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Generalized Solutions to a Cauchy Problem for a Nonconservative Hyperbolic System
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
207, 361]387 Ž1997.
AY975274
Generalized Solutions to a Cauchy Problem for a Nonconservative Hyperbolic System K. T. Joseph School of Mathematics, TIFR, Homi Bhabha Road, Bombay 400 005, India Submitted by Wolfgang L. Wendland Received January 9, 1995
In this paper we show the existence of generalized solutions, in the sense of Colombeau, to a Cauchy problem for a nonconservative system of hyperbolic equations, which has applications in elastodynamics. Q 1997 Academic Press
INTRODUCTION This paper is devoted to the study of a Cauchy problem for a strictly hyperbolic, nonconservative system of equations, namely,
u t s t
qu
qu
u x
s x
y
y k2
s x u
s 0, s 0,
Ž 0.1.
s Ž x, 0 . s s 0 Ž x . .
Ž 0.2.
x
in x g R 2 , t ) 0, with initial conditions u Ž x, 0 . s u 0 Ž x . ,
The system Ž0.1. is a simplified model, which comes for applications in elastodynamics, where u is the velocity, s is the stress, and k ) 0, is the speed of propagation of the elastic waves. The details can be found in w2x and the references given there. It is well known that global smooth solutions do not exist for the problem Ž0.1. and Ž0.2., even when the initial data are smooth; discontinuities in u and s appear in finite time. When u and s are discontinuous the product u sx , which appear in the second equation of the system Ž0.1., 361 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
362
K. T. JOSEPH
does not make sense in the standard theory of distributions. To overcome such difficulties, Colombeau w3x introduced a new notion of generalized functions. In recent works w1, 2, 4]9, 11, 12, 14x and many other references in w4x and w12x, it is recognized by many authors that the algebra of generalized functions of Colombeau is a convenient setup to seek global solutions, where such difficulties arise. Further this approach takes into account the microscopic structure of the shocks Ždiscontinuities . of the solution. Cauret et al. w2x studied the system Ž0.1. numerically, proved the convergence of the scheme, and constructed generalized functions from the numerical scheme which are solutions of Ž0.1.. Moreover it was shown that when one seeks travelling wave solutions, u Ž x, t . s Ž u r y u l . H Ž x y ct . q u l ,
s Ž x, t . s Ž sr y s l . K Ž x y ct . q s l , where H and K are two Heaviside generalized functions and u l , u r , s l , sr , and c are constants, the system Ž0.1. admits an infinite number of jump conditions for the shock, depending upon the microscopic shapes of the jumps in H and K. In this connection we also mention the work of Noussair w11x, who used the Godunov scheme, and the work of Berger w1x, who used a nonconservative Lax]Friedrichs scheme to study the system. More recently, Colombeau w7x and Prignet w14x did numerical tests of the system
u
qu
t s t
qu
u
y
x
s x
yk
2
s x u x
s e1 s e2
2u x2 2s x2
, ,
Ž 0.3.
with special initial data and studied its solutions Ž u e 1 , e 2 , s e 1 , e 2 ., when e 1 , e 2 ª 0, with l s e 1re 2 fixed. In their work computations were done for various Žsmall. values of e 1 and e 2 , keeping the ratio l fixed. These calculations were repeated for different values of l, namely, l s 60, 4, 1, 1r4, 1r60 and they arrived at the conclusion that the limit of Ž u e 1 , e 2 , s e 2 , e 2 . as e 1 , e 2 ª 0, keeping l s e 1re 2 fixed, depends on l and is a solution of the system
u t s t
qu
qu
u x
s x
y
y k2
s x u x
f 0, f 0,
Ž 0.4.
NONCONSERVATIVE HYPERBOLIC SYSTEM
363
in the algebra of generalized functions of Colombeau. When l s 1, i.e., when e 1 s e 2 s e , this corresponds to the case where Ž u, s . s Ž u e , s e . 0 - e - 1 has the same microscopic shape for the discontinuity. The cases l ª 0 and l ª ` corresponds to the extreme cases in which s is replaced by f in the first and second equations, respectively, of the system Ž0.1.. Here s means equality as elements of the algebra of generalized functions and f means association in the sense of Colombeau. In this article we consider the case e 1 s e 2 s e , in Ž0.3. with initial conditions Ž0.2. and using standard arguments, show that for each fixed e ) 0, there exists a unique C` global solution Ž u e , ¨ e . if u 0 Ž x . and ¨ 0 Ž x . are in S Ž R1 ., the space of Schwartz class functions. Later the a priori estimates show that if u 0 Ž x . and ¨ 0 Ž x . are in H 2 Ž R1 . l BV Ž R1 . then there exist C 2 solutions. The main contributions in this article are the estimate Ž2.3. on the higher derivatives of Ž u e , ¨ e . in sup norm and an existence result, Theorem 3.1, for global generalized solution of Ž0.1. and Ž0.2., for rather arbitrary u 0 and ¨ 0 in G Ž R1 .. The paper is organized in the following way. In Section 1, we briefly review the definition of the algebra of generalized functions and the notion of equality and association. In Section 2 we show the existence of a global smooth solution for the viscous system and get estimates on higher derivatives of the solution. Finally in Section 3 we construct a generalized solution of Ž0.1. and Ž0.2..
1. THE ALGEBRA OF GENERALIZED FUNCTIONS OF COLOMBEAU In this section we quickly recall the definition of the algebra of generalized functions in V T s Ž x, t .: y` - x - `, 0 - t - T 4 , for T ) 0, denoted by G Ž V T .. We take the point of view of Colombeau and Oberguggenberger w5x. Let u s Ž u e . 0 - e - 1 be an element of the infinite product of smooth functions in V T : Ž C`Ž V T ..Ž0, 1.. We say u is moderate if it has the following property: For every compact subset K of V T and all nonnegative integers j and l, 'N ) 0 s.t.
j l ue t j xl
L`Ž K .
s O Ž eyN .
as e ª 0.
The set of all moderate elements of Ž C`Ž V T ..Ž0, 1. is denoted by EM Ž V T .. A sequence u s Ž u e . is called null if for all compact subsets K of V T and
364
K. T. JOSEPH
for all nonnegative integers j and l and for all M ) 0,
j l ue t j xl
L`Ž K .
s OŽ e M .
as e ª 0.
The set of all null elements is denoted by N Ž V T .. It is easy to see that EM Ž V T . is an algebra with partial derivatives, the operations being defined pointwise, and N Ž V T . is an ideal which is closed under differentiation. The quotient space, denoted by G Ž VT . s
EM Ž V T . N Ž VT .
,
is an algebra with partial derivatives, the operations being defined on representatives. G Ž V T . is called the algebra of generalized functions. Two elements u and ¨ in G Ž V T . are said to be associated, if for some Žand hence all. representative sequences Ž u e . 0 - e - 1 and Ž ¨ e . 0 - e - 1 of u and ¨ , we have ue y ¨e ª 0 in the sense of distributions in V T as e ª 0 and is denoted by u f ¨ . We emphasize that this notion is different from the notion of equality in G Ž V T ., which means that u y ¨ g N Ž V T ., or in other words,
j l Ž ue y ¨ e . tj l
L`Ž K .
s OŽ e M .
as e ª 0
for all M, for j, l nonnegative integers, and for all compact sets K ; V T . 2. EXISTENCE OF SMOOTH GLOBAL SOLUTION FOR THE VISCOUS SYSTEM In this section we consider the system Ž0.3. with e 1 s e 2 s e , i.e.,
ue t s e t
q ue
qu
e
ue x
s e x
s e
y
yk
x
2
ue x
se se
2 ue x2 2s e x2
, ,
Ž 2.1.
in y` - x - `, t ) 0, with initial conditions u e Ž x, 0 . s u 0 Ž x . ,
s e Ž x, 0 . s s 0 Ž x . .
Ž 2.2.
365
NONCONSERVATIVE HYPERBOLIC SYSTEM
We assume
Ž H.
¨ 0 Ž x . , u 0 Ž x . g H 1 Ž R1 . `
`
Hy` ¨
and X 0
Hy` u Ž x .
X 0
Ž x . dx - ` and
dx - `.
We shall prove THEOREM 2.1. Ži. For each T ) 0, there exists a unique weak solution Ž u e , s e . in H V T . for Ž2.1. and Ž2.2., if ¨ 0 Ž x . and u 0 Ž x . satisfy hypothesis ŽH.. Žii. Let ¨ 0 Ž x . and u 0 Ž x . g S Ž R1 . be the space of Schwartz functions and 0 - e F 1. Let Ž u e , s e . be the solution of Ž2.1. and Ž2.2.. Then, gi¨ en l and j nonnegati¨ e integers, ' constant CT and positi¨ e integers N0 and p 0 such that 1Ž
sup VT
l j t x l
u e Ž x, t . F j
CT
e
N0
,
sup VT
l j t x l
j
s e Ž x, t . F
CT
e N0
, Ž 2.3.
where CT depends polynomially on 5 u 0 5 H p , 5 ¨ 0 5 H p for 0 F p F p 0 , 5 u 0 5 L1 , 5 ¨ 0 5 L1 , 5 u 0 5 BV , 5 ¨ 0 5 BV , and T. Here N0 and p 0 depend only on l and j. The proof of this theorem is broken up into lemmas. For brevity of notation we often suppress the dependence of e and write s , u, w, and ¨ for s e , u e , w e , and ¨ e . First we write Ž2.1. in a convenient form. Let ¨ s s y ku and w s s q ku, where k ) 0 is the constant given in Ž0.1.. Then, Ž2.1. can be written as ¨t q
wt q
ž ž
w y¨ 2k w y¨ 2k
q k ¨x s e ¨xx ,
/ /
y k wx s e wx x .
Ž 2.4.
The initial conditions Ž2.2. become ¨ Ž x, 0 . s ¨ 0 Ž x . s s 0 y ku 0 ;
w Ž x, 0 . s w 0 Ž x . s s 0 q ku 0 . Ž 2.5.
Since ¨ 0 and w 0 are in H 1 Ž R1 ., they are continuous bounded functions on R1. To prove part Ži. of Theorem 2.1, in a standard way, we define a sequence of functions Ž ¨ nŽ x, t ., wnŽ x, t .., n s 0, 1, 2, . . . , iteratively,
Ž ¨ 0 Ž x, t . , w 0 Ž x, t . . s Ž ¨ 0 Ž x . , w 0 Ž x . . ,
366
K. T. JOSEPH
and for n s 1, 2, 3, . . . , let Ž ¨ nŽ x, t ., wnŽ x, t .. be defined by the solution of linear problems
¨n t wn t
q q
ž ž
wny1 y ¨ ny1 2k wny1 y ¨ ny1 2k
qk yk
¨n
/ /
x
wn x
se se
2 ¨n x2 2 wn x2
, ,
Ž 2.6. n
with initial conditions ¨ n Ž x, 0 . s ¨ 0 Ž x . ,
wn Ž x, 0 . s w 0 Ž x . .
Ž 2.7. n
First we assume ¨ 0 Ž x . and w 0 Ž x . are in S Ž R1 .. By linear theory, there exists a unique C` solution Ž ¨ 1 , w 1 . to Ž2.6.1 and Ž2.7.1; see Friedman w10x. Further, the solution and its derivatives decay to zero as < x < ª `. Moreover by maximum principle, sup ¨ 1 Ž x, t . s 5 ¨ 0 5 L`Ž R 1 . ,
sup w 1 Ž x, t . s 5 w 0 5 L`Ž R 1 . .
VT
VT
Iteratively, we get a unique C` solution Ž ¨ n , wn . of the problem Ž2.6. n ] Ž2.7. n such that they and their derivatives decay to zero as < x < ª `. Further sup ¨ n Ž x, t . s 5 ¨ 0 5 L`Ž R 1 . ,
sup wn Ž x, t . s 5 w 0 5 L`Ž R 1 . .
VT
Ž 2.8.
VT
Denote
lŽ1xn, t . s lŽ2xn, t . s
wn Ž x, t . y ¨ n Ž x, t . 2k wn Ž x, t . y ¨ n Ž x, t . 2k
q k and y k.
Ž 2.9.
By Ž2.8., we have that there exists a constant l G 1 such that sup l i n Ž x, t . F l VT
for i s 1, 2 and n s 0, 1, 2, . . . .
Ž 2.10.
With these we shall prove the following lemma. LEMMA 2.1 5 Ž ¨ n , wn . 5 H 1 Ž D T . F C Ž e , l , T . 5 Ž ¨ 0 w 0 . 5 H 1 Ž R 1 . .
Ž 2.11.
Proof. First we take ¨ 0 and w 0 in S Ž R1 .. Multiplying the first equation of Ž2.6. n by ¨ n and the second equation of Ž2.6. by wn , integrating w.r.t. x,
367
NONCONSERVATIVE HYPERBOLIC SYSTEM
integrating parts, and adding we get 1 d
`
H Ž¨ 2 dt y` s ye
2 n
Ž x, t . q wn2 Ž x, t . . dx `
Hy`
ž
`
1 ny1
Ž x, t . ¨ n Ž x, t .
`
Hy`l
y
dx q
/
x
Hy`l
y
2
¨ n Ž x, t .
2 ny1
Hy`
ž
wn Ž x, t .
¨ n Ž x, t . x
wn Ž x, t .
Ž x, t .
`
x
x
2
/
dx
dx
wn Ž x, t . dx.
Ž 2.12.
Now using Ž2.10. and Schwarz inequality we get, `
Hy`l
1 ny1
x
¨n
Fl F
¨ n Ž x, t .
Ž x, t . x
L2 Ž R 1 .
e ¨n
5 ¨ n 5 L2 Ž R 1 .
2
x
2
¨ n Ž x, t . dx
L2 Ž R 1 .
q
l2 2e
5 ¨ n 5 2L2 Ž R 1 . .
Ž 2.13.
Similarly, `
Hy`l
2 ny1
Fl F
wn Ž x, t .
Ž x, t .
x
wn x
L2 Ž R 1 .
e wn 2
x
wn Ž x, t . dx
5 wn 5 L2 Ž R 1 .
2 L2 Ž R 1 .
q
l2 2e
5 wn 5 2L2 Ž R 1 . .
Ž 2.14.
Using Ž2.13. and Ž2.14. in Ž2.12. we get d dt
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . qe
F
l2 e
¨nŽ t . x
2 L2 Ž R 1 .
q
wn Ž t . x
2 L2 Ž R 1 .
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 1L2 Ž R 1 . .
Ž 2.15.
368
K. T. JOSEPH
It follows from Ž2.15. that d
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 ¨ n Ž t . 5 2L2 Ž R 1 . F
l2
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 .
e
dt and integrating this we get
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . F e Ž l
2
5 ¨ 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . .
r e .t
Substituting this in the RHS of Ž2.15. and integrating from 0 to t we get 5 ¨nŽ t .
5 2L2 Ž R 1 .
q 5 wn Ž t .
5 2L2 Ž R 1 .
qe
t
H0
2
¨nŽ s.
ž
x
F Ž 5 u 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . . 1 q
l2
L2 Ž R 1 .
q
t ŽŽ l 2 r e . s.
He e 0
wn Ž s . x
2 L2 Ž R 1 .
/
ds
ds .
After integration we have 5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . t
qe
H0
¨nŽ s.
ž
x
2 L2 Ž R 1 .
wn Ž s .
q
x
F Ž 5 ¨ 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . . e ŽŽ l
2
2 L2 Ž R 1 .
t .r e .
/
ds
.
Ž 2.16.
Now differentiating Ž2.6. n w.r.t. x, multiplying the resulting equations by ¨ x and wx , integrating by parts, and adding we get 1 d 2 dt
`
Hy`
ž
Hy`
s
`
Hy`
wn Ž x, t .
q
2
/ ž / ž Ž ./ ž Ž ./ x
`
qe
2
¨ n Ž x, t .
2
2 ¨ n x, t
q
x2
l1 ny1
dx
x
2 wn x, t
2
dx
x2
¨ n 2 ¨ n Ž x, t .
dx q
`
H
l2 ny1
x x2 ` Repeating the same argument as before, we get ¨nŽ t . x qe
Fe
2 L2 Ž R 1 .
t
H0
q
wn Ž t . x
2 ¨nŽ s.
Ž l 2 r e .t
x2 d¨ 0 dx
x
x2
dx.
2 L2 Ž R 1 .
2 L2 Ž R 1 .
ds q
2 L2 Ž R 1 .
wn 2 wn Ž x, t .
q
dw 0 dx
t
H0
2 wn Ž s . x2
L2 Ž R 1 .
ds
2 L2 Ž R 1 .
.
Ž 2.17.
369
NONCONSERVATIVE HYPERBOLIC SYSTEM
Now to estimate 5 ¨ nr t 5 2L2 Ž R 1 . q 5 wnr t 5 2L2 Ž R 1 . , we note that
¨n
se
ž / t
2
2
2 ¨n
2
ž / x2
se
ž / t
2
x
x x2
,
2
2 wn
2
q l22 ny1 Ž x, t .
ž / x2
wn
y 2 el2 ny1 Ž x, t .
x
wn
2
ž / x
2 wn
?
2
ž /
¨n 2 ¨n
y 2 el1 ny1 Ž x, t .
wn
¨n
q l12 ny1 Ž x, t .
x2
.
Now using, the estimates
¨n 2 ¨n
2 el1 n Ž x, t . 2 el2 n Ž x, t .
Fe2
x x2 wn 2 wn x
Fe
x2
2
2
2 ¨n
q l2
¨n
2
ž / ž / ž / ž / x2
2
2 wn
q l2
x2
x2
wn
, 2
x2
in the above equations and adding the resulting expressions and integrating, we get 2
¨nŽ s.
t
H0
s F 2e
L2 Ž R 1 .
2
wn Ž s .
q
s
H0
x2
q 2l
2
t
H0
L2 Ž R 1 .
¨ Ž s. x
ds
L2 Ž R 1 .
2 ¨nŽ s.
t
2
q
2
2 wn Ž s . x2 wn Ž s .
L2 Ž R 1 .
x
L2 Ž R 1 .
ds
2 L2 Ž R 1 .
ds.
Ž 2.18.
Now using Ž2.17. in Ž2.18. we get t
H0
2
¨nŽ s. s F 2 eŽ l
L2 Ž R 1 .
2
e .t
q
wn Ž s .
Ž e q l2 .
s d¨ 0 dx
2 L2 Ž R 1 .
ds
2 L2 Ž R 1 .
q
dw 0 dx
2 L2 Ž R 1 .
.
Ž 2.19.
370
K. T. JOSEPH
Hence from Ž2.16., Ž2.17., and Ž2.19. we have, for each T ) 0, 5 Ž ¨ n , wn . 5 2H 1 Ž D T . F 2 e Ž l
2
e
r e .T
l2
5 Ž ¨ 0 , w 0 . 5 2H 1 Ž R 1 . .
q e q l2
Since S Ž R1 . is dense in H 1 Ž R1 . it follows that Ž2.11. is true for ¨ 0 , and w 0 are in H 1 Ž R1 .. The proof of the lemma is complete. In the next lemma we prove that the map Ž ¨ ny 1 , wny1 . ª Ž ¨ n , wn . is a contraction map in L`Ž R1 = w0, T0 x., where T0 depends only on 5 ¨ 0 5 L` , ` < X Ž .< ` < X Ž .< 5 w 0 5 L` , Hy` ¨ 0 x dx, Hy` w 0 x dx, and e . LEMMA 2.2. For n G 1 and t G 0,
Ži.
`
Hy` `
Hy`
¨ n Ž x, t . x wn Ž x, t . x
`
X 0
dx F
Hy` ¨
dx F
Hy` w Ž x .
`
Ž x . dx,
X 0
dx.
Ž 2.20. 1
For n s 2, 3, . . . ,
Žii.
5 Ž ¨ n y ¨ ny1 , wn y wny1 . 5 L`Ž D T
0
.
F 5 Ž ¨ ny1 y ¨ ny2 , wny1 y wny2 . 5 L`Ž D T 0 . 1 2
Ž 2.20. 2
for T0 s 1r9C 2 , where Cs
X X ` 2 l q Ž 1r2 k . Ž Hy` ¨ 0 Ž x . dx q H w 0 Ž x . dx .
Ž pe .
1r2
.
Proof. To prove Ži. we make use of an idea of Oleinik w13x to write `
Hy`
¨ n Ž x, t . x
dx s
y iq1Ž t .
Ý Ž y1. iH Ž .
¨ n Ž x, t . x
yi t
i
dx,
Ž 2.21.
where yi Ž t .: i s 0, " 1, " 2, . . . 4 are the points x Žfor fixed t ., where Ž ¨ nr x .Ž x, t . changes sign. We choose the ordering of the index such that on the interval Ž y 0 Ž t ., y 1Ž t .., Ž ¨ nr x .Ž x, t . ) 0. Now from Ž2.21. we get d
`
H dt y`
¨n x
i Ž x, t . dx s Ý Ž y1. H
i
y iq1Ž t .
yiŽ t .
t
ž
¨ n Ž x, t . x
/
dx Ž 2.22.
371
NONCONSERVATIVE HYPERBOLIC SYSTEM
because Ž ¨ nr x .Ž yi Ž t ., t . s 0. Differentiating the first equation of Ž2.6. n w.r.t. x we get
t
¨n
ž / x
q
x
ž
¨n
l1 ny1 Ž x, t .
x
/
se
2 x
¨n
2
ž / x
.
Now multiplying this equation by Žy1. i and then integrating from yi to yiq1 w.r.t. x and using Ž ¨ nr x .Ž x, t . s 0 at x s yi Ž t . and yiq1Ž t . we get
Ž y1.
i
Hy
¨n
t
x
yiq1 i
½
s e Ž y1 .
Ž x, t . dx
i
¨n
x
q Ž y1 .
iq1
ž /Ž Ž. . ž / Ž Ž . .5 yiq1 t , t
x
¨n
x
x
yi t , t
.
Ž 2.23.
By the choice of the ordering of the indices of yi 4 , Ž ¨ Ž x, t ..r x is positive on Ž yiy1Ž t ., yi Ž t .. and negative on Ž yi Ž t ., yiq1Ž t .. if i is an odd integer. Thus Ž ¨ Ž x, t ..r x is nonincreasing at x s yi Ž t .. In other words Ž 2 ¨ r x 2 .Ž yi Ž t ., t . F 0 if i is an odd integer. By a similar reason Ž 2 ¨ Ž yi Ž t ., t ..r x 2 G 0 if i is an even integer. Hence each of the terms on the RHS of Ž2.23. is nonpositive. This implies that the RHS of Ž2.22. is nonpositive. In other words, d
¨n
`
H dt y`
x
Ž x, t . dx F 0.
Integrating this we get `
Hy`
¨n x
`
d
y`
dx
Ž x, t . dx F H
¨ 0 Ž x . dx.
The proof for wn is similar. To prove Žii., first we note from Ž2.6. n and Ž2.7. n that ¨ n satisfies the integral equation ¨ n Ž x, t . s
1
`
Ž 4p t e . y
t
`
1r2
H0 Hy`
Hy`exp
ž
yŽ x y y . 4 te
2
/
¨ 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
=l1 ny1
ž
4p Ž t y s . e
¨n y
Ž y, s . dy ds
1r2
/
372
K. T. JOSEPH
and hence, for n G 2, ¨ n Ž x, t . y ¨ ny1 Ž x, t .
sy
t
`
H0 Hy`
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
= l1 ny1 Ž y, s .
y
/
1r2
¨ n Ž y, s . y l1 ny2 Ž y, s .
¨ ny1 Ž y, s . dy ds.
y
Now
l1 ny1
¨n x
s
ž
y l1 ny2
¨ ny1 x
wny 1 y ¨ ny1 2k
q
ž
qk
wny 1 y wny2 2k
¨n
/ž
x
y
¨ ny1
¨ ny1 y ¨ ny2
y
/
x
2k
¨ ny1
/
x
and hence ¨ n Ž x, t . y ¨ ny1 Ž x, t . s A q B,
Ž 2.24.
where t
A Ž x, t . s y
`
H0 Hy`
½
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e =
ž
wny 1 y ¨ ny1 2k
/
1r2
qk
/
Ž ¨ n y ¨ ny1 . y
5
dy ds
Ž 2.25. and t
B Ž x, t . s y
`
H0 Hy`
=
ž
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
wny 1 y wny2 2k
y
/
1r2
Ž ¨ ny1 y ¨ ny2 . ¨ ny1 2k
/
y
dy ds.
Ž 2.26.
373
NONCONSERVATIVE HYPERBOLIC SYSTEM
Integrating by parts w.r.t. x, we get
A Ž x, t . s
ž
H0 Hy`
=
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
`
t
4p Ž t y s . e
wny 1 y ¨ ny1
ž
/Ž
2k `
t
y
H0 Hy` =
ž
/
y
1r2
¨ n y ¨ ny1 . Ž y, s . dy ds 2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
wny 1 y ¨ ny1 2k
/
1r2
Ž x y y. 2Ž t y s . e
q k Ž ¨ n y ¨ ny1 . Ž y, s . dy ds.
/
Taking sup D t < ¨ n y ¨ ny1 < outside the integral and using part Ži. of the 0 lemma, we get for 0 F t F t 0 F T, A Ž x, t . F sup ¨ n y ¨ ny1 Dt0
`
H Ž¨ 2 k y`
H0
Ž x . q wX0 Ž x . . dx
1r2
4p Ž t y s . e
q l sup ¨ n y ¨ ny1 Dt 0
F
X 0
ds
t
=
1
t 01r2
Ž pe .
1r2
½
2l q
1 2k
1
p 1r2 `
žH
y`
`
t
H0 Hy` X
< z < eyz
Ž t y s. e
¨ 0 Ž x . dx q
= sup ¨ n y ¨ ny1 .
2
`
1r2
X 0
dz ds
Hy` w Ž x .
dx
/5
Ž 2.27.
Dt 0
Here and in what follows, we assume that T0 F T, where T0 is given in the second part of Lemma 2.2. By a similar argument as above, we get
B Ž x, t . F
t 01r2
Ž pe .
1r2
`
Hy` ¨
X 0
Ž x . dx ?
1 2k
= 5 wny1 y wny2 5 L`Ž D t 0 . q 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . 4 . Ž 2.28.
374
K. T. JOSEPH
From Ž2.24. ] Ž2.28., we get for 0 F t F t 0 F T, ¨ n Ž x, t . y ¨ ny1 Ž x, t .
F
t 01r2
Ž pe .
½
1r2
2l q
1 2k
`
žH
y`
X
¨ 0 Ž x . dx q
`
X 0
Hy` w Ž x .
dx
/5
=5 ¨ n y ¨ ny1 5 L`Ž D t 0 . q
t 01r2
Ž pe .
1
1r2
2k
ž
`
Hy` ¨
X 0
Ž x . dx
/
= 5 wny1 y wny2 5 L`Ž D t 0 . q 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . 4 .
Ž 2.29.
Similarly we get for 0 F t F t 0 F T, wn Ž x, t . y wny1 Ž x, t . F
t 01r2
Ž pe .
½
1r2
2l q
1 2k
ž
`
Hy` ¨
X 0
`
Ž x . dx q H
y`
wX0 Ž x . dx
/5
=5 wn y wny1 5 L`Ž D t 0 . q
t 01r2
Ž pe .
1r2
1
`
X 0
H w Ž x. 2 k y`
dx
= 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . q 5 wny1 y wny2 5 L`Ž D t 0 . 4 .
Ž 2.30.
Thus we have, from Ž2.29. and Ž2.30., 5 ¨ n y ¨ ny1 5 L`Ž D t . q 5 wn y wny1 5 L`Ž D t . 0 0 F t 01r2 C 5 ¨ n y ¨ ny1 5 L`Ž D t 0 . q 5 wn y wny1 5 L`Ž D t 0 . 4 q Ct 01r2 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . q 5 wny1 y wny2 5 L`Ž D t 0 . 4 , Ž 2.31. where Cs
` < X ` < X 2 l q Ž 1r2 k . Ž Hy` ¨ 0 Ž x . < dx q Hy` w 0 Ž x . < dx .
Ž pe .
1r2
.
Taking t 0 s T0 s 1r9C 2 in Ž2.31., the estimate Ž2.20. 2 follows. The proof of Lemma 2.2 is complete
NONCONSERVATIVE HYPERBOLIC SYSTEM
375
Proof of Theorem 2.1. From the estimate Ž2.20. 2 it follows that, there exists a continuous function Ž ¨ , w . on D T 0 s.t. Ž ¨ n , wn . converges uniformly to Ž ¨ , w . on D T 0 . In order to prove Ž ¨ n , wn . converges uniformly on D T first we consider the relation for T0 F t F 2T0 , ¨ n Ž x, t . y ¨ ny1 Ž x, t .
s
1 4p Ž t y T 0 . e `
=
Hy`exp
y
`
t
HT Hy`
ž
1r2
yŽ x y y .
2
4 Ž t y T0 . e
/
¨ n Ž y, T0 . y ¨ ny1 Ž y, T0 . dy 2
exp Ž y Ž x y y . . r Ž 4 Ž t y T0 . y 5se .
ž
4p Ž t y T 0 y s . e
0
= l1 ny1 Ž y, s .
x
/
1r2
¨ n Ž y, s . y l1 ny2 Ž y, s .
x
¨ ny1 Ž y, s . dy ds,
Ž 2.32. and a similar expression for wn y wny1. Since 5 ¨ n Ž x, t . 5 L` F 5 ¨ 0 5 L` , 5 wn Ž x, t . 5 L` F 5 w 0 5 L` , `
Hy`
x
`
¨ n Ž x, t . dx F
H¨
X 0
Ž x . dx,
w Ž x, t . dx F wX0 Ž x . dx, x n following the same arguments which led to Ž2.20. 2 we get 5 ¨ n y ¨ ny1 5 L`Ž D T , 2T . q 5 wn y wny1 5 L`Ž D T , 2T .
Hy`
0
F
1 2
H
0
0
5 ¨ ny1 y ¨ ny2 5 L`Ž D ŽT
0 , 2T 0
.
0
q 5 wny1 y wny2 5 L`Ž D T 0 , 2T 0 .
q 32 5 ¨ n Ž x, T0 . y ¨ ny1 Ž x, T0 . 5 L`Ž R 1 . q5 wn Ž x, T0 . y Ž wny1 Ž x, T0 . . 5 L`Ž R 1 . , where we used the notation D T 0 , 2T 0 s Ž x, t .: y` - x - `, T0 F t F 2T0 4 . Iterating the above inequality and using the estimate Ž2.20. 2 we get 5 ¨ n y ¨ ny1 5 L`Ž D T , 2T 0 . q wn y wny1 5 L`Ž D T , 2T . 0
F 3 Ž n y 2. Ž q Ž 12 .
ny 2
0
1 2
.
ny 2
0
5 ¨ 2 y ¨ 1 5 L`Ž D T . q 5 w 2 y w 1 5 L`Ž D T . 0 0
5 ¨ 2 y ¨ 1 5 L`Ž D T
, . 0 2T 0
q 5 w 2 y w 1 5 L`Ž D T 0 , 2T 0 . .
376
K. T. JOSEPH
Now using the estimate Ž2.8. in the above inequality, we get 5 ¨ n y ¨ ny1 5 L`Ž D T F 8 n Ž 12 .
0
ny 2
, 2T 0 .
q 5 wn y wny1 5 L`Ž D T 0 , 2T 0 .
5 ¨ 0 < L` q 5 w 0 5 L` .
Ž 2.33.
It follows from Ž2.33., that for n ) m, 5 ¨ n y ¨ m 5 L`Ž D T
0
, 2T 0 .
q 5 wn y wm 5 L`Ž D T 0 , 2T 0 . n
F 8 Ž 5 ¨ 0 5 L` q 5 w 0 5 L` .
Ý j Ž 12 .
jy2
,
jsm
which implies that there exists Ž ¨ , w . g C 0 Ž D T 0 , 2T0 . such that as n ª `, Ž ¨ n , wn . ª Ž ¨ , w . uniformly on D T , 2T . This procedure can be repeated a 0 0 finite number of times, w2T0 , 3T0 x, w3T0 , 4T0 x, etc., so that we get finally Ž ¨ n ,wn . converges to a continuous function Ž ¨ , w . uniformly in D T for each T ) 0 fixed. However, because of the estimate Ž2.11., there exists a subsequence Ž ¨ n k , wn k . converges weakly in H 1 Ž D T . and strongly in L2 Ž D T . to function Ž ¨ , w . g H 1 Ž D T .. Since Ž ¨ n , wn . converges uniformly in D T to Ž ¨ , w ., Ž ¨ , w . s Ž ¨ , w . and the whole sequence converges weakly in H 1 Ž D T . and strongly in L2 and Ž ¨ , w . g H 1 Ž D T . l C 0 Ž D T .. Now Ž ¨ , w . is a weak solution follows easily by passing to the limit in T
`
H0 Hy` T
`
H0 Hy`
y¨ n ywn
f t f t
q q
ž ž
wny 1 y ¨ ny1 2k wny 1 y ¨ ny1 2k
qk yk
¨n
/ /
x
wn x
f dx dt s e
2f
Hx
f dx dt s e
2
2f
Hx
2
¨ dx dt,
w dx dt.
Also we have, for 0 F t F T, ¨ Ž x, t . s
1
`
Ž 4p t e . y
t
`
1r2
H0 Hy` =
y
Hy`exp
ž
yŽ x y y . 4 te
2
/
¨ 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
¨ Ž y, s . dy ds,
1r2
/
ž
w y¨ 2k
qk
/
377
NONCONSERVATIVE HYPERBOLIC SYSTEM
1
w Ž x, t . s
`
Ž 4p t e . t
y
1r2
`
H0 Hy` =
y
Hy`
exp
ž
yŽ x y y .
2
/
4 te
w 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
1r2
/
ž
w y¨ 2k
yk
w Ž y, s . dy ds.
/
Ž 2.34.
By using the same arguments of the proof of estimate Ž2.20. 2 we can prove uniqueness for Ž ¨ , w . which satisfies the integral equation for a given ¨ 0 and w 0 in H 1 Ž R1 . l BV Ž R1 .. Also in a standard way we can prove Ž ¨ , w . is C` from the integral equation Ž2.34., provided ¨ 0 and w 0 are in S Ž R1 .. This fact also follows from the estimate Ž2.3. that we are going to prove. Next we prove the estimate Ž2.3.. Let Ž ¨ e , w e . be the solution of Ž2.1. and Ž2.2., with ¨ 0 and w 0 in S Ž R1 ., whose existence and uniqueness we have just shown. First we note that ¨ e Ž x, t . and w e Ž x, t . satisfy the estimates 5 ¨ e Ž x, t . 5 L` F 5 ¨ 0 < L` , `
Hy`
¨ e Ž x, t . x
dx F
`
Hy` ¨
X 0
5 w e Ž x, t . 5 L` F 5 w 0 5 L` , `
Ž x . dx,
Hy`
w e Ž x, t . x
F wX0 Ž x . dx,
so that in the original variables u e and ¨ e we have
5 u e 5 L` F
1
Ž 5 s0 5 L
`
k
q k 5 u 0 5 L` . ,
5 s e 5 L` F Ž 5 s 0 5 L` q k 5 u 0 5 . ,
H
ue x
H
Ž x, t . dx F
s e x
1 k
`
žH
y` `
Ž x, t . dx F H
y`
`
s 0X Ž x . dx q k
X 0
Hy` u Ž x .
s 0X Ž x . dx q k
X 0
H u Ž x.
/
dx ,
dx.
Ž 2.35.
From now on we work with the original unknown variables Ž u e , s e . and hence we consider equations Ž2.1.. Differentiating the first equation of
378
K. T. JOSEPH
Ž2.1. w.r.t. x, j times, multiplying it by j u er x j, and integrating by parts, we get 1 d 2 dt
2
j u e Ž x, t .
`
Hy`
dx q e
Hy`
xj `
sy
Hy`
q
1
xj
2
jq1 u e Ž x, t .
dx
x jq1
js e Ž x, t . jq1 u e Ž x, t .
dx
x jq1
j u e Ž x, t . jq1 u e Ž x, t . 2
`
Hy`
2
`
xj
x jq1
dx.
Ž 2.36.
Now recall the standard calculus inequality: For u, ¨ g H j Ž R1 . l L`Ž R1 ., 0 F s F j,
s Ž u¨ . xs
L2 Ž R 1 .
F C j 5 u 5 L`Ž R 1 .
s¨ xs
L2 Ž R 1 .
su
q 5 ¨ 5 L`Ž R 1 .
xs
L2 Ž R 1 .
,
where C j depends only on j. Using this we get, for 0 F s F j, `
Hy`
s u2 xs
2
dx F C j2 5 u 5 2L`
su xs
2 L2 Ž R 1 .
.
Ž 2.37.
Here and in what follows, C j denotes a constant which depends only on j, but need not be the same as the one used before. By Schwarz inequality applied on each of the terms of the RHS of Ž2.36. and then using Ž2.37., we get 1 d 2 dt
`
Hy` F
j u e Ž x, t .
2
dx q e
Hy`
xj
js e Ž x, t . xj
jq1 u e Ž x, t .
`
2
x jq1
dx
jq1 u e L2 Ž R 1 .
q C j 5 u e 5 L`Ž D T .
x jq1
L2 Ž R 1 .
jue Ž t . xj
jq1 u e Ž t . x jq1
L2 Ž R 1 .
L2 Ž R 1 .
.
Using ab F a2er4 q b 2re we obtain from above 1 d 2 dt
`
Hy` F
jue Ž t . xj
2 L2 Ž R 1 .
1
js e Ž t .
e
xj
q
L2 Ž R 1 .
e jq1 u e Ž t . 2
x jq1
q
C j2 5 u e 5 2L`Ž D T .
2
jue Ž t . xj
2 L2 Ž R 1 .
.
379
NONCONSERVATIVE HYPERBOLIC SYSTEM
Integrating this from 0 to t we get 2
jue Ž t . xj F
L2 Ž R 1 .
2
e
H0
q
e
qe
H0
x jq1
2
ds
L2 Ž R 1 .
2
js e Ž s .
t
2
jq1 u Ž s .
t
xj
L2 Ž R 1 .
t
C j2 5 u e 5 2L`Ž D T .
H0
ds 2
jue Ž s . xj
j u0
ds q
L2 Ž R 1 .
xj
2 L2 Ž R 1 .
. Ž 2.38.
From this we have t
H0
jq1 u e Ž s . x jq1
2 L2 Ž R 1 .
2
1 j u0 Ž x .
ds F
e
xj 2
q
e
q
L2 Ž R 1 .
2
t
e2
H0
js e Ž s . xj
jue Ž s .
C j2 5 u e 5 2L`Ž D T . 2
t
2C j2 5 u e 5 2L`Ž D T .
k
H0
xj
2 L2 Ž R 1 .
ds
2 L2 Ž R 1 .
ds.
Iterating this we get t
H0
jq1 u e Ž s . x jq1
2
ds F
j
1
e q q
ž
Ý ks0
j
2
Ý
e2
ž
e2
ks0
ž
/
x jyk
2C j2 5 u e 5 2L`Ž D T .
e2
C j2 5 u e 5 2L`Ž D T . 2
k
t
/H
/
t
H0 5 u
e
2 L2 Ž R 1 .
jyks e Ž s .
0
jq1
2
e
jyk u 0
x jyk
2 L2 Ž R 1 .
Ž s . 5 2L2 Ž R 1 . .
Hence for 0 - e F 1, we have t
H0
jq1 u e Ž s . x jq1 F
1
e
2Ž jq2.
2 L2 Ž R 1 .
ds
ž 1 q 2C
= 5 u 0 5 2H j q 2
j 2 5 e 52` L Ž DT . j u
/
t
H0 5 s Ž s . 5
2 Hj
ds q
t
H0 5 u Ž s . 5
2 L2 Ž R 1 .
ds . Ž 2.39.
380
K. T. JOSEPH
From Ž2.38. and Ž2.39. we get
jue Ž t . xj
2 L2 Ž R 1 .
ž 1 q 2C F
j 2 5 e 52` L Ž DT . j u
/
e 2 jq2 T
= 5 u 0 5 2H j q 2
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds . 0
Ž 2.40. By using the inequality sup f 2 Ž x . F x
1
df
2
2
dx
L2 Ž R 1 .
q 5 f 5 2L2 Ž R 1 . ,
with f Ž x . s Ž jy1 u e Ž x, t ..r x jy1, for each fixed t and using Ž2.40. with j and j replaced by j y 1, we get
sup
2
jy1 u e Ž x, t . x jy1
VT
ž 1 q 2C F
j 2 5 e 52` L Ž DT . j u
e
/
2 jq2
T
= 5 u 0 5 2H j q 2
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds . Ž 2.41. 0
Differentiating the second equation of Ž2.1. j times w.r.t. x, multiplying it with js er x j, integrating w.r.t. x, and integrating by parts, we get 1 d
js e Ž t .
2 dt
xj s yk 2 Fk
2
q
2 L2 Ž R 1 .
jq1s e Ž t .
qe
x jq1
j u e jq1s e
H x
j
x jq1
jue xj
dx q
2 L2 Ž R 1 .
jy1
Hx
u sx . jy1 Ž
jq1s e L2 Ž R 1 .
x jq1
jy1 Ž u esxe . x jy1
L2 Ž R 1 .
jq1s e L2 Ž R 1 .
x jq1
L2 Ž R 1 .
.
jq1 x jq1
s e dx
381
NONCONSERVATIVE HYPERBOLIC SYSTEM
As before, we get d
js e Ž t .
dt
xj F
2 L2 Ž R 1 .
2 k 4 jue
e
xj
2
jq1s e Ž t .
qe
x jq1
2 L2 Ž R 1 .
L2 Ž R 1 .
2 jy1 Ž u esxe .
q
e
xj
2 L2 Ž R 1 .
.
Ž 2.42.
Now we recall the calculus inequality,
jy1 x jy1
2
s
ž / u
x
jy1
L2 Ž R 1 .
F
C j2
Ý
ku
2
xk
L`Ž R 1 .
ks0
s
jyk 2
x
L2 Ž R 1 .
,
where C j depends only on j. Using Ž2.41. in the above inequality, we get, for 0 F t F T, 2
jy1 u e Ž t . s e Ž t . x
x jy1
L2 Ž R 1 .
C j2 1 q C j2 5 u e 5 2L`Ž D T .
ž
F
j
/
e 2 jq2 T
= 5 u 0 5 2H j q 2
H0 5 s
e
5 s e Ž t . 5 2H j T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds
Ž 2.43.
0
for suitable C j which depends only on j. Now integrating Ž2.42. from 0 to t and using Ž2.39. and Ž2.43. in the resulting expression we have, for 0 F t F T,
js e Ž t . xj F
2 L2 Ž R 1 .
qe
t
H0
jq1s e Ž s . x jq1
2 k 4 1 q 2C j2 5 u e 5 L`Ž D T .
ž
e
L2 Ž R 1 .
ds
j
/
e 2 jq2
= 5 u 0 5 2H j q 2
T
H0 5 s
q
2
C j2
T
H0 5 s
e
Ž s.
= 5 u 0 5 2H j q 2
5 2H j T
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds 0
ž 1 q 2C ds
j 2 5 e 52` L Ž DT . j u
e
e
/
2 jq2
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . sds q 5 s 0 5 2H j 0
382
K. T. JOSEPH
F Žk q 4
ž 1 q 2C q 1.
C j2
j 2 5 e 52` L Ž DT . j u
e
/
2 jq3
= 1 q 5 u e 5 2H j q 5 s 0 5 2H j q T
= 1q2
H0
5 s e Ž s . 5 2H j q
T
H0 5 u T
žH
0
e
Ž s . 5 2L2 Ž R 1 . 2
5 s e Ž s . 5 2H j ds
/
.
Ž 2.44.
Thus we have, with A s Ž k 4 q C j2 q 1 . 1 q 2C j2 5 u e 5 2L`Ž D T .
ž
? 1 q 5 u 0 5 2H j q 5 s 0 5 2H j q
ž
T
H0
2
jq1s e Ž s .
ds F
x jq1
A
T
H0 5 u
1q
e 2 jq4
e
T
H0
j
/
Ž s . 5 2L2 Ž R 1 . ds ,
/
Ž 2.45.
2
5 s e Ž s . 5 2H j ds ,
so that T
H0
5 s e Ž t . 5 2H jq 1 F Ž j q 1 .
A
1q
e 2 jq4
T
H0
2
5 s e Ž s . 5 2H j ds .
From this we get 1q
T
H0
5 s e Ž s . 5 2H jq 1 ds F
2 Ž j q 1.
A 1q
e 2 jq4
T
H0
2
5 s Ž s . 5 2H j ds .
Iterating this we get
ž
1q
T
H0 5 s
F
ž
e
Ž s . 5 2H jq 1 ds
2 Ž j q 1. A
e 2 jq4
/ jq1
/ ž
1q
T
H0 5 s
2 Ž jq1 .
e
Ž s . 5 2L2 Ž R 1 . ds
/
.
Hence T
H0
Ž 5 s e Ž s . 5 2H jq 1 ds . F
ž
2 Ž j q 1. A
e 2 jq4
jq1
/ ž
1q
T
H0
2 Ž jq1 .
5 s e Ž s . 5 2L2 Ž R 1 . ds
/
.
Ž 2.46.
383
NONCONSERVATIVE HYPERBOLIC SYSTEM
Using Ž2.46. in Ž2.40. and Ž2.44., we get
jue Ž t . xj
2 L2 Ž R 1 .
F
e 2Ž jq1.Ž jq2.
= 1q 1q
ž
js e Ž t . xj
2 L2 Ž R 1 .
F
jq1
4 Ž j q 1. A
T
H0
2 Ž jq1 . 2
5 s e Ž s . 5 2L2 Ž R 1 . ds
/
,
2 Ž jq2 .
4 Ž j q 1. A
1q
e 4Ž jq1.Ž jq2.
T
H0
5 s e Ž s . 5 2L2 Ž R 1 . ds
4 Ž jq1 .
,
Ž 2.47. Where A is given by Ž2.45.. Now the RHS of Ž2.47. contains T
H0 5 u
e
Ž s . 5 2L2 Ž R 1 . dx and
T
H0 5 s
e
Ž s . 5 2L2 Ž R 1 . ds.
To estimate these we estimate first the L1-norm of u e Ž x, t . and s e Ž x, t . w.r.t. x. As in the proof of the first part of Lemma 2.2, we write `
Hy` u
e
i Ž x, t . dx s Ý Ž y1. H
y iq1Ž t . e
u Ž y, t . dy,
yiŽ t .
i
where yi Ž t .: i s 0, " 1, " 2, . . . .4 are points x, where u e Ž x, t . changes sign. We call the ordering such that on Ž y 0 Ž t ., y 1Ž t .., u e Ž x, t . ) 0. Now as in that proof, we get from the first equation of Ž2.1., d
`
H u dt y`
e
i Ž x, t . dx F Ý Ž y1. H
y iq1Ž t .
y
yiŽ t .
i
s e
`
Ž y, t . dy F H
y`
sxe Ž x, t . dx.
Similarly from the second equation of Ž2.1., we get d
`
H s dt y`
e
Ž x, t . dx F 5 u e Ž x, t . 5 L`Ž V T . `
Hy` s
=
x
e
`
Ž x, t . dx q k 2H
y`
u ex Ž x, t . dx.
384
K. T. JOSEPH
Using the estimates Ž2.35. in the above inequalities and integrating we get, 0 F t F T, `
Hy` u
e
`
Ž x, t . dx F H
q `
Hy` s
u 0 Ž x . dx
y`
e
T
`
Hy` s
k
`
`
Ž x . dx q kH
y`
s 0 Ž x . dx q
Ž x, t . dx F H
y`
=
X 0
`
Hy` s
X 0
T
uX0 Ž x . dx ,
5 s 0 5 L` q k 5 u 0 5 L` q k 2
k
`
Ž x . dx q kH
y`
uX0 Ž x . dx
and hence `
Hy` u
e
Ž x, t .
2
dx F 5 u e Ž x, t . 5 Le 5 u e Ž x, t . 5 L1 Ž R 1 . F
1
5 s 0 5 L` q k 5 u 0 5 L`
k
= 5 u 0 5 L1 q
T k
ž
`
Hy` s
X 0
dx q k
`
X 0
Hy` u
dx
/ Ž 2.48.
and `
Hy` s
e
Ž x, t .
2
dx F
½
5 s 0 5 L` q k 5 u 0 5 Le 5 s 0 5 L1 q
=
T k
k 2 q 5 s0 5 L
`
`
Hy` s
X 0
q k 5 u 0 5 L` 4 `
Ž x . dx q kH
y`
5
uX0 Ž x . dx .
Ž 2.49.
Using Ž2.48. and Ž2.49. in Ž2.47., we get for j s 0, 1, 2, . . . , sup 0FtFT
sup 0FtFT
jue Ž t . xj js e Ž t . xj
2 L2 Ž R 1 .
F
2 L2 Ž R 1 .
F
CT2
e 4Ž jq1.Ž jq2. CT2
e 4Ž jq1.Ž jq2.
,
,
Ž 2.50.
385
NONCONSERVATIVE HYPERBOLIC SYSTEM
where CT is a polynomial of L1 norm, L` norm, BV norm, and H j norm of u 0 Ž x . and s 0 Ž x . and T, whose coefficients depends only on j. Now using sup
X
xgR 0FtFT
ž
jy1 u e Ž x, t . x jy1
2
`
F sup
/
0FtFT
Hy`
ž
/ /
x jy1
`
q
2
jy1 u e Ž x, t .
Hy`
j u e Ž x, t .
ž
xj
dx 2
dx
and the same inequality with u replaced by s , we get, for j s 1, 2, . . . , sup
jy1 u e Ž x, t . x
xgR 1 0FtFT
sup
jy1
jy1s e Ž x, t . x jy1
xgR 1 0FtFT
F
F
e
CT 2Ž jq2.Ž jq1. CT
e 2Ž jq2.Ž jq1.
,
,
Ž 2.51.
where CT is a polynomial of L1 norm, L` norm, BV norm, and H j norm of u 0 Ž x . and s 0 Ž x . and T, whose coefficients depends only on j. This gives the estimate Ž2.3. for l s 0, j s 0, 1, 2, . . . . Now applying the differential operator lqjr t l x j on both sides of the equation Ž2.1., we get
lqjq1 u e t lq1 x j lqjq1s e
se se
lqjq2 u e t l x jq2 lqjq2s e
y y
1 lqjq1 Ž u e . 2
2
t l jq1
lqj Ž u e Ž s
e
r x .
q
.
lqjq1s e t l x jq1
q k2
,
lqjq1 u e
. Ž 2.52. t lq1 x j t l x jq2 tl x j t l x jq1 When l s 0, the RHS of Ž2.52. does not contain any t derivatives. Hence jq1 u er t x j and jq1s er t x j can be estimated using Ž2.1. in Ž2.52.. Thus we get Ž2.3. when l s 1, j s 0, 1, 2, . . . . When l s 1, the RHS of Ž2.52. can be estimated using Ž2.3. with l s 1 and j s 0, 1, 2, . . . . This gives Ž2.3. for l s 2, j s 0, 1, 2, . . . . Proceeding inductively, we get the estimate Ž2.3.. The proof of Theorem 2.1 is complete.
3. GENERALIZED SOLUTIONS FOR Ž0.1. AND Ž0.2. In this section we solve the problem u t q uu x y sx f 0,
st q u sx y k 2 u x f 0,
Ž 3.1.
386
K. T. JOSEPH
with initial conditions u Ž x, 0 . s u 0 ,
s Ž x, 0 . s s 0 .
Ž 3.2.
We assume u 0 s Ž u 0e . 0 - t F 1 and s 0 s Ž s 0e . 0 - e F 1 are in G Ž R1 ., the algebra of the generalized function of Colombeau, and u 0e and s 0e satisfy the estimate, for j s 1, 2, . . . , C2 C1 j 5 u 0e 5 H j F l , 5 f 0e 5 H j F m j , e j e j du 0e Ž x .
H
dx
Hu
e 0
F
Ž x . dx F
C
e
p
C
d s 0e Ž x .
,
H
,
Hs
dx e 0
F
C
ep
,
C
Ž x . dx F
, Ž 3.3. ep for some constants C1 j , C2 j , C and for some integers l j , m j , p. Here we remark that all tempered distributions can be represented by Ž3.3., i.e., for all T g S Ž R1 . there exists u 0e satisfying Ž3.3. with Ž u 0e . f T. With these assumptions, we shall prove the following theorem.
e
p
THEOREM 3.1. Let Ž u, s . s Ž u e , s e . 0 - e - 1 be the solution of Ž2.1. and Ž2.2. with initial data Ž u 0 , s 0 . s Ž u 0e , s 0e .. Then Ž u, s . g G Ž V T . for each fixed T ) 0 and is a solution of Ž3.1. and Ž3.2. as e ª 0. Proof. From the assumptions Ž3.3. on Ž u 0 , s 0 . and the estimate Ž2.3. and the fact that the constant C appearing in Ž2.3. depends polynomially on H j norms, L1 norms, L` norms, and BV norms of initial data Ž u 0e , s 0e ., we easily get Ž u, s . g G Ž V T .. Now to show that Ž u, s . satisfy Ž3.1. we note that
HH Ž u HH Ž s
t
e
e t
q u e u ex y sxe . f dx dt s e
HHf
q u esxe y k 2 u ex . f dx dt s e
HHf
xxu
e
xxs
e
dx dt, dx dt,
for any f Ž x, t . g C0`Ž R1 = Ž0, T ... Now the RHS of the above equations goes to zero as e ª 0, since u e and s e are bounded. The proof of the theorem is complete. ACKNOWLEDGMENTS The author is grateful to Professor J. F. Colombeau for introducing to him the new theory of generalized functions and the problem discussed in this paper. His constant encouragement and valuable suggestions during the course of this work is gratefully acknowledged. The author is also thankful to the referees for their constructive criticism, which improved the presentation of an early version of the paper.
NONCONSERVATIVE HYPERBOLIC SYSTEM
387
REFERENCES 1. F. Berger, ‘‘Methodes Numeriques Utilisant des Multplications de Distributions,’’ These, Lyon]St. Etinne, 1992. 2. J. J. Cauret, J. F. Colombeau, and A. Y. Le Roux, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equation, J. Math. Anal. Appl. 139 Ž1989., 552]573. 3. J. F. Colombeau, ‘‘New Generalized Functions and Multiplication of Distributions,’’ North-Holland, Amsterdam, 1984. 4. J. F. Colombeau, ‘‘Multiplication of Distributions, a Tool in Mathematics, Numerical Engineering and Theoretical Physics,’’ Lecture Notes in Mathematics, Vol. 1532, Springer-Verlag, Berlin, 1992. 5. J. F. Colombeau and Oberguggenberger, Hyperbolic systems with a compatible quadratic term, delta waves and multiplication of distributions, Comm. Partial Differential Equations 15 Ž1990., 905]938. 6. J. F. Colombeau and A. Heibig, Generalized solutions to Cauchy problems, Monatsh. Math. 117 Ž1994., 33]49. 7. J. F. Colombeau, Multiplication of distributions: A graduate course, application to theoretical and numerical solutions of partial differential equations, Lyon 1993. 8. J. F. Colombeau A. Heibig, and M. Oberguggenberger, The Cauchy problem in a space of generalized functions I, II, C. R. Acad. Sci. Paris Ser. I Math. 317 Ž1993., 851]855; 319 Ž1994., 1179]1183. 9. J. F. Colombeau, A. Y. Le Roux, A. Noussair, and B. Perrot, Microscopic profiles of shock waves and ambiguities in multiplications of distributions, SIAM J. Numer. Anal. 26 Ž1989., 871]883. 10. A. Friedman, ‘‘Partial Differential Equations of Parabolic Type,’’ Prentice-Hall, Englewood Cliffs, N.J., 1964. 11. A. Noussair, ‘‘Conception et Validation de Schemas Numeriques pour l’Elastoplasticite Dynamique en Milieu Non Homogene,’’ Thesis, Bordeaux, 1989. 12. M. Oberguggenberger, ‘‘Multiplication of Distributions and Applications to PDEs,’’ Pittman Research Notes in Math, Vol. 259, Longman, Harlow, 1992. 13. O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12 Ž1957., 3]73, wEnglish translation Amer. Math. Soc. Transl. Ser. 2 26 Ž1957., 95]172. 14. A. Prignet, Numerical Methods for uniqueness of jump conditions of a quasilinear hyperbolic system in the space of generalized functions, Preprint, Lyon, 1994.
207, 361]387 Ž1997.
AY975274
Generalized Solutions to a Cauchy Problem for a Nonconservative Hyperbolic System K. T. Joseph School of Mathematics, TIFR, Homi Bhabha Road, Bombay 400 005, India Submitted by Wolfgang L. Wendland Received January 9, 1995
In this paper we show the existence of generalized solutions, in the sense of Colombeau, to a Cauchy problem for a nonconservative system of hyperbolic equations, which has applications in elastodynamics. Q 1997 Academic Press
INTRODUCTION This paper is devoted to the study of a Cauchy problem for a strictly hyperbolic, nonconservative system of equations, namely,
u t s t
qu
qu
u x
s x
y
y k2
s x u
s 0, s 0,
Ž 0.1.
s Ž x, 0 . s s 0 Ž x . .
Ž 0.2.
x
in x g R 2 , t ) 0, with initial conditions u Ž x, 0 . s u 0 Ž x . ,
The system Ž0.1. is a simplified model, which comes for applications in elastodynamics, where u is the velocity, s is the stress, and k ) 0, is the speed of propagation of the elastic waves. The details can be found in w2x and the references given there. It is well known that global smooth solutions do not exist for the problem Ž0.1. and Ž0.2., even when the initial data are smooth; discontinuities in u and s appear in finite time. When u and s are discontinuous the product u sx , which appear in the second equation of the system Ž0.1., 361 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
362
K. T. JOSEPH
does not make sense in the standard theory of distributions. To overcome such difficulties, Colombeau w3x introduced a new notion of generalized functions. In recent works w1, 2, 4]9, 11, 12, 14x and many other references in w4x and w12x, it is recognized by many authors that the algebra of generalized functions of Colombeau is a convenient setup to seek global solutions, where such difficulties arise. Further this approach takes into account the microscopic structure of the shocks Ždiscontinuities . of the solution. Cauret et al. w2x studied the system Ž0.1. numerically, proved the convergence of the scheme, and constructed generalized functions from the numerical scheme which are solutions of Ž0.1.. Moreover it was shown that when one seeks travelling wave solutions, u Ž x, t . s Ž u r y u l . H Ž x y ct . q u l ,
s Ž x, t . s Ž sr y s l . K Ž x y ct . q s l , where H and K are two Heaviside generalized functions and u l , u r , s l , sr , and c are constants, the system Ž0.1. admits an infinite number of jump conditions for the shock, depending upon the microscopic shapes of the jumps in H and K. In this connection we also mention the work of Noussair w11x, who used the Godunov scheme, and the work of Berger w1x, who used a nonconservative Lax]Friedrichs scheme to study the system. More recently, Colombeau w7x and Prignet w14x did numerical tests of the system
u
qu
t s t
qu
u
y
x
s x
yk
2
s x u x
s e1 s e2
2u x2 2s x2
, ,
Ž 0.3.
with special initial data and studied its solutions Ž u e 1 , e 2 , s e 1 , e 2 ., when e 1 , e 2 ª 0, with l s e 1re 2 fixed. In their work computations were done for various Žsmall. values of e 1 and e 2 , keeping the ratio l fixed. These calculations were repeated for different values of l, namely, l s 60, 4, 1, 1r4, 1r60 and they arrived at the conclusion that the limit of Ž u e 1 , e 2 , s e 2 , e 2 . as e 1 , e 2 ª 0, keeping l s e 1re 2 fixed, depends on l and is a solution of the system
u t s t
qu
qu
u x
s x
y
y k2
s x u x
f 0, f 0,
Ž 0.4.
NONCONSERVATIVE HYPERBOLIC SYSTEM
363
in the algebra of generalized functions of Colombeau. When l s 1, i.e., when e 1 s e 2 s e , this corresponds to the case where Ž u, s . s Ž u e , s e . 0 - e - 1 has the same microscopic shape for the discontinuity. The cases l ª 0 and l ª ` corresponds to the extreme cases in which s is replaced by f in the first and second equations, respectively, of the system Ž0.1.. Here s means equality as elements of the algebra of generalized functions and f means association in the sense of Colombeau. In this article we consider the case e 1 s e 2 s e , in Ž0.3. with initial conditions Ž0.2. and using standard arguments, show that for each fixed e ) 0, there exists a unique C` global solution Ž u e , ¨ e . if u 0 Ž x . and ¨ 0 Ž x . are in S Ž R1 ., the space of Schwartz class functions. Later the a priori estimates show that if u 0 Ž x . and ¨ 0 Ž x . are in H 2 Ž R1 . l BV Ž R1 . then there exist C 2 solutions. The main contributions in this article are the estimate Ž2.3. on the higher derivatives of Ž u e , ¨ e . in sup norm and an existence result, Theorem 3.1, for global generalized solution of Ž0.1. and Ž0.2., for rather arbitrary u 0 and ¨ 0 in G Ž R1 .. The paper is organized in the following way. In Section 1, we briefly review the definition of the algebra of generalized functions and the notion of equality and association. In Section 2 we show the existence of a global smooth solution for the viscous system and get estimates on higher derivatives of the solution. Finally in Section 3 we construct a generalized solution of Ž0.1. and Ž0.2..
1. THE ALGEBRA OF GENERALIZED FUNCTIONS OF COLOMBEAU In this section we quickly recall the definition of the algebra of generalized functions in V T s Ž x, t .: y` - x - `, 0 - t - T 4 , for T ) 0, denoted by G Ž V T .. We take the point of view of Colombeau and Oberguggenberger w5x. Let u s Ž u e . 0 - e - 1 be an element of the infinite product of smooth functions in V T : Ž C`Ž V T ..Ž0, 1.. We say u is moderate if it has the following property: For every compact subset K of V T and all nonnegative integers j and l, 'N ) 0 s.t.
j l ue t j xl
L`Ž K .
s O Ž eyN .
as e ª 0.
The set of all moderate elements of Ž C`Ž V T ..Ž0, 1. is denoted by EM Ž V T .. A sequence u s Ž u e . is called null if for all compact subsets K of V T and
364
K. T. JOSEPH
for all nonnegative integers j and l and for all M ) 0,
j l ue t j xl
L`Ž K .
s OŽ e M .
as e ª 0.
The set of all null elements is denoted by N Ž V T .. It is easy to see that EM Ž V T . is an algebra with partial derivatives, the operations being defined pointwise, and N Ž V T . is an ideal which is closed under differentiation. The quotient space, denoted by G Ž VT . s
EM Ž V T . N Ž VT .
,
is an algebra with partial derivatives, the operations being defined on representatives. G Ž V T . is called the algebra of generalized functions. Two elements u and ¨ in G Ž V T . are said to be associated, if for some Žand hence all. representative sequences Ž u e . 0 - e - 1 and Ž ¨ e . 0 - e - 1 of u and ¨ , we have ue y ¨e ª 0 in the sense of distributions in V T as e ª 0 and is denoted by u f ¨ . We emphasize that this notion is different from the notion of equality in G Ž V T ., which means that u y ¨ g N Ž V T ., or in other words,
j l Ž ue y ¨ e . tj l
L`Ž K .
s OŽ e M .
as e ª 0
for all M, for j, l nonnegative integers, and for all compact sets K ; V T . 2. EXISTENCE OF SMOOTH GLOBAL SOLUTION FOR THE VISCOUS SYSTEM In this section we consider the system Ž0.3. with e 1 s e 2 s e , i.e.,
ue t s e t
q ue
qu
e
ue x
s e x
s e
y
yk
x
2
ue x
se se
2 ue x2 2s e x2
, ,
Ž 2.1.
in y` - x - `, t ) 0, with initial conditions u e Ž x, 0 . s u 0 Ž x . ,
s e Ž x, 0 . s s 0 Ž x . .
Ž 2.2.
365
NONCONSERVATIVE HYPERBOLIC SYSTEM
We assume
Ž H.
¨ 0 Ž x . , u 0 Ž x . g H 1 Ž R1 . `
`
Hy` ¨
and X 0
Hy` u Ž x .
X 0
Ž x . dx - ` and
dx - `.
We shall prove THEOREM 2.1. Ži. For each T ) 0, there exists a unique weak solution Ž u e , s e . in H V T . for Ž2.1. and Ž2.2., if ¨ 0 Ž x . and u 0 Ž x . satisfy hypothesis ŽH.. Žii. Let ¨ 0 Ž x . and u 0 Ž x . g S Ž R1 . be the space of Schwartz functions and 0 - e F 1. Let Ž u e , s e . be the solution of Ž2.1. and Ž2.2.. Then, gi¨ en l and j nonnegati¨ e integers, ' constant CT and positi¨ e integers N0 and p 0 such that 1Ž
sup VT
l j t x l
u e Ž x, t . F j
CT
e
N0
,
sup VT
l j t x l
j
s e Ž x, t . F
CT
e N0
, Ž 2.3.
where CT depends polynomially on 5 u 0 5 H p , 5 ¨ 0 5 H p for 0 F p F p 0 , 5 u 0 5 L1 , 5 ¨ 0 5 L1 , 5 u 0 5 BV , 5 ¨ 0 5 BV , and T. Here N0 and p 0 depend only on l and j. The proof of this theorem is broken up into lemmas. For brevity of notation we often suppress the dependence of e and write s , u, w, and ¨ for s e , u e , w e , and ¨ e . First we write Ž2.1. in a convenient form. Let ¨ s s y ku and w s s q ku, where k ) 0 is the constant given in Ž0.1.. Then, Ž2.1. can be written as ¨t q
wt q
ž ž
w y¨ 2k w y¨ 2k
q k ¨x s e ¨xx ,
/ /
y k wx s e wx x .
Ž 2.4.
The initial conditions Ž2.2. become ¨ Ž x, 0 . s ¨ 0 Ž x . s s 0 y ku 0 ;
w Ž x, 0 . s w 0 Ž x . s s 0 q ku 0 . Ž 2.5.
Since ¨ 0 and w 0 are in H 1 Ž R1 ., they are continuous bounded functions on R1. To prove part Ži. of Theorem 2.1, in a standard way, we define a sequence of functions Ž ¨ nŽ x, t ., wnŽ x, t .., n s 0, 1, 2, . . . , iteratively,
Ž ¨ 0 Ž x, t . , w 0 Ž x, t . . s Ž ¨ 0 Ž x . , w 0 Ž x . . ,
366
K. T. JOSEPH
and for n s 1, 2, 3, . . . , let Ž ¨ nŽ x, t ., wnŽ x, t .. be defined by the solution of linear problems
¨n t wn t
q q
ž ž
wny1 y ¨ ny1 2k wny1 y ¨ ny1 2k
qk yk
¨n
/ /
x
wn x
se se
2 ¨n x2 2 wn x2
, ,
Ž 2.6. n
with initial conditions ¨ n Ž x, 0 . s ¨ 0 Ž x . ,
wn Ž x, 0 . s w 0 Ž x . .
Ž 2.7. n
First we assume ¨ 0 Ž x . and w 0 Ž x . are in S Ž R1 .. By linear theory, there exists a unique C` solution Ž ¨ 1 , w 1 . to Ž2.6.1 and Ž2.7.1; see Friedman w10x. Further, the solution and its derivatives decay to zero as < x < ª `. Moreover by maximum principle, sup ¨ 1 Ž x, t . s 5 ¨ 0 5 L`Ž R 1 . ,
sup w 1 Ž x, t . s 5 w 0 5 L`Ž R 1 . .
VT
VT
Iteratively, we get a unique C` solution Ž ¨ n , wn . of the problem Ž2.6. n ] Ž2.7. n such that they and their derivatives decay to zero as < x < ª `. Further sup ¨ n Ž x, t . s 5 ¨ 0 5 L`Ž R 1 . ,
sup wn Ž x, t . s 5 w 0 5 L`Ž R 1 . .
VT
Ž 2.8.
VT
Denote
lŽ1xn, t . s lŽ2xn, t . s
wn Ž x, t . y ¨ n Ž x, t . 2k wn Ž x, t . y ¨ n Ž x, t . 2k
q k and y k.
Ž 2.9.
By Ž2.8., we have that there exists a constant l G 1 such that sup l i n Ž x, t . F l VT
for i s 1, 2 and n s 0, 1, 2, . . . .
Ž 2.10.
With these we shall prove the following lemma. LEMMA 2.1 5 Ž ¨ n , wn . 5 H 1 Ž D T . F C Ž e , l , T . 5 Ž ¨ 0 w 0 . 5 H 1 Ž R 1 . .
Ž 2.11.
Proof. First we take ¨ 0 and w 0 in S Ž R1 .. Multiplying the first equation of Ž2.6. n by ¨ n and the second equation of Ž2.6. by wn , integrating w.r.t. x,
367
NONCONSERVATIVE HYPERBOLIC SYSTEM
integrating parts, and adding we get 1 d
`
H Ž¨ 2 dt y` s ye
2 n
Ž x, t . q wn2 Ž x, t . . dx `
Hy`
ž
`
1 ny1
Ž x, t . ¨ n Ž x, t .
`
Hy`l
y
dx q
/
x
Hy`l
y
2
¨ n Ž x, t .
2 ny1
Hy`
ž
wn Ž x, t .
¨ n Ž x, t . x
wn Ž x, t .
Ž x, t .
`
x
x
2
/
dx
dx
wn Ž x, t . dx.
Ž 2.12.
Now using Ž2.10. and Schwarz inequality we get, `
Hy`l
1 ny1
x
¨n
Fl F
¨ n Ž x, t .
Ž x, t . x
L2 Ž R 1 .
e ¨n
5 ¨ n 5 L2 Ž R 1 .
2
x
2
¨ n Ž x, t . dx
L2 Ž R 1 .
q
l2 2e
5 ¨ n 5 2L2 Ž R 1 . .
Ž 2.13.
Similarly, `
Hy`l
2 ny1
Fl F
wn Ž x, t .
Ž x, t .
x
wn x
L2 Ž R 1 .
e wn 2
x
wn Ž x, t . dx
5 wn 5 L2 Ž R 1 .
2 L2 Ž R 1 .
q
l2 2e
5 wn 5 2L2 Ž R 1 . .
Ž 2.14.
Using Ž2.13. and Ž2.14. in Ž2.12. we get d dt
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . qe
F
l2 e
¨nŽ t . x
2 L2 Ž R 1 .
q
wn Ž t . x
2 L2 Ž R 1 .
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 1L2 Ž R 1 . .
Ž 2.15.
368
K. T. JOSEPH
It follows from Ž2.15. that d
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 ¨ n Ž t . 5 2L2 Ž R 1 . F
l2
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 .
e
dt and integrating this we get
5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . F e Ž l
2
5 ¨ 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . .
r e .t
Substituting this in the RHS of Ž2.15. and integrating from 0 to t we get 5 ¨nŽ t .
5 2L2 Ž R 1 .
q 5 wn Ž t .
5 2L2 Ž R 1 .
qe
t
H0
2
¨nŽ s.
ž
x
F Ž 5 u 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . . 1 q
l2
L2 Ž R 1 .
q
t ŽŽ l 2 r e . s.
He e 0
wn Ž s . x
2 L2 Ž R 1 .
/
ds
ds .
After integration we have 5 ¨ n Ž t . 5 2L2 Ž R 1 . q 5 wn Ž t . 5 2L2 Ž R 1 . t
qe
H0
¨nŽ s.
ž
x
2 L2 Ž R 1 .
wn Ž s .
q
x
F Ž 5 ¨ 0 5 2L2 Ž R 1 . q 5 w 0 5 2L2 Ž R 1 . . e ŽŽ l
2
2 L2 Ž R 1 .
t .r e .
/
ds
.
Ž 2.16.
Now differentiating Ž2.6. n w.r.t. x, multiplying the resulting equations by ¨ x and wx , integrating by parts, and adding we get 1 d 2 dt
`
Hy`
ž
Hy`
s
`
Hy`
wn Ž x, t .
q
2
/ ž / ž Ž ./ ž Ž ./ x
`
qe
2
¨ n Ž x, t .
2
2 ¨ n x, t
q
x2
l1 ny1
dx
x
2 wn x, t
2
dx
x2
¨ n 2 ¨ n Ž x, t .
dx q
`
H
l2 ny1
x x2 ` Repeating the same argument as before, we get ¨nŽ t . x qe
Fe
2 L2 Ž R 1 .
t
H0
q
wn Ž t . x
2 ¨nŽ s.
Ž l 2 r e .t
x2 d¨ 0 dx
x
x2
dx.
2 L2 Ž R 1 .
2 L2 Ž R 1 .
ds q
2 L2 Ž R 1 .
wn 2 wn Ž x, t .
q
dw 0 dx
t
H0
2 wn Ž s . x2
L2 Ž R 1 .
ds
2 L2 Ž R 1 .
.
Ž 2.17.
369
NONCONSERVATIVE HYPERBOLIC SYSTEM
Now to estimate 5 ¨ nr t 5 2L2 Ž R 1 . q 5 wnr t 5 2L2 Ž R 1 . , we note that
¨n
se
ž / t
2
2
2 ¨n
2
ž / x2
se
ž / t
2
x
x x2
,
2
2 wn
2
q l22 ny1 Ž x, t .
ž / x2
wn
y 2 el2 ny1 Ž x, t .
x
wn
2
ž / x
2 wn
?
2
ž /
¨n 2 ¨n
y 2 el1 ny1 Ž x, t .
wn
¨n
q l12 ny1 Ž x, t .
x2
.
Now using, the estimates
¨n 2 ¨n
2 el1 n Ž x, t . 2 el2 n Ž x, t .
Fe2
x x2 wn 2 wn x
Fe
x2
2
2
2 ¨n
q l2
¨n
2
ž / ž / ž / ž / x2
2
2 wn
q l2
x2
x2
wn
, 2
x2
in the above equations and adding the resulting expressions and integrating, we get 2
¨nŽ s.
t
H0
s F 2e
L2 Ž R 1 .
2
wn Ž s .
q
s
H0
x2
q 2l
2
t
H0
L2 Ž R 1 .
¨ Ž s. x
ds
L2 Ž R 1 .
2 ¨nŽ s.
t
2
q
2
2 wn Ž s . x2 wn Ž s .
L2 Ž R 1 .
x
L2 Ž R 1 .
ds
2 L2 Ž R 1 .
ds.
Ž 2.18.
Now using Ž2.17. in Ž2.18. we get t
H0
2
¨nŽ s. s F 2 eŽ l
L2 Ž R 1 .
2
e .t
q
wn Ž s .
Ž e q l2 .
s d¨ 0 dx
2 L2 Ž R 1 .
ds
2 L2 Ž R 1 .
q
dw 0 dx
2 L2 Ž R 1 .
.
Ž 2.19.
370
K. T. JOSEPH
Hence from Ž2.16., Ž2.17., and Ž2.19. we have, for each T ) 0, 5 Ž ¨ n , wn . 5 2H 1 Ž D T . F 2 e Ž l
2
e
r e .T
l2
5 Ž ¨ 0 , w 0 . 5 2H 1 Ž R 1 . .
q e q l2
Since S Ž R1 . is dense in H 1 Ž R1 . it follows that Ž2.11. is true for ¨ 0 , and w 0 are in H 1 Ž R1 .. The proof of the lemma is complete. In the next lemma we prove that the map Ž ¨ ny 1 , wny1 . ª Ž ¨ n , wn . is a contraction map in L`Ž R1 = w0, T0 x., where T0 depends only on 5 ¨ 0 5 L` , ` < X Ž .< ` < X Ž .< 5 w 0 5 L` , Hy` ¨ 0 x dx, Hy` w 0 x dx, and e . LEMMA 2.2. For n G 1 and t G 0,
Ži.
`
Hy` `
Hy`
¨ n Ž x, t . x wn Ž x, t . x
`
X 0
dx F
Hy` ¨
dx F
Hy` w Ž x .
`
Ž x . dx,
X 0
dx.
Ž 2.20. 1
For n s 2, 3, . . . ,
Žii.
5 Ž ¨ n y ¨ ny1 , wn y wny1 . 5 L`Ž D T
0
.
F 5 Ž ¨ ny1 y ¨ ny2 , wny1 y wny2 . 5 L`Ž D T 0 . 1 2
Ž 2.20. 2
for T0 s 1r9C 2 , where Cs
X X ` 2 l q Ž 1r2 k . Ž Hy` ¨ 0 Ž x . dx q H w 0 Ž x . dx .
Ž pe .
1r2
.
Proof. To prove Ži. we make use of an idea of Oleinik w13x to write `
Hy`
¨ n Ž x, t . x
dx s
y iq1Ž t .
Ý Ž y1. iH Ž .
¨ n Ž x, t . x
yi t
i
dx,
Ž 2.21.
where yi Ž t .: i s 0, " 1, " 2, . . . 4 are the points x Žfor fixed t ., where Ž ¨ nr x .Ž x, t . changes sign. We choose the ordering of the index such that on the interval Ž y 0 Ž t ., y 1Ž t .., Ž ¨ nr x .Ž x, t . ) 0. Now from Ž2.21. we get d
`
H dt y`
¨n x
i Ž x, t . dx s Ý Ž y1. H
i
y iq1Ž t .
yiŽ t .
t
ž
¨ n Ž x, t . x
/
dx Ž 2.22.
371
NONCONSERVATIVE HYPERBOLIC SYSTEM
because Ž ¨ nr x .Ž yi Ž t ., t . s 0. Differentiating the first equation of Ž2.6. n w.r.t. x we get
t
¨n
ž / x
q
x
ž
¨n
l1 ny1 Ž x, t .
x
/
se
2 x
¨n
2
ž / x
.
Now multiplying this equation by Žy1. i and then integrating from yi to yiq1 w.r.t. x and using Ž ¨ nr x .Ž x, t . s 0 at x s yi Ž t . and yiq1Ž t . we get
Ž y1.
i
Hy
¨n
t
x
yiq1 i
½
s e Ž y1 .
Ž x, t . dx
i
¨n
x
q Ž y1 .
iq1
ž /Ž Ž. . ž / Ž Ž . .5 yiq1 t , t
x
¨n
x
x
yi t , t
.
Ž 2.23.
By the choice of the ordering of the indices of yi 4 , Ž ¨ Ž x, t ..r x is positive on Ž yiy1Ž t ., yi Ž t .. and negative on Ž yi Ž t ., yiq1Ž t .. if i is an odd integer. Thus Ž ¨ Ž x, t ..r x is nonincreasing at x s yi Ž t .. In other words Ž 2 ¨ r x 2 .Ž yi Ž t ., t . F 0 if i is an odd integer. By a similar reason Ž 2 ¨ Ž yi Ž t ., t ..r x 2 G 0 if i is an even integer. Hence each of the terms on the RHS of Ž2.23. is nonpositive. This implies that the RHS of Ž2.22. is nonpositive. In other words, d
¨n
`
H dt y`
x
Ž x, t . dx F 0.
Integrating this we get `
Hy`
¨n x
`
d
y`
dx
Ž x, t . dx F H
¨ 0 Ž x . dx.
The proof for wn is similar. To prove Žii., first we note from Ž2.6. n and Ž2.7. n that ¨ n satisfies the integral equation ¨ n Ž x, t . s
1
`
Ž 4p t e . y
t
`
1r2
H0 Hy`
Hy`exp
ž
yŽ x y y . 4 te
2
/
¨ 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
=l1 ny1
ž
4p Ž t y s . e
¨n y
Ž y, s . dy ds
1r2
/
372
K. T. JOSEPH
and hence, for n G 2, ¨ n Ž x, t . y ¨ ny1 Ž x, t .
sy
t
`
H0 Hy`
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
= l1 ny1 Ž y, s .
y
/
1r2
¨ n Ž y, s . y l1 ny2 Ž y, s .
¨ ny1 Ž y, s . dy ds.
y
Now
l1 ny1
¨n x
s
ž
y l1 ny2
¨ ny1 x
wny 1 y ¨ ny1 2k
q
ž
qk
wny 1 y wny2 2k
¨n
/ž
x
y
¨ ny1
¨ ny1 y ¨ ny2
y
/
x
2k
¨ ny1
/
x
and hence ¨ n Ž x, t . y ¨ ny1 Ž x, t . s A q B,
Ž 2.24.
where t
A Ž x, t . s y
`
H0 Hy`
½
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e =
ž
wny 1 y ¨ ny1 2k
/
1r2
qk
/
Ž ¨ n y ¨ ny1 . y
5
dy ds
Ž 2.25. and t
B Ž x, t . s y
`
H0 Hy`
=
ž
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
wny 1 y wny2 2k
y
/
1r2
Ž ¨ ny1 y ¨ ny2 . ¨ ny1 2k
/
y
dy ds.
Ž 2.26.
373
NONCONSERVATIVE HYPERBOLIC SYSTEM
Integrating by parts w.r.t. x, we get
A Ž x, t . s
ž
H0 Hy`
=
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
`
t
4p Ž t y s . e
wny 1 y ¨ ny1
ž
/Ž
2k `
t
y
H0 Hy` =
ž
/
y
1r2
¨ n y ¨ ny1 . Ž y, s . dy ds 2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
wny 1 y ¨ ny1 2k
/
1r2
Ž x y y. 2Ž t y s . e
q k Ž ¨ n y ¨ ny1 . Ž y, s . dy ds.
/
Taking sup D t < ¨ n y ¨ ny1 < outside the integral and using part Ži. of the 0 lemma, we get for 0 F t F t 0 F T, A Ž x, t . F sup ¨ n y ¨ ny1 Dt0
`
H Ž¨ 2 k y`
H0
Ž x . q wX0 Ž x . . dx
1r2
4p Ž t y s . e
q l sup ¨ n y ¨ ny1 Dt 0
F
X 0
ds
t
=
1
t 01r2
Ž pe .
1r2
½
2l q
1 2k
1
p 1r2 `
žH
y`
`
t
H0 Hy` X
< z < eyz
Ž t y s. e
¨ 0 Ž x . dx q
= sup ¨ n y ¨ ny1 .
2
`
1r2
X 0
dz ds
Hy` w Ž x .
dx
/5
Ž 2.27.
Dt 0
Here and in what follows, we assume that T0 F T, where T0 is given in the second part of Lemma 2.2. By a similar argument as above, we get
B Ž x, t . F
t 01r2
Ž pe .
1r2
`
Hy` ¨
X 0
Ž x . dx ?
1 2k
= 5 wny1 y wny2 5 L`Ž D t 0 . q 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . 4 . Ž 2.28.
374
K. T. JOSEPH
From Ž2.24. ] Ž2.28., we get for 0 F t F t 0 F T, ¨ n Ž x, t . y ¨ ny1 Ž x, t .
F
t 01r2
Ž pe .
½
1r2
2l q
1 2k
`
žH
y`
X
¨ 0 Ž x . dx q
`
X 0
Hy` w Ž x .
dx
/5
=5 ¨ n y ¨ ny1 5 L`Ž D t 0 . q
t 01r2
Ž pe .
1
1r2
2k
ž
`
Hy` ¨
X 0
Ž x . dx
/
= 5 wny1 y wny2 5 L`Ž D t 0 . q 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . 4 .
Ž 2.29.
Similarly we get for 0 F t F t 0 F T, wn Ž x, t . y wny1 Ž x, t . F
t 01r2
Ž pe .
½
1r2
2l q
1 2k
ž
`
Hy` ¨
X 0
`
Ž x . dx q H
y`
wX0 Ž x . dx
/5
=5 wn y wny1 5 L`Ž D t 0 . q
t 01r2
Ž pe .
1r2
1
`
X 0
H w Ž x. 2 k y`
dx
= 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . q 5 wny1 y wny2 5 L`Ž D t 0 . 4 .
Ž 2.30.
Thus we have, from Ž2.29. and Ž2.30., 5 ¨ n y ¨ ny1 5 L`Ž D t . q 5 wn y wny1 5 L`Ž D t . 0 0 F t 01r2 C 5 ¨ n y ¨ ny1 5 L`Ž D t 0 . q 5 wn y wny1 5 L`Ž D t 0 . 4 q Ct 01r2 5 ¨ ny1 y ¨ ny2 5 L`Ž D t 0 . q 5 wny1 y wny2 5 L`Ž D t 0 . 4 , Ž 2.31. where Cs
` < X ` < X 2 l q Ž 1r2 k . Ž Hy` ¨ 0 Ž x . < dx q Hy` w 0 Ž x . < dx .
Ž pe .
1r2
.
Taking t 0 s T0 s 1r9C 2 in Ž2.31., the estimate Ž2.20. 2 follows. The proof of Lemma 2.2 is complete
NONCONSERVATIVE HYPERBOLIC SYSTEM
375
Proof of Theorem 2.1. From the estimate Ž2.20. 2 it follows that, there exists a continuous function Ž ¨ , w . on D T 0 s.t. Ž ¨ n , wn . converges uniformly to Ž ¨ , w . on D T 0 . In order to prove Ž ¨ n , wn . converges uniformly on D T first we consider the relation for T0 F t F 2T0 , ¨ n Ž x, t . y ¨ ny1 Ž x, t .
s
1 4p Ž t y T 0 . e `
=
Hy`exp
y
`
t
HT Hy`
ž
1r2
yŽ x y y .
2
4 Ž t y T0 . e
/
¨ n Ž y, T0 . y ¨ ny1 Ž y, T0 . dy 2
exp Ž y Ž x y y . . r Ž 4 Ž t y T0 . y 5se .
ž
4p Ž t y T 0 y s . e
0
= l1 ny1 Ž y, s .
x
/
1r2
¨ n Ž y, s . y l1 ny2 Ž y, s .
x
¨ ny1 Ž y, s . dy ds,
Ž 2.32. and a similar expression for wn y wny1. Since 5 ¨ n Ž x, t . 5 L` F 5 ¨ 0 5 L` , 5 wn Ž x, t . 5 L` F 5 w 0 5 L` , `
Hy`
x
`
¨ n Ž x, t . dx F
H¨
X 0
Ž x . dx,
w Ž x, t . dx F wX0 Ž x . dx, x n following the same arguments which led to Ž2.20. 2 we get 5 ¨ n y ¨ ny1 5 L`Ž D T , 2T . q 5 wn y wny1 5 L`Ž D T , 2T .
Hy`
0
F
1 2
H
0
0
5 ¨ ny1 y ¨ ny2 5 L`Ž D ŽT
0 , 2T 0
.
0
q 5 wny1 y wny2 5 L`Ž D T 0 , 2T 0 .
q 32 5 ¨ n Ž x, T0 . y ¨ ny1 Ž x, T0 . 5 L`Ž R 1 . q5 wn Ž x, T0 . y Ž wny1 Ž x, T0 . . 5 L`Ž R 1 . , where we used the notation D T 0 , 2T 0 s Ž x, t .: y` - x - `, T0 F t F 2T0 4 . Iterating the above inequality and using the estimate Ž2.20. 2 we get 5 ¨ n y ¨ ny1 5 L`Ž D T , 2T 0 . q wn y wny1 5 L`Ž D T , 2T . 0
F 3 Ž n y 2. Ž q Ž 12 .
ny 2
0
1 2
.
ny 2
0
5 ¨ 2 y ¨ 1 5 L`Ž D T . q 5 w 2 y w 1 5 L`Ž D T . 0 0
5 ¨ 2 y ¨ 1 5 L`Ž D T
, . 0 2T 0
q 5 w 2 y w 1 5 L`Ž D T 0 , 2T 0 . .
376
K. T. JOSEPH
Now using the estimate Ž2.8. in the above inequality, we get 5 ¨ n y ¨ ny1 5 L`Ž D T F 8 n Ž 12 .
0
ny 2
, 2T 0 .
q 5 wn y wny1 5 L`Ž D T 0 , 2T 0 .
5 ¨ 0 < L` q 5 w 0 5 L` .
Ž 2.33.
It follows from Ž2.33., that for n ) m, 5 ¨ n y ¨ m 5 L`Ž D T
0
, 2T 0 .
q 5 wn y wm 5 L`Ž D T 0 , 2T 0 . n
F 8 Ž 5 ¨ 0 5 L` q 5 w 0 5 L` .
Ý j Ž 12 .
jy2
,
jsm
which implies that there exists Ž ¨ , w . g C 0 Ž D T 0 , 2T0 . such that as n ª `, Ž ¨ n , wn . ª Ž ¨ , w . uniformly on D T , 2T . This procedure can be repeated a 0 0 finite number of times, w2T0 , 3T0 x, w3T0 , 4T0 x, etc., so that we get finally Ž ¨ n ,wn . converges to a continuous function Ž ¨ , w . uniformly in D T for each T ) 0 fixed. However, because of the estimate Ž2.11., there exists a subsequence Ž ¨ n k , wn k . converges weakly in H 1 Ž D T . and strongly in L2 Ž D T . to function Ž ¨ , w . g H 1 Ž D T .. Since Ž ¨ n , wn . converges uniformly in D T to Ž ¨ , w ., Ž ¨ , w . s Ž ¨ , w . and the whole sequence converges weakly in H 1 Ž D T . and strongly in L2 and Ž ¨ , w . g H 1 Ž D T . l C 0 Ž D T .. Now Ž ¨ , w . is a weak solution follows easily by passing to the limit in T
`
H0 Hy` T
`
H0 Hy`
y¨ n ywn
f t f t
q q
ž ž
wny 1 y ¨ ny1 2k wny 1 y ¨ ny1 2k
qk yk
¨n
/ /
x
wn x
f dx dt s e
2f
Hx
f dx dt s e
2
2f
Hx
2
¨ dx dt,
w dx dt.
Also we have, for 0 F t F T, ¨ Ž x, t . s
1
`
Ž 4p t e . y
t
`
1r2
H0 Hy` =
y
Hy`exp
ž
yŽ x y y . 4 te
2
/
¨ 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
¨ Ž y, s . dy ds,
1r2
/
ž
w y¨ 2k
qk
/
377
NONCONSERVATIVE HYPERBOLIC SYSTEM
1
w Ž x, t . s
`
Ž 4p t e . t
y
1r2
`
H0 Hy` =
y
Hy`
exp
ž
yŽ x y y .
2
/
4 te
w 0 Ž y . dy
2
exp Ž y Ž x y y . . r Ž 4 Ž t y s . e .
ž
4p Ž t y s . e
1r2
/
ž
w y¨ 2k
yk
w Ž y, s . dy ds.
/
Ž 2.34.
By using the same arguments of the proof of estimate Ž2.20. 2 we can prove uniqueness for Ž ¨ , w . which satisfies the integral equation for a given ¨ 0 and w 0 in H 1 Ž R1 . l BV Ž R1 .. Also in a standard way we can prove Ž ¨ , w . is C` from the integral equation Ž2.34., provided ¨ 0 and w 0 are in S Ž R1 .. This fact also follows from the estimate Ž2.3. that we are going to prove. Next we prove the estimate Ž2.3.. Let Ž ¨ e , w e . be the solution of Ž2.1. and Ž2.2., with ¨ 0 and w 0 in S Ž R1 ., whose existence and uniqueness we have just shown. First we note that ¨ e Ž x, t . and w e Ž x, t . satisfy the estimates 5 ¨ e Ž x, t . 5 L` F 5 ¨ 0 < L` , `
Hy`
¨ e Ž x, t . x
dx F
`
Hy` ¨
X 0
5 w e Ž x, t . 5 L` F 5 w 0 5 L` , `
Ž x . dx,
Hy`
w e Ž x, t . x
F wX0 Ž x . dx,
so that in the original variables u e and ¨ e we have
5 u e 5 L` F
1
Ž 5 s0 5 L
`
k
q k 5 u 0 5 L` . ,
5 s e 5 L` F Ž 5 s 0 5 L` q k 5 u 0 5 . ,
H
ue x
H
Ž x, t . dx F
s e x
1 k
`
žH
y` `
Ž x, t . dx F H
y`
`
s 0X Ž x . dx q k
X 0
Hy` u Ž x .
s 0X Ž x . dx q k
X 0
H u Ž x.
/
dx ,
dx.
Ž 2.35.
From now on we work with the original unknown variables Ž u e , s e . and hence we consider equations Ž2.1.. Differentiating the first equation of
378
K. T. JOSEPH
Ž2.1. w.r.t. x, j times, multiplying it by j u er x j, and integrating by parts, we get 1 d 2 dt
2
j u e Ž x, t .
`
Hy`
dx q e
Hy`
xj `
sy
Hy`
q
1
xj
2
jq1 u e Ž x, t .
dx
x jq1
js e Ž x, t . jq1 u e Ž x, t .
dx
x jq1
j u e Ž x, t . jq1 u e Ž x, t . 2
`
Hy`
2
`
xj
x jq1
dx.
Ž 2.36.
Now recall the standard calculus inequality: For u, ¨ g H j Ž R1 . l L`Ž R1 ., 0 F s F j,
s Ž u¨ . xs
L2 Ž R 1 .
F C j 5 u 5 L`Ž R 1 .
s¨ xs
L2 Ž R 1 .
su
q 5 ¨ 5 L`Ž R 1 .
xs
L2 Ž R 1 .
,
where C j depends only on j. Using this we get, for 0 F s F j, `
Hy`
s u2 xs
2
dx F C j2 5 u 5 2L`
su xs
2 L2 Ž R 1 .
.
Ž 2.37.
Here and in what follows, C j denotes a constant which depends only on j, but need not be the same as the one used before. By Schwarz inequality applied on each of the terms of the RHS of Ž2.36. and then using Ž2.37., we get 1 d 2 dt
`
Hy` F
j u e Ž x, t .
2
dx q e
Hy`
xj
js e Ž x, t . xj
jq1 u e Ž x, t .
`
2
x jq1
dx
jq1 u e L2 Ž R 1 .
q C j 5 u e 5 L`Ž D T .
x jq1
L2 Ž R 1 .
jue Ž t . xj
jq1 u e Ž t . x jq1
L2 Ž R 1 .
L2 Ž R 1 .
.
Using ab F a2er4 q b 2re we obtain from above 1 d 2 dt
`
Hy` F
jue Ž t . xj
2 L2 Ž R 1 .
1
js e Ž t .
e
xj
q
L2 Ž R 1 .
e jq1 u e Ž t . 2
x jq1
q
C j2 5 u e 5 2L`Ž D T .
2
jue Ž t . xj
2 L2 Ž R 1 .
.
379
NONCONSERVATIVE HYPERBOLIC SYSTEM
Integrating this from 0 to t we get 2
jue Ž t . xj F
L2 Ž R 1 .
2
e
H0
q
e
qe
H0
x jq1
2
ds
L2 Ž R 1 .
2
js e Ž s .
t
2
jq1 u Ž s .
t
xj
L2 Ž R 1 .
t
C j2 5 u e 5 2L`Ž D T .
H0
ds 2
jue Ž s . xj
j u0
ds q
L2 Ž R 1 .
xj
2 L2 Ž R 1 .
. Ž 2.38.
From this we have t
H0
jq1 u e Ž s . x jq1
2 L2 Ž R 1 .
2
1 j u0 Ž x .
ds F
e
xj 2
q
e
q
L2 Ž R 1 .
2
t
e2
H0
js e Ž s . xj
jue Ž s .
C j2 5 u e 5 2L`Ž D T . 2
t
2C j2 5 u e 5 2L`Ž D T .
k
H0
xj
2 L2 Ž R 1 .
ds
2 L2 Ž R 1 .
ds.
Iterating this we get t
H0
jq1 u e Ž s . x jq1
2
ds F
j
1
e q q
ž
Ý ks0
j
2
Ý
e2
ž
e2
ks0
ž
/
x jyk
2C j2 5 u e 5 2L`Ž D T .
e2
C j2 5 u e 5 2L`Ž D T . 2
k
t
/H
/
t
H0 5 u
e
2 L2 Ž R 1 .
jyks e Ž s .
0
jq1
2
e
jyk u 0
x jyk
2 L2 Ž R 1 .
Ž s . 5 2L2 Ž R 1 . .
Hence for 0 - e F 1, we have t
H0
jq1 u e Ž s . x jq1 F
1
e
2Ž jq2.
2 L2 Ž R 1 .
ds
ž 1 q 2C
= 5 u 0 5 2H j q 2
j 2 5 e 52` L Ž DT . j u
/
t
H0 5 s Ž s . 5
2 Hj
ds q
t
H0 5 u Ž s . 5
2 L2 Ž R 1 .
ds . Ž 2.39.
380
K. T. JOSEPH
From Ž2.38. and Ž2.39. we get
jue Ž t . xj
2 L2 Ž R 1 .
ž 1 q 2C F
j 2 5 e 52` L Ž DT . j u
/
e 2 jq2 T
= 5 u 0 5 2H j q 2
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds . 0
Ž 2.40. By using the inequality sup f 2 Ž x . F x
1
df
2
2
dx
L2 Ž R 1 .
q 5 f 5 2L2 Ž R 1 . ,
with f Ž x . s Ž jy1 u e Ž x, t ..r x jy1, for each fixed t and using Ž2.40. with j and j replaced by j y 1, we get
sup
2
jy1 u e Ž x, t . x jy1
VT
ž 1 q 2C F
j 2 5 e 52` L Ž DT . j u
e
/
2 jq2
T
= 5 u 0 5 2H j q 2
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds . Ž 2.41. 0
Differentiating the second equation of Ž2.1. j times w.r.t. x, multiplying it with js er x j, integrating w.r.t. x, and integrating by parts, we get 1 d
js e Ž t .
2 dt
xj s yk 2 Fk
2
q
2 L2 Ž R 1 .
jq1s e Ž t .
qe
x jq1
j u e jq1s e
H x
j
x jq1
jue xj
dx q
2 L2 Ž R 1 .
jy1
Hx
u sx . jy1 Ž
jq1s e L2 Ž R 1 .
x jq1
jy1 Ž u esxe . x jy1
L2 Ž R 1 .
jq1s e L2 Ž R 1 .
x jq1
L2 Ž R 1 .
.
jq1 x jq1
s e dx
381
NONCONSERVATIVE HYPERBOLIC SYSTEM
As before, we get d
js e Ž t .
dt
xj F
2 L2 Ž R 1 .
2 k 4 jue
e
xj
2
jq1s e Ž t .
qe
x jq1
2 L2 Ž R 1 .
L2 Ž R 1 .
2 jy1 Ž u esxe .
q
e
xj
2 L2 Ž R 1 .
.
Ž 2.42.
Now we recall the calculus inequality,
jy1 x jy1
2
s
ž / u
x
jy1
L2 Ž R 1 .
F
C j2
Ý
ku
2
xk
L`Ž R 1 .
ks0
s
jyk 2
x
L2 Ž R 1 .
,
where C j depends only on j. Using Ž2.41. in the above inequality, we get, for 0 F t F T, 2
jy1 u e Ž t . s e Ž t . x
x jy1
L2 Ž R 1 .
C j2 1 q C j2 5 u e 5 2L`Ž D T .
ž
F
j
/
e 2 jq2 T
= 5 u 0 5 2H j q 2
H0 5 s
e
5 s e Ž t . 5 2H j T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds
Ž 2.43.
0
for suitable C j which depends only on j. Now integrating Ž2.42. from 0 to t and using Ž2.39. and Ž2.43. in the resulting expression we have, for 0 F t F T,
js e Ž t . xj F
2 L2 Ž R 1 .
qe
t
H0
jq1s e Ž s . x jq1
2 k 4 1 q 2C j2 5 u e 5 L`Ž D T .
ž
e
L2 Ž R 1 .
ds
j
/
e 2 jq2
= 5 u 0 5 2H j q 2
T
H0 5 s
q
2
C j2
T
H0 5 s
e
Ž s.
= 5 u 0 5 2H j q 2
5 2H j T
H0 5 s
e
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . ds 0
ž 1 q 2C ds
j 2 5 e 52` L Ž DT . j u
e
e
/
2 jq2
T
Ž s . 5 2H j ds q H 5 u e Ž s . 5 2L2 Ž R 1 . sds q 5 s 0 5 2H j 0
382
K. T. JOSEPH
F Žk q 4
ž 1 q 2C q 1.
C j2
j 2 5 e 52` L Ž DT . j u
e
/
2 jq3
= 1 q 5 u e 5 2H j q 5 s 0 5 2H j q T
= 1q2
H0
5 s e Ž s . 5 2H j q
T
H0 5 u T
žH
0
e
Ž s . 5 2L2 Ž R 1 . 2
5 s e Ž s . 5 2H j ds
/
.
Ž 2.44.
Thus we have, with A s Ž k 4 q C j2 q 1 . 1 q 2C j2 5 u e 5 2L`Ž D T .
ž
? 1 q 5 u 0 5 2H j q 5 s 0 5 2H j q
ž
T
H0
2
jq1s e Ž s .
ds F
x jq1
A
T
H0 5 u
1q
e 2 jq4
e
T
H0
j
/
Ž s . 5 2L2 Ž R 1 . ds ,
/
Ž 2.45.
2
5 s e Ž s . 5 2H j ds ,
so that T
H0
5 s e Ž t . 5 2H jq 1 F Ž j q 1 .
A
1q
e 2 jq4
T
H0
2
5 s e Ž s . 5 2H j ds .
From this we get 1q
T
H0
5 s e Ž s . 5 2H jq 1 ds F
2 Ž j q 1.
A 1q
e 2 jq4
T
H0
2
5 s Ž s . 5 2H j ds .
Iterating this we get
ž
1q
T
H0 5 s
F
ž
e
Ž s . 5 2H jq 1 ds
2 Ž j q 1. A
e 2 jq4
/ jq1
/ ž
1q
T
H0 5 s
2 Ž jq1 .
e
Ž s . 5 2L2 Ž R 1 . ds
/
.
Hence T
H0
Ž 5 s e Ž s . 5 2H jq 1 ds . F
ž
2 Ž j q 1. A
e 2 jq4
jq1
/ ž
1q
T
H0
2 Ž jq1 .
5 s e Ž s . 5 2L2 Ž R 1 . ds
/
.
Ž 2.46.
383
NONCONSERVATIVE HYPERBOLIC SYSTEM
Using Ž2.46. in Ž2.40. and Ž2.44., we get
jue Ž t . xj
2 L2 Ž R 1 .
F
e 2Ž jq1.Ž jq2.
= 1q 1q
ž
js e Ž t . xj
2 L2 Ž R 1 .
F
jq1
4 Ž j q 1. A
T
H0
2 Ž jq1 . 2
5 s e Ž s . 5 2L2 Ž R 1 . ds
/
,
2 Ž jq2 .
4 Ž j q 1. A
1q
e 4Ž jq1.Ž jq2.
T
H0
5 s e Ž s . 5 2L2 Ž R 1 . ds
4 Ž jq1 .
,
Ž 2.47. Where A is given by Ž2.45.. Now the RHS of Ž2.47. contains T
H0 5 u
e
Ž s . 5 2L2 Ž R 1 . dx and
T
H0 5 s
e
Ž s . 5 2L2 Ž R 1 . ds.
To estimate these we estimate first the L1-norm of u e Ž x, t . and s e Ž x, t . w.r.t. x. As in the proof of the first part of Lemma 2.2, we write `
Hy` u
e
i Ž x, t . dx s Ý Ž y1. H
y iq1Ž t . e
u Ž y, t . dy,
yiŽ t .
i
where yi Ž t .: i s 0, " 1, " 2, . . . .4 are points x, where u e Ž x, t . changes sign. We call the ordering such that on Ž y 0 Ž t ., y 1Ž t .., u e Ž x, t . ) 0. Now as in that proof, we get from the first equation of Ž2.1., d
`
H u dt y`
e
i Ž x, t . dx F Ý Ž y1. H
y iq1Ž t .
y
yiŽ t .
i
s e
`
Ž y, t . dy F H
y`
sxe Ž x, t . dx.
Similarly from the second equation of Ž2.1., we get d
`
H s dt y`
e
Ž x, t . dx F 5 u e Ž x, t . 5 L`Ž V T . `
Hy` s
=
x
e
`
Ž x, t . dx q k 2H
y`
u ex Ž x, t . dx.
384
K. T. JOSEPH
Using the estimates Ž2.35. in the above inequalities and integrating we get, 0 F t F T, `
Hy` u
e
`
Ž x, t . dx F H
q `
Hy` s
u 0 Ž x . dx
y`
e
T
`
Hy` s
k
`
`
Ž x . dx q kH
y`
s 0 Ž x . dx q
Ž x, t . dx F H
y`
=
X 0
`
Hy` s
X 0
T
uX0 Ž x . dx ,
5 s 0 5 L` q k 5 u 0 5 L` q k 2
k
`
Ž x . dx q kH
y`
uX0 Ž x . dx
and hence `
Hy` u
e
Ž x, t .
2
dx F 5 u e Ž x, t . 5 Le 5 u e Ž x, t . 5 L1 Ž R 1 . F
1
5 s 0 5 L` q k 5 u 0 5 L`
k
= 5 u 0 5 L1 q
T k
ž
`
Hy` s
X 0
dx q k
`
X 0
Hy` u
dx
/ Ž 2.48.
and `
Hy` s
e
Ž x, t .
2
dx F
½
5 s 0 5 L` q k 5 u 0 5 Le 5 s 0 5 L1 q
=
T k
k 2 q 5 s0 5 L
`
`
Hy` s
X 0
q k 5 u 0 5 L` 4 `
Ž x . dx q kH
y`
5
uX0 Ž x . dx .
Ž 2.49.
Using Ž2.48. and Ž2.49. in Ž2.47., we get for j s 0, 1, 2, . . . , sup 0FtFT
sup 0FtFT
jue Ž t . xj js e Ž t . xj
2 L2 Ž R 1 .
F
2 L2 Ž R 1 .
F
CT2
e 4Ž jq1.Ž jq2. CT2
e 4Ž jq1.Ž jq2.
,
,
Ž 2.50.
385
NONCONSERVATIVE HYPERBOLIC SYSTEM
where CT is a polynomial of L1 norm, L` norm, BV norm, and H j norm of u 0 Ž x . and s 0 Ž x . and T, whose coefficients depends only on j. Now using sup
X
xgR 0FtFT
ž
jy1 u e Ž x, t . x jy1
2
`
F sup
/
0FtFT
Hy`
ž
/ /
x jy1
`
q
2
jy1 u e Ž x, t .
Hy`
j u e Ž x, t .
ž
xj
dx 2
dx
and the same inequality with u replaced by s , we get, for j s 1, 2, . . . , sup
jy1 u e Ž x, t . x
xgR 1 0FtFT
sup
jy1
jy1s e Ž x, t . x jy1
xgR 1 0FtFT
F
F
e
CT 2Ž jq2.Ž jq1. CT
e 2Ž jq2.Ž jq1.
,
,
Ž 2.51.
where CT is a polynomial of L1 norm, L` norm, BV norm, and H j norm of u 0 Ž x . and s 0 Ž x . and T, whose coefficients depends only on j. This gives the estimate Ž2.3. for l s 0, j s 0, 1, 2, . . . . Now applying the differential operator lqjr t l x j on both sides of the equation Ž2.1., we get
lqjq1 u e t lq1 x j lqjq1s e
se se
lqjq2 u e t l x jq2 lqjq2s e
y y
1 lqjq1 Ž u e . 2
2
t l jq1
lqj Ž u e Ž s
e
r x .
q
.
lqjq1s e t l x jq1
q k2
,
lqjq1 u e
. Ž 2.52. t lq1 x j t l x jq2 tl x j t l x jq1 When l s 0, the RHS of Ž2.52. does not contain any t derivatives. Hence jq1 u er t x j and jq1s er t x j can be estimated using Ž2.1. in Ž2.52.. Thus we get Ž2.3. when l s 1, j s 0, 1, 2, . . . . When l s 1, the RHS of Ž2.52. can be estimated using Ž2.3. with l s 1 and j s 0, 1, 2, . . . . This gives Ž2.3. for l s 2, j s 0, 1, 2, . . . . Proceeding inductively, we get the estimate Ž2.3.. The proof of Theorem 2.1 is complete.
3. GENERALIZED SOLUTIONS FOR Ž0.1. AND Ž0.2. In this section we solve the problem u t q uu x y sx f 0,
st q u sx y k 2 u x f 0,
Ž 3.1.
386
K. T. JOSEPH
with initial conditions u Ž x, 0 . s u 0 ,
s Ž x, 0 . s s 0 .
Ž 3.2.
We assume u 0 s Ž u 0e . 0 - t F 1 and s 0 s Ž s 0e . 0 - e F 1 are in G Ž R1 ., the algebra of the generalized function of Colombeau, and u 0e and s 0e satisfy the estimate, for j s 1, 2, . . . , C2 C1 j 5 u 0e 5 H j F l , 5 f 0e 5 H j F m j , e j e j du 0e Ž x .
H
dx
Hu
e 0
F
Ž x . dx F
C
e
p
C
d s 0e Ž x .
,
H
,
Hs
dx e 0
F
C
ep
,
C
Ž x . dx F
, Ž 3.3. ep for some constants C1 j , C2 j , C and for some integers l j , m j , p. Here we remark that all tempered distributions can be represented by Ž3.3., i.e., for all T g S Ž R1 . there exists u 0e satisfying Ž3.3. with Ž u 0e . f T. With these assumptions, we shall prove the following theorem.
e
p
THEOREM 3.1. Let Ž u, s . s Ž u e , s e . 0 - e - 1 be the solution of Ž2.1. and Ž2.2. with initial data Ž u 0 , s 0 . s Ž u 0e , s 0e .. Then Ž u, s . g G Ž V T . for each fixed T ) 0 and is a solution of Ž3.1. and Ž3.2. as e ª 0. Proof. From the assumptions Ž3.3. on Ž u 0 , s 0 . and the estimate Ž2.3. and the fact that the constant C appearing in Ž2.3. depends polynomially on H j norms, L1 norms, L` norms, and BV norms of initial data Ž u 0e , s 0e ., we easily get Ž u, s . g G Ž V T .. Now to show that Ž u, s . satisfy Ž3.1. we note that
HH Ž u HH Ž s
t
e
e t
q u e u ex y sxe . f dx dt s e
HHf
q u esxe y k 2 u ex . f dx dt s e
HHf
xxu
e
xxs
e
dx dt, dx dt,
for any f Ž x, t . g C0`Ž R1 = Ž0, T ... Now the RHS of the above equations goes to zero as e ª 0, since u e and s e are bounded. The proof of the theorem is complete. ACKNOWLEDGMENTS The author is grateful to Professor J. F. Colombeau for introducing to him the new theory of generalized functions and the problem discussed in this paper. His constant encouragement and valuable suggestions during the course of this work is gratefully acknowledged. The author is also thankful to the referees for their constructive criticism, which improved the presentation of an early version of the paper.
NONCONSERVATIVE HYPERBOLIC SYSTEM
387
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