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

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518 16. Nonlinear Hyperbolic Equations

We can write (10.4) in quasi-linear form as

(10.8)







DK.v/ 0





;

where, for b 2 R

(10.9) DK.v/b D

.jvj

/b C

.jvj

/.v  b/v;

that is,

(10.10) DK .v / D

.jvj

/I C

.jvj

/jvj

;

being the orthogonal projection of R

onto the line spanned by v (if v ¤ 0).

Writing (10.8)as

(10.11) U

 A.U /U

D 0;

for U D .v; w/

, we see that the eigenvalues of A.U / are given by

(10.12) Spec A.u/ Df˙



W 

2 Spec DK.v /g:

Now, if k D 1; DK.v/ is scalar:

(10.13) DK .v / D

.v/:

As long as F

.v/ > 0, the system (10.8) is strictly hyperbolic, with characteristic

speeds



D˙

.v/:

In this case, the system (10.4) describes longitudinal vibrations of a string. The

Riemann problem for this system was considered in (7.55)–(7.60), and DiPerna’s

global existence theorem was applied in the discussion of Fig. 9.2.

On the other hand, if k>1,then

(10.14) Spec DK.v / D

.jvj

/ C

.jvj

/jvj

;

where the ﬁrst listed eigenvalue has multiplicity k  1 and the last one has mul-

tiplicity 1. We can rewrite these eigenvalues, using the notion of “tension” T.r/,

deﬁned so that

(10.15) F

.v/ D T.jvj/

jvj

;

10. Vibrating strings revisited 519

FIGURE 10.1 Graph of f

FIGURE 10.2 Graph of T

that is, so 2f

.jvj

/ D T.jvj/=jvj. A calculation gives

.jvj

/ C 4f

.jvj

/jvj

D T

.jvj/;

(10.16) Spec DK.v / D

(



jvj



jvj

;



jvj



)

The basic expected behavior of the function f.r/is discussed in  7, around the

formula (7.60). The function f.r/ should be expected to behave as in Fig. 10.1.

For r larger than a certain a; f .r/ should increase. On the other hand, also f.r/

should get very large as r & 0, since the material of the string would resist

compression. This means for the tension T.r/ that T.r/ > 0 for r>aand

T.r/ <0for 0<r<a. On the other hand, we expect T.r/to increase whenever

r increases, so T

.r/ > 0 for all r. Such behavior of the tension is depicted in

Fig. 10.2.

520 16. Nonlinear Hyperbolic Equations

We conclude that when jvj >a,then

(10.17) Spec A.U / D

(

T.jvj/

jvj

; ˙



jvj



)

consists of real numbers. If k D 2, we have four eigenvalues. These are all distinct

as long as f

.jvj

/ ¤ 0, that is, as long as the function f.r/ is strongly convex.

If this convexity fails somewhere on jvj >a, then the system (10.4) will not be

strictly hyperbolic, but it will be symmetrizable hyperbolic, as long as jvj >a.

On the other hand, when jvj <a, then Spec A.U /, which is still given by

(10.17), has two purely imaginary elements, as well as two real elements (the

former being eigenvalues with multiplicity k  1). Thus (10.4) is not hyperbolic

in the region jvj <a.

Let us concentrate for now on the region jvj >a,where(10.4) is hyperbolic,

and examine whether it is genuinely nonlinear. We consider the case k D 2.Let

us denote the two eigenvalues of DK.v/ givenby(10.16)by

.v/:

(10.18) 

.v/ D

T.jvj/

jvj

;

.v/ D



jvj



Thus

(10.19)

.jvj

/>0H) 

.v/ > 

.v/;

.jvj

/<0H) 

.v/ < 

.v/:

From (10.10) we see that we can take as eigenvectors of DK.v /

(10.20) r

.v/ D v; r

.v/ D Jv;

where J W R

! R

is counterclockwise rotation by 90

. It follows that eigen-

vectors of A.U / corresponding to the eigenvalues

(10.21) 

j ˙

D˙



.v/

are given by

(10.22) r

j ˙



.v/



j ˙

.v/



10. Vibrating strings revisited 521

Thus

(10.23) r

j ˙

r

j ˙

D˙r

.v/ r



.v/ D˙



.v/

.v/ r

.v/:

Now, by (10.20), if we use polar coordinates .r; / on R

, with r

D v

then

(10.24) R

D r

@

;

(10.25) r

r

.v/ D 0; r

r

.v/ D

.jvj/jvj:

Thus we see that within the hyperbolic region jvj >a,(10.4) has two linearly de-

generate ﬁelds and two ﬁelds that are genuinely nonlinear as long as T

.jvj/ ¤ 0.

We now describe an interesting complication that arises in the treatment of

the system (10.4)whenk  2. Namely, even if initial data lie entirely in the

hyperbolic region, the solution might not stay in the hyperbolic region. Consider

the following simple example, with k D 2.Weletv.0;x/;w.0;x/be initial data

for a purely longitudinal wave, so that the motion of the string is conﬁned to the

-axis. Thus, we take

(10.26) v.0; x/ D



.0; x/



;w.0;x/D



.0; x/



For all t  0, the solution will have the form

(10.27) v.t; x/ D



.t; x/



;w.t;x/D



.t; x/



;

where the pair .v

/ satisﬁes the k D 1 case of (10.4).

Suppose K.v/ is given by (10.7)andT.jvj/ D 2f



jvj



jvj has the behavior

indicated in Fig. 10.2; also, let us assume f is real analytic on .0; 1/. Introduce

another variable ,andlet



.t;x;/;w

.t;x;/



solve the k D 1 case of (10.4),

with initial data

(10.28)

.0;x;/ D a C ;

.0;x;/ Db sin x;

where b is some positive constant. By the Cauchy–Kowalewsky theorem, there is

a T>0such that there is a unique, real-analytic solution deﬁned for jtj <T,for

all x 2 R (periodic of period 2), and for all jja=2. Note that

(10.29) @

.0;0;/Db:

522 16. Nonlinear Hyperbolic Equations

It follows easily from the implicit function theorem that, for all >0sufﬁciently

small, there exists t./ 2 .0; T / such that

(10.30) v



t./;0;



<a:

This is a well-behaved solution to the longitudinal wave problem, but the solution

so produced to the k D 2 case of (10.4), having the form (10.27), clearly has the

property that

(10.31)



v.t./;0;/;w.t./;0;/





.t./; 0; /; 0Iw

.t./; 0; /; 0



does not belong to the hyperbolic region, despite the fact that the initial data do.

Note that, for the system under consideration, solutions to the Riemann prob-

lem for .v

/ have the behavior discussed in  7, illustrated by Fig. 7.4 there.

For example, the situation illustrated in Fig. 10.3 can arise. Here, we have Rie-

mann data

(10.32) U

D .v

>a; v

>a;

but the intermediate state U

has the form

(10.33) U

D .v

<a:

This is also a well-behaved solution to the longitudinal wave problem, but the

solution so produced to the k D 2 case of (10.4) is the following. We have the

Riemann problem

(10.34) U

D .v

;0Iw

;0/

;0Iw

;0/

;

FIGURE 10.3 Connecting U

to U

10. Vibrating strings revisited 523

and a weak solution to (10.4), involving two jumps, with intermediate state

(10.35) U

D .v

;0Iw

;0/

Clearly, U

does not belong to the hyperbolic region for the k D 2 case of (10.4),

even though U

and U

do belong to the hyperbolic region.

M. Shearer, [Sh1, Sh2](seealso[CRS]), has proposed a method for solv-

ing Riemann problems for a class of systems, including the system for vibrating

strings considered here, which leaves the hyperbolic region invariant. The solu-

tion produced by this method to the Riemann problem described in the preceding

paragraph has an intermediate state U

different from the one described above.

The situation is depicted in Fig. 10.4.HereU

D .v

/ with v

<0(in

fact, v

< a). The physical interpretation is that the string develops a kink and

doubles back on itself. For motion strictly conﬁned to one dimension, this would

not be allowable, as it involves interpenetration of different string segments. If

the string has two or more dimensions in which to move, one thinks of these seg-

ments as lying side by side; however, one does not ask which segment lies on

which side; for example, for k D 2, one does not ask whether the conﬁguration is

as in Fig. 10.5AorasinFig.10.5B.

Of course, there is only one real-analytic solution with the initial data (10.26)–

(10.28); there is no real-analytic modiﬁcation that stays in the hyperbolic region.

FIGURE 10.4 More Complex Connection

524 16. Nonlinear Hyperbolic Equations

FIGURE 10.5 Crinkled Strings

In [PS] there is a study of the behavior of approximations of such solutions via

Glimm’s scheme. It is found that typically these approximations do not converge,

even weakly, to the smooth solution.

Further material on systems that change type can be found in [KS]. Also, pa-

pers in [KK3] deal with systems for which strict hyperbolicity can fail, as can

happen in the hyperbolic region for (10.4)iff

.r/ D 0 for some r>a

Exercises

1. Work out the equations for radially symmetric vibrations of a two-dimensional mem-

brane in R

. Perform an analysis parallel to that done in this section for the vibrating

string system.

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