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

Подождите немного. Документ загружается.

438 16. Nonlinear Hyperbolic Equations

This proposition applies to the equations of prescribed Gaussian curvature,

for a surface S that is the graph of y D u.x/; x 2   R

, under certain

circumstances. The Gauss curvature K.x/ is related to u.x/ via the PDE

(3.31) det H.u/  K.x/



1 Cjruj



.nC2/=2

D 0;

where H.u/ is the Hessian matrix,

(3.32) H.u/ D .@

u/:

Note that, if F.u/ D det H.u/,then

(3.33) DF .u/v D TrŒC.u/H.v/;

where C.u/ is the cofactor matrix of H.u/,so

(3.34) H.u/C.u/ D Œdet H.u/I:

Of course, (3.31) is elliptic if K>0. Suppose K is negative and on the hypersur-

face † Dfx

D 0g Cauchy data are prescribed, u D f.x

/; @

u D g.x

/; x

;:::;x

n1

/.Then@

u D @

f on † for 1  j; k  n 1; @

u D @

on † for 1  j  n  1,andthen(3.31) uniquely speciﬁes @

u, hence H.u/,on

†, provided det H.f / ¤ 0. If the matrix H.u/ has signature .n  1; 1/,andif

† is spacelike for its quadratic form, then (3.31) is a hyperbolic Monge–Ampere

equation, and Proposition 3.4 applies.

We next treat quasi-linear equations of degree m,

(3.35) @

u D

m1

j D0

.t;x;D

m1

u;D

u C C.t;x;D

m1

u/;

with initial conditions

(3.36) u.0/ D f

u.0/ D f

;:::;@

m1

u.0/ D f

m1

Here, A

.t;x;w;D

/ is a differential operator, homogeneous of degree m  j .

Assume u takesvaluesinR

, but for simplicity we suppose the operators A

have

scalar coefﬁcients. We will produce a ﬁrst-order system for v D .v

;:::;v

m1

with

(3.37) v

D ƒ

m1

u;:::;v

D ƒ

mj 1

u;:::;v

m1

D @

m1

3. Second-order and higher-order hyperbolic systems 439

We have

(3.38)

D ƒv

;

m2

D ƒv

m1

;

m1

.t;x;Pv;D

/ƒ

1Cj m

C C.t;x;Pv/;

where Pv D D

m1

u (i.e., @

u D @

j C1m

), so P 2 OP S

. Note that the

operator A

.t;x;Pv;D

/ƒ

1Cj m

is of order 1. The initial condition for (3.38)

(3.39) v

.0/ D ƒ

m1

;:::;v

.0/ D ƒ

mj 1

;:::;v

m1

.0/ D f

m1

The system (3.38)hastheform

(3.40) @

v D L.t; x; P v; D/v C G.t; x; P v/;

where L is an m  m matrix of pseudodifferential operators, which are scalar

(though each entry acts on K-vectors). Note that the eigenvalues of the principal

symbol of L are i



.t;x;v;/,where D 



are the roots of the characteristic

equation

(3.41) 



m1

j D0

.t;x;Pv;/

D 0:

We will make the hypothesis of strict hyperbolicity, that for  ¤ 0 this equation

has m distinct real roots, so L.t; x; P v; / has m distinct purely imaginary eigen-

values. Consequently, as in Proposition 2.2, there exists a symmetrizer, an m m,

matrix-valued function R.t; x; w; /, homogeneous of degree 0 in  and smooth

in its arguments, such that, for  ¤ 0,

(3.42)

R.t; x; w; / is positive-deﬁnite,

R.t;x;w;/L.t; x;w;/ is skew-adjoint.

Note that, given r 2 .0; 1/ n Z

(3.43)

v 2 C

1Cr

H) L.t; x; P v; / 2 C

1Cr

and

R.t; x; P v; / 2 C

1Cr

From here, an argument directly parallel to (2.21)–(2.39) establishes the solvabil-

ity of (3.38)–(3.39). We have the following result:

440 16. Nonlinear Hyperbolic Equations

Theorem 3.5. If (3.35) is strictly hyperbolic, and we prescribe initial data f

sCm1j

.M /; s > n=2 C 1, then there is a unique local solution

u 2 C.I;H

sCm1

.M // \ C

m1

.I; H

.M //;

which persists as long as, for some r>0,

ku.t/k

mCr

CCk@

m1

u.t/k

1Cr

is bounded.

In [Tay] it is shown that the solution persists as long as

ku.t/k



CCk@

m1

u.t/k



is bounded.

While there is a relative abundance of second-order hyperbolic equations and

systems arising in various situations, particularly in mathematical physics, com-

pared to the higher-order case, nevertheless there is value in studying higher-order

equations, in addition to the fact that such study arises as a “natural” extension of

the second-order case. We mention as an example the appearance of a third-order,

quasi-linear hyperbolic equation, arising from the study of relativistic ﬂuid mo-

tion; this will be discussed in  6 and 8 of Chap. 18.

Exercises

1. Formulate and prove a ﬁnite propagation speed result for solutions to (3.1).

2. Recall Exercise 2 of  2, dealing with the equation (2.42) for compressible ﬂuid ﬂow

when v has the special form v Dr

'.t;x/. Show that ' satisﬁes the second-order

PDE

(3.44) @

.r'/ C

j 1

.r'/ D 0;

where r' D .'

; r

'/ and the functions H

are given by

(3.45)

.r'/ DK





;

.r'/ D .@

'/H

.r'/; j  1:

Here, K is the inverse function of h, deﬁned by:

y D h./ ”  D K.y/:

Examine the hyperbolicity of this PDE.

3. Consider three-dimensional Minkowski space R

1;2

Df.t;x;y/g, with metric ds

dt

C dx

C dy

.LetS beasurfaceinR

1;2

,givenby

y D u.t; x/:

Exercises 441

Show that the condition for S to be a minimal surface in R

1;2

is that

(3.46) .1 C u

 2.u

 u

 .1  u

D 0:

Show that this is hyperbolic provided u

<1C u

, and that this holds provided

the induced metric tensor on S has signature .1; 1/.(Hint:Toget(3.46), adapt the

calculations used to produce the minimal surface equation (7.6) in Chap. 14.)

Exercises on nonlinear Klein–Gordon equations, and variants

In these exercises we consider the initial-value problem for semilinear hyperbolic equations

of the form

(3.47) u

 u C m

u D f.u/;

for real-valued u. Here,  is the Laplace operator on a compact Riemannian manifold M ,

or on R

. We assume m>0,andsetƒ D

 C m

1. Show that, for s>0, sufﬁciently smooth solutions to (3.47) satisfy

(3.48)



kƒ

sC1

Ckƒ



D 2



f.u/; ƒ



2. Using arguments such as those that arose in proving Proposition 1.5, show that smooth

solutions to (3.47) persist as long as ku.t/k

can be bounded.

3. Note that the s D 0 case of (3.48) can be written as

(3.49)



kruk

C m

kuk

Cku



D 2

f.u/ dV:

Thus, if f.u/ D g

.u/,wehave

(3.50) kru.t/k

C m

ku.t/k

Cku

.t/k





u.t/



dV D const.

Deduce that

(3.51) g  0 H) ku.t/k

Cku

.t/k

 const.

4. Deduce that (3.47) is globally solvable for nice initial data, provided that f.u/ D g

.u/

with g.u/  0 and that dim M D n D 1.

5. Note that the s D 1 case of (3.48) can be written as

(3.52)

kLuk

Ckru

C m

D 2



rf.u/; ru



C 2m



f.u/; u



;

where L D C m

. Assume dim M D n D 3, so that, by Proposition 2.2 of

Chap. 13,

.M /  L

.M /:

442 16. Nonlinear Hyperbolic Equations

Deduce that the right side of (3.52)isthen

(3.53)

krf.u/k

Ckru

C m

kf.u/k

C m

 C kf

.u/k

kLuk

C m

kf.u/k

Ckru

C m

(Hint: To estimate kf

.u/ruk

,usekvwk

kvk

kwk

with 2p

D 6,so

2p D 3:)

6. If f.u/ Du

,thenkf

.u/k

 9kuk

 C kuk

and kf.u/k

kuk

Making use of (3.51) to estimate kuk

, demonstrate global solvability for

(3.54) u

 u C m

u Du

;

with nice initial data, given that dim M D n D 3.

For further material on nonlinear Klein–Gordon equations, including treatments of

(3.54) with u

replaced by u

,see[Gril, Ra,Re,Seg, St,Str].

In Exercises 7–12, we consider the equation (3.47) under the hypotheses

(3.55) f.0/ D 0; jf

.`/

.u/jC

;` 1:

An example is f.u/ D sin u;then(3.47) is called the sine–Gordon equation.

7. Show that if u is a sufﬁciently smooth solution to (3.47), and we take the “energy”

E.t/ Dkƒu.t/k

Cku

.t/k

,then

 C C CE.t/;

and hence

(3.56) ku.t/k

 C.t/:

This partially extends Exercise 3,inthatf.u/ D g

.u/, with g.u/  C

juj

8. Deduce that (3.47) is globally solvable for nice initial data (given (3.55)), provided

that n D 1.

In Exercises 9–11, assume that n  2 and that u.0/ D u

2 H

/; u

.0/ D u

s1

/; s > n=2 C 1.

9. Establish an estimate of the form

(3.57) ku.t/k

 C.t/;

and deduce that (3.47) is globally solvable (given (3.55)), provided n D 2 or 3.(Hint:

Write u.t/ D v.t/ C w.t/,wherev.t/ solves

 .  m

/v D 0; v.0/ D u

.0/ D u

;

and

(3.58) w.t/ D

sin.t  s/ƒ



u.s/



ds:

Exercises 443

To get (3.57) from this, establish the estimate

(3.59) kf.u.t//k

 C.t/;

from (3.56).)

10. Suppose n D 4. Show that

(3.60) ku.t/k

 C.t/;

and deduce that (3.47) is globally solvable (given (3.55)), provided n D 4.

(Hint: Start with k@

f.u/k

 C

kruk

, and use the Sobolev

estimate

(3.61) k'k

 C kr'k

;pD

n  2

;

to deduce that

(3.62) n D 4 H) k@

f.u/k

 C kuk

C C kuk

Then use (3.58) to estimate kw.t/k

11. Show that (3.60) also holds when n D 5. Deduce that (3.47) is globally solvable

(given (3.55)), provided n D 5.(Hint: Start with k@

f.u/k

 C

kruk

, and apply (3.61), with p D 5=3,toget

(3.63) n D 5 H) kf.u/k

2;5=3

 C.t/:

Use the Sobolev imbedding result H

;p

/  L

np=.np/

/ to deduce

(3.64) kf.u.t//k

21=2

 C.t/:

Use (3.58) to deduce

(3.65) n D 5 H) ku.t/k

2C1=2

 C.t/:

Iterate this argument, to get (3.60).)

12. Derive results on the global existence of weak solutions to (3.47), under the hypothesis

(3.55), analogous to those in Exercises 12 and 13 of  1.

For further results on the equation (3.47), under hypotheses like (3.55), but more

general, see [BW]and[Str].

Exercises on wave maps

In these exercises, we consider the initial-value problem for semilinear hyperbolic

systems of the form

(3.66) u

 u D B.x; u; ru/;

where B.x; u;p/is smooth in .x; u/ and a quadratic form in p. Here,  is the Laplace

operator on a compact Riemannian manifold X, u.t; x/ takesvaluesinR

,andru D

t;x

444 16. Nonlinear Hyperbolic Equations

1. Show that, for s  0, sufﬁciently smooth solutions to (3.66) satisfy

(3.67)



u.t/k

Ck@

u.t/k



D 2



;ƒ

B.x; u; ru/



2. Using arguments such as those that arose in proving Proposition 1.5, show that smooth

solutions to (3.66) persist as long as ku.t/k

Ck@

u.t/k

can be bounded.

3. Suppose that, for t 2 I; u.t; x/ solves (3.66)andu W I  X ! N ,whereN is a

submanifold of R

. Suppose also that, for all .t; x/ıI  X,

(3.68) B.x; u; ru/ ? T

Show that

(3.69) e.t; x/ D

;E.t/D

e.t; x/ dV .x/;

satisﬁes

(3.70)

D 0:

(Hint: The hypothesis (3.68) implies u

 B.x; u; ru/ D 0.Thenuse(3.67), with

s D 0:)

In Exercises 4–6, suppose X is the ﬂat torus T

, or perhaps X D R

. Assume

(3.68) holds. Deﬁne

(3.71) m

.t; x/ D u

 @

4. Show that

(3.72)



D 0:

(Hint: Start with @

e D u

 u

u r

, and use the equation (3.66); then use

 B.x; u; ru/ D 0:)

5. Show that, for each j D 1;:::;n,

(3.73)



u @

u/  @

u @



(Hint:Use@

u  B.x; u; ru/ to get @

D u  @

u C u

 @

; then compute @

and subtract.)

The considerations of Exercises 1–5 apply to the “wave map” equation

(3.74) u

 u D .u/.ru; ru/;

4. Equations in the complex domain and the Cauchy–Kowalewsky theorem 445

where ru Dr

t;x

u and .u/.ru; ru/ is as in the harmonic map equation (10.25)

of Chap. 14. Indeed, (3.74) is the analogue of the harmonic map equation for a map

u W M ! N when N is Riemannian but M is Lorentzian.

6. Suppose n D 1.ThenX D S

(or R

). Show that (3.74) has a global smooth solution,

for smooth Cauchy data, u.0/ D f; u

.0/ D g, satisfying f W X ! N; g.x/ 2

f.x/

N .

(Hint: In this case, (3.72)–(3.73)imply@

e @

e D 0, which gives a pointwise bound

for e.t; x/:) This argument follows [Sha].

For results in higher dimensions, including global weak solutions and singularity

formation, see [Sha], and references given therein.

4. Equations in the complex domain

and the Cauchy–Kowalewsky theorem

Consider an mth order, nonlinear system of PDE of the form

(4.1)

D A.t; x; D

u;D

m1

u;:::;D

m1

u/;

u.0; x/ D g

.x/; : : : ; @

m1

u.0; x/ D g

m1

.x/:

The Cauchy–Kowalewsky theorem is the following:

Theorem 4.1. If A is real analytic in its arguments and g

are real analytic, for

x 2 O  R

, then there is a unique u.t; x/ that is real analytic for x 2 O



O;tnear 0, and satisﬁes (4.1).

We established the linear case of this in Chap. 6. Here, in order to prove

Theorem 4.1, we use a method of Garabedian [Gb1, Gb2], to transmutate (4.1)

into a symmetric hyperbolic system for a function of .t;x;y/.Tobegin,bya

simple argument, it sufﬁces to consider a general ﬁrst-order, quasi-linear, N  N

system, of the form

(4.2)

j D1

.t; x; u/

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

We assume that A

and f are real analytic in their arguments, and we use these

symbols also to denote the holomorphic extensions of these functions. Similarly,

we assume g is analytic, with holomorphic extension g.z/.Wewanttosolve(4.2)

for u which is real analytic, that is, we want to extend u to u.t;x;y/,soastobe

holomorphic in z D x C iy,sothat

(4.3)

C i

D 0:

446 16. Nonlinear Hyperbolic Equations

Now, multiplying this by iB

and adding to (4.2), we have

(4.4)

j D1



C iB





j D1

C f.t;x;u/:

We arrange for this to be symmetric hyperbolic by taking

(4.5) B

.t; z; u/ D





 A



Thus we have a local smooth solution to (4.4), given smooth initial data

u.0;x;y/ D g.x; y/.Now,ifg.x; y/ is holomorphic for .x; y/ 2 U ,wewantto

show that u.t;x;y/ is holomorphic for .x; y/ 2 U

 U if t is close to 0.Tosee

this, set

(4.6) v



C i



Then

(4.7)



j D1



C iB







j D1



j D1





j D1





C @



f.t;z; u/:

Since A

.t; z; u/ and f.t;z; u/ are holomorphic in z and u,



.t; z; u/ D





D C

.t; z; u/v



;

and similarly @



f.t;z; u/ D F.t;z; u/v



. Thus

(4.8)



j D1



C iB







j D1



C Ev



j D1

j

;

with

E D

.t; z; u/ C F.t;z; u/; G

j

D i@



This is a symmetric hyperbolic, .N n/  .N n/ system for v D .v





W 1   

N; 1    n/. The hypothesis that g.x; y/ is holomorphic for .x; y/ 2 U means

v.0; x; y/ D 0 for .x; y/ 2 U . Thus, by ﬁnite propagation speed, v.t; x; y/ D 0

on a neighborhood of f0gU .

Exercises 447

Thus we have a solution to (4.2) which is holomorphic in x C iy, under the

hypotheses of Theorem 4.1. We have not yet established analyticity in t; in fact, so

far we have not used the analyticity of A

and f in t.Wedothisnow.Asabove,

we use A

;f,andu also to denote the holomorphic extensions to  D t C is.

Having u for s D 0, and desiring @u=@t Di@u=@s, we produce u.t;s;x;y/ as

the solution to

(4.9)

D i

j D1

C if; u.t;0;x;y/ D solution to (4.4):

Applying iB

to (4.3) and adding to (4.9), we get

(4.10)

j D1



C iB





j D1

C if;

which we arrange to be symmetric hyperbolic by taking

(4.11) B

.t;s;x;y/ D





C A



To see that the solution to (4.9) is holomorphic in t Cis,let

(4.12) w D



C i



By the initial condition for u at s D 0 given in (4.9), we have w D 0 for s D 0.

Meanwhile, parallel to (4.7), w satisﬁes a symmetric hyperbolic system, so u is

holomorphicin t Cis. This establishes the Cauchy–Kowalewsky theorem for (4.2),

and the general case (4.1) follows easily.

There are other proofs of the Cauchy–Kowalewsky theorem. Some work by

estimating the terms in the power series of u.t; x/ about .0; x

/. Such proofs are

often presented near the beginning of PDE books, as they are elementary, though

many students have grumbled that going through this somewhat elaborate argu-

ment at such an early stage is rather painful. The proof presented above reﬂects

an aesthetic sensibility that prefers the use of complex function theory to power-

series arguments. Another sort of proof, with a similar aesthetic, is given in [Nir];