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

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428 16. Nonlinear Hyperbolic Equations

We now introduce a more general notion of symmetrizer, following Lax [L1],

which will bring in pseudodifferential operators. We will say that a function

R.t; u;x;/, smooth on R  R

 T



M n 0, homogeneous of degree 0 in ,

is a symmetrizer for (2.10) provided

(2.16) R.t; u;x;/is a positive-deﬁnite, K  K matrix

and

(2.17) R.t; u;x;/

.t; x; u/

is self-adjoint;

for each .t; u;x;/.Wethensay(2.10) is symmetrizable. One reason for the im-

portance of this notion is the following:

Proposition 2.2. Whenever (2.10) is strictly hyperbolic, it is symmetrizable.

Proof. If we denote the eigenvalues of L.t; u;x;/ D

.t; x; u/



.t; u;x;/ <<

.t; u;x;/;

then the 



are well-deﬁned C

-functions of .t; u;x;/, homogeneous of degree

1 in .IfP



.t; u;x;/are the projections onto the 



-eigenspaces of L,

(2.18) P



2i





  L.t; u;x;/



1

d;

then P



is smooth and homogeneous of degree 0 in .Then

(2.19) R.t; u;x;/ D

.t; u;x;/



.t; u;x;/

gives the desired symmetrizer.

We will use results on pseudodifferential operators with nonregular symbols,

developed in Chap. 13,  9. Note that

(2.20) u 2 C

1Cr

H) R 2 C

1Cr

;

where the symbol class on the right is deﬁned as in (9.46) of Chap.13. Now, with

R D R.t; u;x;D/,set

(2.21) Q D

.R C R



/ C Kƒ

1

;

where K>0is chosen so that Q is a positive-deﬁnite operator on L

2. Symmetrizable hyperbolic systems 429

We will work with approximate solutions u

to (2.10), given by (1.4), with

(2.22) L

v D

.t;x;J

Given j˛jDk, we want to obtain estimates on .D

.t/; Q

.t//,whereQ

arises by the process above, from R

D R.t; J

;x;/. We begin with

(2.23)





D 2





C .D

D 2 Re





C 2K



;ƒ

1



C 2 Re.D

For the last term, we have the estimate

(2.24) j.D

/jC



.t/k



.t/k

We can write the ﬁrst term on the (far) right side of (2.23) as twice the real part

(2.25) .R

/ C .R

The last term has an easy estimate. We write the ﬁrst term in (2.25)as

(2.26)

/ C .R

ŒD

J

C.ŒR

L

Note that as long as (2.20) holds, with r>0;R

also has symbol in C

1Cr

and we have, by Proposition 9.9 of Chap. 13,

(2.27) R

D R

C R

2 OP S

1;ı

2 OP C

1Cr

.1Cr/ı

1;ı

Furthermore, by (9.42) of Chap.13,

(2.28) D

.x; / 2 S

1;ı

; j˛jD1

if r>0.In(2.27)and(2.28) we have uniform bounds for " 2 .0; 1.Takeı close

enough to 1 that .1 C r/ı  1.WethenhaveŒR

 bounded in L.H

1

upon applying Proposition 9.10 of Chap. 13 to R

. Hence we have

(2.29) ŒR

 bounded in L.H

k1

430 16. Nonlinear Hyperbolic Equations

with bound given in terms of ku

.t/k

1Cr

. Now Moser estimates yield

(2.30) kL

k1

 C





C C





k1

Consequently, we deduce

(2.31)



ŒR

L



 C



.t/k

1Cr



.t/k

Moving to the second term in (2.26), note that, for L D

.t; x; u/@

(2.32) ŒD

;L D

ŒD

.t; x; u/ @

By the Moser estimate, as in (1.13), we have

(2.33)



ŒD

;Lv



 C

Lip

kvk

CkB

kvk

Lip

Hence the second term in (2.26) is bounded by C





It remains to estimate the ﬁrst term in (2.26). We claim that

(2.34) j.R

v; v/jC





kvk

To see this, parallel to (2.27), we can write

(2.35) L

D L

C L

2 OPS

1;ı

2 OP C

1Cr

1.1Cr/ı

1;ı

;

and, parallel to (2.28),

(2.36) D

.x; / 2 S

1;ı

; j˛jD1:

Now, provided .1 C r/ı  1,

(2.37) R

D R

mod L.L

and

(2.38)





DR

mod OP S

1;ı

;

so we have (2.34).

Our analysis of (2.23) is complete; we have, for any r>0,

(2.39)

/  C



.t/k

1Cr



.t/k

; j˛jDk:

Exercises 431

From here we can parallel the rest of the argument of  1, to prove the following:

Theorem 2.3. If (2.10) is symmetrizable, in particular if it is strictly hyperbolic,

the initial-value problem, with u.0/ D f 2 H

.M /, has a unique local solution

u 2 C.I;H

.M // whenever k>n=2C 1.

We have the following slightly weakened analogue of the persistence result,

Proposition 1.5:

Proposition 2.4. Suppose u 2 C



Œ0; T /; H

.M /



;k>n=2C 1, and assume

u solves the symmetrizable hyperbolic system (2.10)fort 2 .0; T /. Assume also

that, for some r>0,

(2.40) ku.t/k

1Cr

.M /

 K<1;

for t 2 Œ0; T /. Then there exists T

>T such that u extends to a solution to (2.10),

belonging to C



Œ0; T

/; H

.M /



For the proof of this (and also for the proof of the part of Theorem 2.3 asserting

that whenever f 2 H

.M /,thenu is continuous, not just bounded, in t, with

values in H

.M /), one estimates



u.t/; QD

u.t/



in place of (1.40). Then estimates parallel to (2.24)–(2.39) arise, as the reader can

verify, yielding the bound

(2.41)

j˛jk



u.t/; QD

u.t/



 C



ku.t/k

1Cr



ku.t/k

If we use this in place of (1.46), the proof of Proposition 1.5 can be parallelled to

establish Proposition 2.4.

It follows that the result given in Corollary 1.6, on the local existence of C

solutions, extends to the case of symmetrizable hyperbolic systems (2.10).

We mention that actually Proposition 2.4 can be sharpened to the level of

Proposition 1.5. In fact, they can both be improved; the norms C

1Cr

.M / and

.M / appearing in the statements of these results can be weakened to the Zyg-

mund norm C



.M /. A proof, which is somewhat more complicated than the proof

of the result established here, can be found in Chap. 5 of [Tay].

Exercises

1. Show that, for smooth solutions, (2.11) is equivalent to

(2.42)



C div.v/ D 0;

v C grad h./ D 0;

432 16. Nonlinear Hyperbolic Equations

assuming p D p./. Here, h./ satisﬁes

./ D 

1

./:

2. Assume v is a solution to (2.42) of the form v Dr

'.t; x/, for some real-valued '.

One says v deﬁnes a potential ﬂow. Show that if ' and  vanish at inﬁnity appropriately

and h.0/ D 0,then

(2.43) '

C h./ D 0:

This is part of Bernoulli’s law for compressible ﬂuid ﬂow. Compare with (5.45).

3. Set m D v, the momentum density. Show that, for smooth solutions, (2.42) is equiva-

lent to

(2.44)



C div m D 0;

C div.

1

m ˝ m/ C grad p./ D 0:

(Hint: Make use of the identity div.u ˝v/ D .div v/u Cr

u:)

4. Show that a symmetrizer for the system (2.44)isgivenby



./ C 

1

jvj

v

;vD



Reconsider this problem after doing Exercise 4 in  8, in light of formulas (8.26)–(8.29)

for one space dimension, and of formula (5.53) in general.

5. Consider the one space variable case of (2.10):

(2.45) u

D B.t; x; u/u

C g.t; x; u/; u.0/ D f:

Show that if this is strictly hyperbolic, that is, B.t; x; u/ is a K  K matrix-valued

function whose eigenvalues 



.t; x; u/ are all real and distinct, then (2.45)issym-

metrizable in the easy sense deﬁned after (2.10). (Hint: Eliminate the s from the proof

of Proposition 2.2.)

3. Second-order and higher-order hyperbolic systems

We begin our discussion of second-order equations with quasi-linear systems, of

the form

(3.1)



j;k

.t;x;D

u/@

u 

.t;x;D

u/@

D C.t;x;D

u/:

For now, we assume A

and B

are scalar, though we allow u to take values

in R

.HereD

u stands for .u; u

; r

u/, which we also denote W D .u; u

; u

;

:::;u

/,so

3. Second-order and higher-order hyperbolic systems 433

(3.2) u

; u

;1 j  n:

We obtain a ﬁrst-order system for W , namely

(3.3)

D u

;

.t;x;W/@

C C.t;x;W /;

D @

;

which is a system of the form

(3.4)

.t;x;W/@

W C g.t; x; W /:

We can apply to each side the matrix

(3.5) R D

1

(tensored with the L  L identity matrix), where A

1

is the inverse of the matrix

A D .A

/. The matrix R is positive-deﬁnite as long as A is, that is, as long as A

is symmetric and

(3.6)

.t;x;W/



 C jj

;C>0:

Under this hypothesis, (3.3) is symmetrizable. Consequently we have:

Proposition 3.1. Under the hypothesis (3.6), if we pick initial data f 2

kC1

.M /, g 2 H

.M /; k > n=2 C 1,then(3.1) has a unique local solu-

tion

(3.7) u 2 C



I;H

kC1

.M /



\ C



I;H

.M /



satisfying u.0/ D f; u

.0/ D g.

Proof. Deﬁne W D .u; u

; u

;:::;u

/, as the solution to (3.3), with initial data

(3.8) u.0/ D f; u

.0/ D g; u

.0/ D @

By Proposition 2.1, we know that there is a unique local solution W 2 C

.I; H

.M //. It remains to show that u possesses all the stated properties. That

434 16. Nonlinear Hyperbolic Equations

u.0/ D f is obvious, and the ﬁrst line of (3.3) yields u

.0/ D u

.0/ D g.Also,

D u

2 C.I;H

/, which gives part of (3.7). The key to completing the proof

is to show that if W satisﬁes (3.3) with initial data (3.8), then in fact u

D @u=@x

on I  M .

To this end, set

D u



Since we know that @u=@t D u

, applying @=@t to each side yields



D 0;

by the last line of (3.3). Since u

.0/ D @

u.0/ by (3.8), it follows that v

D 0,so

indeed u

D @

u. Then substituting u

for u

and @

u for u

in the middle line of

(3.3) yields the desired equation (3.1)foru.

Finally, since u

2 C.I;H

/,wehaver

u 2 C.I;H

/, and consequently

u 2 C.I;H

kC1

As in  1,weﬁrsttakeM D T

. Parallel to Exercise 7 in  1, we can establish

a ﬁnite propagation speed result and then, as in Exercise 8 of  1, obtain a local

solution to (3.1) for other M .

We note that (3.6) is stronger than the natural hypothesis of strict hyperbolicity,

which is that, for  ¤ 0, the characteristic polynomial

(3.9) 



.t;x;W/

 

j;k

.t;x;W/



has two distinct real roots,  D 



.t;W;x;/. However, in the more general

strictly hyperbolic case, using Cauchy data to deﬁne a Lorentz metric over the

initial surface ft D 0g, we can effect a local coordinate change so that, at t D 0,

/ is positive-deﬁnite, when the PDE is written in these coordinates, and

then the local existence in Proposition 3.1 (and the comment following its proof)

applies.

Let us reformulate this result, in a more invariant fashion. Consider a PDE of

the form

(3.10)

j;k

.t;x;D

u/@

u C F.t;x;D

u/ D 0:

We let u take values in R

but assume a

.t;x;W/ is real-valued. Assume the

matrix .a

/ has an inverse, .a

Proposition 3.2. Assume



.t;x;W/



deﬁnes a Lorentz metric on O and S 

O is a spacelike hypersurface, on which smooth Cauchy data are given:

(3.11) u

D f; Y u

D g;

3. Second-order and higher-order hyperbolic systems 435

where Y is a vector ﬁeld transverse to S. Then the initial-value problem (3.10–

3.11) has a unique smooth solution on some neighborhood of S in O.

In Chap. 18 we will apply this result to Einstein’s gravitational equations.

We now look at a second-order, quasi-linear, L  L system of the form

(3.12)



j;k

.x; D

u/@

u D F.x;D

u/;

where, for each j; k 2f1;:::;ng;A

.x; W / is a smooth, L  L, matrix-valued

function satisfying

(3.13) a

jj

I 

j;k

.x; W /



 a

jj

for some a

2 .0; 1/. This includes equations of vibrating membranes and

elastic solids studied in  1 of Chap. 2. In particular, the condition (3.13) reﬂects

the condition (1.60) of Chap. 2. Note that the system (3.12) might not be strictly

hyperbolic.

Here, using results of Chap. 13,  10, we will write

.x; D

u/@

u  F.x;D

in terms of a paradifferential operator:

(3.14)

j;k

.t;x;D

u/@

u  F.x;D

u/ DM.uIx; D/u C R.u/;

where R.u/ 2 C

and (parallel to (8.20) of Chap. 15) if r>0,

(3.15) u 2 C

2Cr

H) M.uIx; / 2 A

1Cr

1;1

C S

1r

1;1

Thus, given ı 2 .0; 1/, we can use the symbol-smoothing process as in (10.101)–

(10.104) of Chap. 13 to write

(3.16)

M.uIx; / D M

.uIx; / C M

.uIx; /;

.uIx; / 2 A

1Cr

1;ı

.uIx; / 2 S

2.1Cr/ı

1;1

As in (3.13) we have (with perhaps different constants a

)

(3.17) a

jj

I  M

.uIx; /  a

jj

436 16. Nonlinear Hyperbolic Equations

for jj1. We can assume M

.uIx; /  I ,forjj1. Thus, given (3.15),

(3.18) M

.uIx; /

1=2

D G.uIx; / 2 A

1Cr

1;ı

Now let us set

(3.19) H.uIx; D/ D



G.uIx; D/ C G.uIx; D/





C I 2 OP A

1Cr

1;ı

;

which is self-adjoint and positive-deﬁnite and satisﬁes

(3.20) H.uIx; D/

 M.uIx; D/ D B.uIx; D/ 2 OP S

1;ı

C OP S

2.1Cr/ı

1;1

Set E.uIx; D/ D H.uIx; D/

1

2 OPS

1

1;ı

,andset

(3.21) v D H.uIx; D/u;wD u

We have the system

(3.22)

D w;

D H.uIx; D/w C C

.uIx; D/v;

DH.uIx; D/v C C

.uIx; D/v C R.u/;

where

(3.23)

.uIx; D/ D @

H.uIx; D/  E.uIx; D/ 2 OPS

1;ı

;

.uIx; D/ D B.uIx; D/E.uIx; D/ 2 OP S

1;ı

C OPS



1;1

;

provided ı is sufﬁciently close to 1 that 1  .1 C r/ı D<0.

Somewhat parallel to (1.4), we obtain solutions to (3.22) as limits of solutions

(3.24)

D J

;

D J

H.J

Ix; D/J

C J

Ix; D/J

;

DJ

H.J

Ix; D/J

C J

Ix; D/J

C R.J

Indeed, setting U

D .u

/, one obtains an estimate

(3.25)



.t/



 C



1Cr





.t/



C 1

;

from which local existence follows, by arguments similar to those used in  1.We

record the result.

3. Second-order and higher-order hyperbolic systems 437

Proposition 3.3. Under the hypothesis (3.13), if we pick initial data f 2

sC1

.M /, g 2 H

.M /, s>n=2C 1,then(3.12) has a unique local solution

(3.26) u 2 C



I;H

sC1

.M /



\ C



I;H

.M /



;

satisfying u.0/ D f; u

.0/ D g.

Having considered some quasi-linear equations, we now look at a completely

nonlinear, second-order equation:

(3.27) u

D F.t;x;D

u;@

u/; u.0/ D f; u

.0/ D g:

Here F D F.t;x;;;/ is smooth in its arguments;  D .

/ D .@

u/,

and so on. We assume u is real-valued. As before, set v D .v

;:::;v

/ D

.u;@

u;:::;@

u/. We obtain for v a quasi-linear system of the form

(3.28)

D F.t;x;D

v/;

j;k



F/.t;x;D

v/ @



F/.t;x;D

v/ @

C G

.t;x;D

v/;

with initial data

(3.29) v.0/ D .f; @

f;:::;@

f/; v

.0/ D .g; @

g;:::;@

g/:

The system (3.28) is not quite of the form (3.1), but the difference is minor. One

can reduce this to a ﬁrst-order system and construct a symmetrizer in the same

fashion, as long as

(3.30) 





F/.t;x;D

v/







F/.t;x;D

v/



has two distinct real roots  for each  ¤ 0. This is the strict hyperbolicity condi-

tion. Proposition 3.1 holds also for (3.28), so we have the following:

Proposition 3.4. If (3.27) is strictly hyperbolic, then given

f 2 H

kC1

.M /; g 2 H

.M /; k >

n C 2;

there is locally a unique solution

u 2 C.I;H

kC1

.M // \ C

.I; H

.M //: