)(P)I] --, 0 as k - m uniformly in the region G’ \ P’. Hence the limit function z(t, x) is defined, continuous and bounded in this region. .‘fherefore the function

404

N.N.

Kuznetsov

is smooth, and since the functions Ii(P) thesis, Fi

are piece-wise

smooth by hypo-

taking the limit in (19) we find that the function q(t)

(y(t)

is piece-wise

smooth. Hence the function

5 (1) = t:.

= t)(t)

is also Piece-wise smooth. Finally, passing to the limit in equation (21) we find that z(t, x) is piece-wise smooth in C’ \ P’. It obviously satisfies equation (14). conditions (15) and, for t < t” condition (M), in this region. To prove the third part of the lemmait is sufficient, as usual, to use equation (21), putting k = m. this proves Lemma4. Suppose that the region G” is bounded by the arcs A%‘,, the characteristics line t=t’: discontinuity

t=

N,, N,, and the segments mr%‘~,, AB t’

@,o,

< t”

B%‘N, of

of the straight

and that it contains the pencil of lines of

. . ., mN,o

(cf.

Fig.

2).

‘I&se

lines intersect

only

FIG. 2.

at the point 0 and have not other commonPoints in the region G’: I& ~$1C G” consist of all the points (t, X) E G” which satisfy the condition t > t’ + E (0 < 6 < t”t’). ive put r” = A til + %‘~WTN,+~,V~B, i&O=L. Lemma 5. If t” - t’ is sufficiently small, there exists in the region GI a function z(t, x) which possesses the following properties:

1) it is continuous and piece-wise

continuously differentiable

in the

Hyperbolic

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linear

of

equations

405

region G” \ R” where R” is the union of the lines Lkand a finite number of piece-wise smooth lines consisting of the arcs of the characteristics of system (1) and having only a finite number of points in commonwith L,, and satisfies equation (14) in this region; 2) on each line L, (k = I, ..*, N,) with the exception of a finite number of points of the last one, it satisfies the continuity condition for I? (l(j), and on r”it satisfies condition (15), where i(P) are given piece-wise smoo$h functions which, generally speaking, are discontinuous at the points wi; 3) it is bounded uniformly with respect to E in G”: E p 09 411 f

23

mhere i? is a constant which is independent of E. Proof. Since in the region Githe lines L, have no commonpoints and the distance between them is not ll!ess than some positive number, the first two parts of the leonnacan be proved by the successive application, a finite number of times, of Lesnna4 to the regions of the form considered there, in the form of the union of which we can put Gs Therefore it is only the proof of the third part of the lenxnawhich we lack. It is sufficient to do this proof in the case when the charactersitfcs NT, IV,, . . . , NT, Nr+;_,, . . ., Ni are inward characteristics for all the lines L, in the given region. Let us fix

tp (t’ + e < tp < t”)

and denote the point of intersection

of the line L, and the straight line t = tp by W . At the points p E L, we introduce, as before, the vectors gtk)( B), q(k)(P) and the matrices F, k) (P). We consider the characteristics Ni(Wk) which leave the Point Wk. They must intersect the line Lk+l for i r + 1, at least when tp - t’ is sufficiently small, for otherwise less than r characteristics would arrive on Lk+l from the left and-less than n - r would arrive on Lk_l from the right, contrary to hypothesis. bet Wi+I (i < r), wk--l (i > r + 1) be the points of intersection the characteristics 1, 2, .1., Wl (i>r+l),

Ni (IV,) with .L~_c~, Lk+

respectively.

of

putting k =

N,, we construct the aggregate of points !I+‘: (i >, r + I), Iv: (i = 1, i = 2, N, - I), WX, (i
W&,_+l(i < r) (the points

W$

@N,+~ lie on r”).

aggregate with the arbitrary points

We supplement this

wi (i < r), wik, (i >

ing them so that they lie above the line

t = tp.

F +

I),

select-

Carrying out a similar

406

N.N.

construction

for the points

j = 1, n; k = 1, NJ,

Kuznetsov

IV;, we construct the points

IV:! (i = 1, n;

then WF’ and so on, Let us introduce the

matrices Gk and Hk, putting

(G&j

=

{

tFkhj 0

(id

rh

(i>

4;

(H&j =

(i d 4. ( ~~~~~~ (i&r + 11,

and the matrices 4’ ), the only non-zero column of which is the i-th column, which is equal to the i-th column of the matrices G,, and also fi: i, which are defined similarly. We introduce the matrices

on the nN,-th order, and the ~~-component vectors

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lit

systems

of

linear

equations

407

where t&...ik

P(l) i,&...c,

wk

=

= H?’’

$I;

1 Wi&.,.im

~

***ik) , p;;,-;.ik = G$’

(i, 1

s

(sz),=ximdz +

Hfrn)

fWip...im

=

s

(sz)x=rrmdz.

tWk" ii...&

According to formula (18), c(k)

,

‘&pi..b

k+l

Gk

fh&b;:,,

at the point w,

x

Gf’

q(k)

i

+

2

HI;‘)

q(k),

i

where, for s < r

Therefore

and

so that

Using the notation introduced above, we can write these relations:

cp = ~$$$I, + 2 pi- wi. i

Applying this to

cbil_ 11 @% 1 * + *,

i

we obtain

N,N. Jluznctsov

408

Let us estimate the terms with p and ~0. We put

Using the definition

of W, we find:

since II@’ 11+ [I Hf’ 11 is

equal

either

to 11Gf’ 11(i < r$, or

to

1)Hlf” 11(i > r), ~th e right-hand side of the last inequality is not greater than WJ (t” -

TV)h (tp) <

wu (t” -

t’) h (tp). Therefore the terms in

relation (22) which contain TUdo not exceed, with respect to their norm, the quantity

The terms containing p are estimated similarly.

They do not exceed

the quantities Qi dTo, with respect to their norm, where zO is the upper bound of the norm of the function given on r”. Therefore, from (22) we obtain the following inwuality:

lit

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of Iinear

systems

Let us choose the number k in this inequality .. the points W?“.zk lie below the straight line i,&. .ik ik+l W,

of the points k-

lies

above it.

409

equations

J’ut

in such a way that all t = t ” and at least tp = t’ +

one

E. Clearly

m as E - m. ~&ma 6. For E > 0 we have

where P(E) is continuous proof. is clear aggregate verified pose that

Let us find

N;-

an estimate

1.

i

for

I/ Si,Si,.

. . si, /I. First

.

.

N,-1

N1--1

x = k-

(IV, -

Then the norm of the matrix

(for K > 1) and similarly instead of I) Si’

in order

and cl(O) = I), and (?, is a constant. of

all,

it

Y

r .-Y (i,i, . . . la) . , . (Q, . . . z,.J (zIliqfl.

q = E (&),

number).

e>O

that the maximumnumber of equal indices in succession in the of indices i,i,. . . ik is equal to P, - 1. For it is easily that the matrix k?i(P,).?i(P,) . . . h’,(P,) = ‘0 if k > !i’ - 1. Supthe aggregate of indices i ,i,. . . ik has the form

.. 7 (111 where

for

for

(PI)

K

=

II X II Si

..

&;;;i,

(K is the whole part of a

1)q

Si,Si,. . . Si,’ does not exceed

0. Ye note that we have written

P2)

(26)

II -

f

*

(1Si 11

IIsi Pv) II

to save space.

Using the definition minants, we have

of F, k) and the Hadamar inequality

for deter-

410

R.N.

Kuznetsov

so that

It is easy to verify that for i, # i,

IlSiSi,II<

7 II%II*

therefore

We can take b> 1. Then, from condition (4), the norms of the matrix ~~espondin~ to sequences of indices other than (26) s,s,* . . s,, satisfy

estimate (27) even more. Therefore

In accordance with (4) we put nN~-%N:-27 = 1 Then

p1 (1 > p1 > 0).

n(.v,-~)(q+l)/(ci-t)b(n’,-zttof1)/(6--l)+f/(q--l)r = zz (1 _ pJ {mW%-l)bl+M’,---a))W-1) <.I _ p2 (0 < FLn < Irl)I if q>l

+(lnS)-’

ln {~Wrl)&i+!(N,-a)

}.

Thus, for those values of k for which the last inequality fied we have

from which, clearly,

inecpalities

Due to Lemma6, ineguality

(24)-(25)

follow.

(23) takes the form

is satis-

This Proves Lemma5.

Hyperbo Zic systems

IICPII< Qc+J%l +

P

of

linear

411

equations

(4 I.800) + OOWQ (f - t’) h Pd.

(28)

Let us suppose that for any sequence {Ed) - 0

(Icp+i II< h @p+i)

('l3+r

= t’ +

ef)7

or, in other words, for any t < ?, where ? is some number sufficiently close to t’

(f < t
IIP (t)II< h (t)

We note that it follows from the definition of g(t) Therefore (29) is equivalent to the inequality

(29) that

i(t - 0) = g(t).

Ilt;(t - O)ll< h (0.

(30)

The function h(t) is piece-wise constant and non-increasing, with a jump at the point t,, only when 115 (t, - 0) II> h (to), uld this contradicts (30). Therefore it follows from proposition (29) that L(T) for t’ < t
h (t) = h (7) and the solution

z(t,

x) is

(1’ < t
bounded.

Therefore, having assumed that 115~11 is not bounded as E - 0 we can find a sequence {Ed) - o such that 11E&+ j/ h (tp+i). In this case (28) gives

If

we

ciently

choose t ” - t’

so small that

Qooa (t” -

small E CL(E)< LI,,and Q,,oo. (t” -

h(r) <

1 -

Qoozo Q. (t” - t’)

05 -

which contradicts our assumption that the third part bf Lemma5.

PO

00 as i - 03 with )I 5p-p (I>

t’) <

t’) + l.~ <

1, 1,

then for suffiso that

(1’< r <%

11cpI\ is not bounded and proves

The existence of precisely the same function z(t, x) in the region containing the pencil of lines of discontinuity with its centre on the upper boundary is proved in exactly the same way. Let us turn to the proof of Lemmasl-3. Lemma1 is a corollary of the first two parts of Lemma4 (cf. the beginning of the proof of Lemma5).

N.N. Kuznetsov

412

Lemma2 is also and 5.

a simple consequence

of the third

parts

of Lemas 4

Let us prove Lemma3. In the region (;’ of Iemma 4 the lines of discontinuity uE are inward characteristics ,? I’!‘, ..., j?n'f'. The new lines of discontinuity arise as a result of the intersection of these characteristics with the other lines L, and are also characteristics. The distance between the lines L, in Ct is not less than some positive ni~ber, so that there is a finite number of these points. It follo!vs that the number of lines of discontinuity is finite. These lines consist of the arcs of the inward characteristics and cannot have more than a finite number of points in common with the lines L,, Let _!),p, be an arc of the characteristic ftj which has a d~scor~tinuity ye and suppose that it has no points in common with the lines L,. Let ? be any point of this arc. We consider the ,arcs ?l?i of the characteristics Iv,(P) (? is an internal point of !‘~P~, .i .f j). Integrating the i-th equation of system (14) along the arc PIE”; we find

sides of the line P:i(~n) Since hi # h. the points “f, Pi are on different and so, putt i ng tPi - tP; + 0 in (31) and using Lemma 2, on the basrs of which the right-hand

side

in (31) tends to zero,

hi (P)l = 0 and this

completes

we obtain

ti#ih

the proof.

Trans

Sb.,

40 (82).

lnted

No. 4, 467-476,

by

R. Feinstein

1.

Godunov. S. K., Mat.

2.

Rozhdestvenskii,

3.

Babeuko, K.I. and 6,&l. fand. I.I., Nauch. Dokl. VYSS&. shk., No. 1,

1956.

B.L., Usp. mat. Nauk. 15, NO. 6(96), 59-117. 1960.

12-18. 1958. 4.

Douglis, A., Communs pure appl. Math.. 14, 267-284. 1961.