Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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508 16. Nonlinear Hyperbolic Equations

4. Show that if there is an entropy-ﬂux pair .; q/ such that  is strongly convex, then the

positive-deﬁnite, L  L matrix .@

=@u

/ is a symmetrizer for (8.56).

In Exercises 5–7,letu

D .v

/ be smooth solutions to

(8.58)

 @

D 0;

 @

K.v

/ D "@

;

for t  0. Assume ">0.

5. If either x 2 S

or the functions in (8.58) decrease fast enough as jxj!1, show that

E.t/ D

w.t;x/

C K



v.t;x/



satisﬁes

D"

.t; x/

dx:

6. If .; q/ is an entropy-ﬂux pair for

(8.59)

 w

D 0;

 K.v/

D 0;

show that

.u

/ C @

q.u

/ D "

@

(8.60)

@

/ D 

and  is convex (which holds for .; q/ given by (8.25)), deduce that

.u

/ C @

q.u

/  "@

.w

7. Now suppose that, as " & 0; u

converges boundedly to u D .v; w/, a weak solution

to (8.59). If .; q/ is an entropy-ﬂux pair and (8.60) holds, with  convex, deduce that

(8.61) .u/

C q.u/

 0;

in the same sense as (8.17). Taking .; q/ as in (8.25), deduce that if u has a jump across

,asin(8.23), then

K.v

/.v

 v

/ 

K./ d 

 w

;

K.v

/.v

 v

/ 

K./ d C

 w

9. Global weak solutions of some 2  2 systems 509

9. Global weak solutions of some 2  2 systems

Here we establish existence, for all t  0, of entropy-satisfying weak solutions to

aclassof2  2 systems of conservation laws in one space variable:

(9.1) u

C F.u/

D 0; u.0; x/ D f.x/:

We will take x 2 S

D R=Z; modiﬁcations for x 2 R are not difﬁcult. We

assume f takes values in a certain convex open set   R

and F W  ! R

is smooth. As before, we set A.u/ D DF .u/,a2  2 matrix-valued function

of u. We assume strict hyperbolicity; namely, A.u/ has real, distinct eigenvalues



.u/<

.u/, with associated eigenvectors r

.u/; r

.u/. We will assume that

 has a global coordinate system .

;

/,where

2 C

./ is a j -Riemann

invariant. In fact, we assume that  maps  diffeomorphically onto a region

R Df W A

<

where 1  A

C1. The assumptions stated in this paragraph will be

called the “standard hypotheses” on (9.1).

We will obtain a solution to (9.1) as a limit of solutions to

(9.2) @

C @

F.u

/ D "@

; u

.0; x/ D f.x/:

Methods of Chap. 15,  1 (particularly Proposition 1.3 there), yield, for any ">0,

a solution u

.t/, deﬁned for 0  t<T."/, given any f 2 L

/, taking values

in a compact subset of . The solution is C

on .0; T ."// S

and continues as

long as we have

(9.3) u

.t; x/ 2 K;

for some compact K  . For now we make the hypothesis that (9.3) holds, for

all t  0. We also have the identity

(9.4) ku

.t/k

C "

.s/k

ds Dkf k

To study the behavior of the solutions u

to (9.2)as" ! 0,weusethethe-

ory of Young measures, developed in  11 of Chap. 13. By Proposition 11.3 of

Chap. 13, there exists a sequence u

D u

, with "

! 0, and an element

.u;/ 2 Y

 S

/ such that

(9.5) u

! .u;/ in Y

 S

510 16. Nonlinear Hyperbolic Equations

By Proposition 11.1 and Corollary 11.2 of Chap. 13,

(9.6) F.u

/ ! F weak



in L

 S

where

(9.7)

F.x/ D

F.y/ d

t;x

.y/; a.e. .t; x/ 2 R

 S

Since "@

! 0 in D

 S

/, this implies

(9.8) @

u C @

F D 0:

To conclude that u is a weak solution to (9.1), we need to show that

F D F.u/,

which will follow if we can show that the convergence u

! .u;/in Y



/ is sharp (i.e.,  D 

), or equivalently, that 

t;x

is a point mass on R

,for

almost every .t; x/ 2 R

 S

Following [DiP4] we use entropy-ﬂux pairs as a tool for examining ,ina

chain of reasoning parallel to, but somewhat more elaborate than, that used to

treat the scalar case in  11 of Chap. 13.

For any smooth entropy-ﬂux pair .; q/,wehave

(9.9) @

.u

/ C @

q.u

/ D "@

.u

/  "@



;

where 

/ is the 2  2 Hessian matrix of second-order partial derivatives of .

We have the identity

(9.10) "

 

dx dt D





f.x/



dx 





.T; x/



dx:

We rewrite (9.9)as

(9.11) @

.u

/ C @

q.u

/ D "@

.u

/  R

;

with

(9.12) R

bounded in L

 S

If  is convex, this follows directly from (9.10), since then the left side of (9.10)

is the integral of a positive quantity. But even if  is not assumed to be convex, we

can appeal to (9.4)tosay

is bounded in L

S

/, and this plus (9.3)

implies (9.12).

Since @

.u

/ D 

, we also deduce from (9.3)–(9.4) that the quantity

.u

/ is bounded in L

 S

/. Hence

(9.13) "@

.u

/ ! 0 in H

1

 S

/; as " ! 0:

9. Global weak solutions of some 2  2 systems 511

Now we can apply Lemma 12.6 of Chap. 13 (Murat’s lemma) to deduce from

(9.11)–(9.13)that

(9.14) @

.u

/ C @

q.u

/ is precompact in H

1

loc

 S

Now, let .

/ and .

/ be any two entropy-ﬂux pairs, and consider the

vector-valued functions

(9.15) v





/; q





/; 



;

where u

is as in (9.5). By (9.14), we have

(9.16) div v

; rot w

precompact in H

1

loc

 S

Also the L

bound on u

implies that v

and w

are bounded in L

S

and a fortiori in L

loc

 S

/. Therefore, we can apply the div-curl lemma,

either in the form developed in the exercises after  8 of Chap.5 or in the form

developed in the exercises after  6 of Chap. 13. We have

(9.17) v

 w

! v  w in D

 S

/; v D .

; q

/; w D .q

; 

In view of the L

-bounds, we hence have

(9.18)



/  

/ ! 

 

weak



in L

 S

Recall that we want to show that any measure  D 

t;x

, arising in the disin-

tegration of the measure  in (9.5), is supported at a point. We are assuming that

there are global coordinates .

;

/ on  consisting of Riemann invariants. Let

(9.19) R Df W a



 

 a

be a minimal rectangle (in -coordinates) containing the support of . The follow-

ing provides the key technical result:

Lemma 9.1. If a



, then each closed vertical side of R must contain a point

where @

=@

D 0.

Proof. We have from (9.18)that

(9.20) h; 

 

iDh; 

ih; q

ih; 

ih; q

for all entropy-ﬂux pairs .

/.Let



.k/; q.k/



be a family of entropy-ﬂux

pairs of the form (8.39), with k 2 R; jkj large, so that .k/ > 0. Thus, for jkj

large, we can deﬁne a probability measure 

512 16. Nonlinear Hyperbolic Equations

(9.21) h

;fiD

h; .k/f i

h; .k/i

We can take a subsequence k

!C1such that

(9.22) 

! 

;

k

! 



; weak



in M./:

In view of the exponential factor e

k

in .k/, it is clear that

(9.23) supp 

 R \f

D a

Now set 

Dh

;

i. We claim that

(9.24) h; q  

iDh

;q 

i;

for every entropy-ﬂux pair .; q/.Toestablishthis,use(9.20) with .

/ D

.; q/ and .

/ D



.k/; q.k/



.Weget

(9.25)

h; q.k/ .k/qi

h; .k/i

Dh; i

h; q.k/i

h; .k/i

h; qi:

Since, by (8.43), q

D 



in the expansion (8.39), we have

(9.26)

h; q.k/i

h; .k/i

h; 

.k/i

h; .k/i

C O.k

1

/ Dh

;

iCO.k

1

Now, letting k D˙k

and passing to the limit yield h

;

iD

for (9.26).

Similarly,

(9.27)

h; q.k/i

h; .k/i

!h

;

i;

so (9.25) yields (9.24) in the limit.

Now, use (9.20) with .

/ D



.k/; q.k/



;.

/ D



.k/; q.k/



Thus

(9.28)

h; .k/q.k/  .k/q.k/i

h; .k/ih; .k/i

h; q.k/i

h; .k/i



h; q.k/i

h; .k/i

The right side converges to 



 

as k D k

!C1. Meanwhile, note that

.k/q.k/  .k/q.k/ D O.k

1

/.Alsoh; .k/ih; .k/i!C1,faster

than e

k.a

a



"/

, by the deﬁnition of R,ifa



. Thus the left side of (9.28)

tends to zero. We deduce that

(9.29) 

D 



9. Global weak solutions of some 2  2 systems 513

The identities (9.24)and(9.29) imply that

(9.30) h

;q 

iDh



;q 

i;

for every entropy-ﬂux pair .; q/. Now with .; q/ D



.k/; q.k/



,wehave

(9.31) h

;q 

iDe



;.q

 



1

C O.k

2

;

where 

1

and q

1

are the second terms in the expansion (8.39). If a



the identity of these two expressions forces h

 



iD0.By(8.53), this

implies

(9.32)



;

@

@

D 0:

Since 

are probability measures and 

>0, this forces @

=@

to change

sign on supp 

, proving the lemma.

Corollary 9.2. If (9.1) is genuinely nonlinear, so @

=@

and @

=@

are both

nowhere vanishing, then  is supported at a point.

We therefore have the following result:

Theorem 9.3. Assume that (9.1) satisﬁes the standard hypotheses and that so-

lutions u

to (9.2) satisfy (9.3). If (9.1) is genuinely nonlinear, then there is a

sequence u

! u, converging boundedly and pointwise a.e., such that u solves

(9.1). Also, u satisﬁes the entropy inequality @

.u/ C @

q.u/  0, for every

entropy-ﬂux pair .; q/ such that  is convex (on a neighborhood of K).

Certain cases of (9.1) that satisfy the standard hypotheses but for which gen-

uine nonlinearity fails, not everywhere on , but just on a curve, are amenable to

treatment via the following extension of Lemma 9.1:

Lemma 9.4. If both characteristic ﬁelds of (9.1) are genuinely nonlinear outside

acurve

D .

/, with strictly monotone, then  is supported at a point.

Proof. By Lemma 9.1, each closed side of the rectangle R must intersect this

curve, so it must go through a pair of opposite vertices of R; call them P and Q.

By (9.32), we see that 

and 



must be supported at these points. Thus (9.24)

and (9.29) imply that

(9.33) q.Q/  

.Q/.Q/ D q.P/  

.P /.P /:

We have the same sort of identity with 

replaced by 

,so

(9.34)





.Q/  

.Q/



.Q/ D





.P /  

.P /



.P/;

514 16. Nonlinear Hyperbolic Equations

for every entropy . In particular, we can take .u/ D u

 Q

;jD 1; 2,to

deduce from (9.34)thatP D Q, since the strict hyperbolicity hypothesis implies



.P /  

.P / ¤ 0. This implies R is a point, so the lemma is proved.

For an example of when this applies, consider the system (7.14), namely,

(9.35)

 w

D 0;

 K.v/

D 0;

which, by (7.16), is strictly hyperbolic provided K

.v/ ¤ 0.By(7.55),

(9.36) r

r

D˙

1=2

;

so we have genuine nonlinearity provided K

.v/ ¤ 0. However, in cases

modeling the transverse vibrations of a string, by (7.12), we might have, for

example,

(9.37) K.v/ D v C av

;

for some positive constant a.ThenK

.v/ D 1 C 3av

>0,butK

.v/ D 6av

vanishes, at v D 0. In this case, Riemann invariants are given by (8.30), that is,

(9.38) 

D w ˙

.s/ ds;

so genuine nonlinearity fails on the line 

D 



(i.e., 

D 

). Thus Lemma 9.4

applies in this case.

To make use of Theorem 9.3 and the analogous consequence of Lemma 9.4,

we need to verify (9.3). The following result of [CCS] is sometimes useful for

this:

Proposition 9.5. Let

O    R

be a compact, convex region whose boundary

consists of a ﬁnite number of level curves 

of Riemann invariants, 

, such that

r

points away from O on 

; more precisely,

(9.39) .u  y/ r

.u/>0; for u 2 

;y2 O:

If f 2 L

/ and f.x/ 2 K  O for all x 2 S

, then, for any ">0,the

solution to

(9.40) @

C @

F.u

/ D "@

; u

.0; x/ D f.x/

exists on Œ0; 1/  S

, and u

.t; x/ 2 O.

9. Global weak solutions of some 2  2 systems 515

Proof. We remark that it sufﬁces to prove the result under the further hypothesis

that f 2 C

/.First,foranyı>0, consider

(9.41) @

"ı

C @

F.u

"ı

/ D "@

"ı

 ır.u

"ı

/; u

"ı

.0; x/ D f.x/;

where we pick y 2 O and take .u/ Dju  yj

. This has a unique local solution.

If we show that u

"ı

.t; x/ 2 O,forall.t; x/, then it has a solution on Œ0; 1/ S

If it is not true that u

"ı

.t; x/ 2 O for all .t; x/, there is a ﬁrst t

>0such that,

for some x

2 S

; u.t

/ 2 @O.Sayu.t

/ lies on the level curve 

.Take

the dot product of (9.41) with r

"ı

/,toget(via(8.32))

(9.42)



"ı

/ C 

"ı



"ı

D "@



"ı

/  ".@

"ı

/  

"ı

 ır

"ı

/ r.u

"ı

Our geometrical hypothesis on O implies

(9.43) .@

"ı

/  

"ı

 0 and r

"ı

/ r.u

"ı

/>0;

at .t

/. Meanwhile, the characterization of .t

/ implies

(9.44) @



"ı

/ D 0; and @



"ı

/  0;

at .t

/. Plugging (9.43)–(9.44)into(9.42) yields @



"ı

/<0at .t

/,an

impossibility. Thus u

"ı

2 O for all .t; x/ 2 Œ0; 1/ S

Now, if (9.40) has a solution on Œ0; T /  S

, analysis of the nonhomogeneous

linear parabolic equation satisﬁed by w

"ı

D u

 u

"ı

shows that u

"ı

! u

Œ0; T / S

,ası ! 0, so it follows that u

.t; x/ 2 O, and hence that (9.40)hasa

global solution, as asserted.

As an example of a case to which Proposition 9.5 applies, consider the system

(9.35), with K.v/ given by (9.37), modeling transverse vibrations of a string.

There are arbitrarily large, invariant regions O in  D R

of the form depicted in

Fig. 9.1. Here, @O D 

[ 

, as depicted, and we take

(9.45)



D w ˙

.s/ ds;



D 

on 

;

D

on 

;



D 



on 

;

D



on 

Another example, with  Df.v; w/ W 0<v<1g, is depicted in Fig. 9.2.

This applies also to the system (9.35), but with K.v/ given by (7.60). It models

longitudinal waves in a string. In this case, there are invariant regions of the form

O containing arbitrary compact subsets of .

516 16. Nonlinear Hyperbolic Equations

FIGURE 9.1 Setting for Proposition 9.5

Since we have seen that Lemma 9.4 applies in these cases, it follows that

the conclusion of Theorem 9.3, that is, the existence of global entropy-satisfying

solutions, holds, given initial data with range in any compact subset of .

Exercises

1. As one speciﬁc way to end the proof of Proposition 9.5, show that w

"ı

D u

"ı

 u

"

satisﬁes

"ı

C @

"ı

/ D "@

"ı

 ır.u

"ı

/ C r.u

"

/; w

"ı

.0; x/ D 0;

FIGURE 9.2 Another Setting for Proposition 9.5

10. Vibrating strings revisited 517

where G

"ı



"ı

C .1  s/u

"



ds. Deduce that there exists K<1 such

that, for "; ı;  2 .0; 1,

"ı

.t/k



"ı

.t/k

C K.ı C /;

granted that u

"ı

.t; x/; u

"

.t; x/ 2 O,forall.t; x/. Use Gronwall’s inequality to es-

timate kw

"ı

.t/k

, showing that, for ﬁxed " 2 .0; 1, it tends to zero as ı;  ! 0,

locally uniformly in t 2 Œ0; 1/. Use this to show that u

"ı

! u

,ası ! 0.

10. Vibrating strings revisited

As we have mentioned, the equation for a string vibrating in R

was derived in

 1 of Chap. 2, from an action integral of the form

(10.1) J.u/ D

“

I 

.t; x/

 F



.t; x/



dx dt;

where x 2  D Œ0; L; t 2 I D .t

/. Assume that the mass density m is a

positive constant. The stationary condition is

(10.2) mu

 F

D 0;

which is a second-order, k  k system. If we set

(10.3) v D u

;wD u

;

we get a ﬁrst-order, .2k/  .2k/ system:

(10.4)

 w

D 0;

 K.v/

D 0;

where

(10.5) K.v/ D

.v/:

Let us assume that F.u

/ is a function of ju

alone:

(10.6) F.u

/ D f.ju

Then K.v/ has the form

(10.7) K.v/ D

.jvj

/v: