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

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318 15. Nonlinear Parabolic Equations

for 0<t 1, so we require n.p  1/=q < 1. Therefore, we have part of the

following result:

Proposition 1.3. Under the hypothesis (1.17), if f 2 L

.M /,thePDE(1.16)

has a unique solution u 2 C.Œ0; T ; L

.M //,provided

(1.22) q  p and q>n.p 1/:

Furthermore, u 2 C

..0; T   M/.

It remains to establish the smoothness. First, replacing L

by L

in (1.21), we

see that, for any t 2 .0; T ; u.t/ 2 L

for all q

<q=.pq=n/.Aspq=n < 1,

this means q

exceeds q by a factor >1. Iterating this gives u.t/ 2 L

,where

exceeds q

j 1

by increasing factors. Once you have q

>np, the next iteration

gives u.t/ 2 C

.M /,forsomer>0. Now, consider the spaces

(1.23) X D C

.M /; Y D H

r1";q

.M /;

where q is chosen very large, and ">0very small. The fact that u 7! F

.u/

is locally Lipschitz from C

.M / to C

.M /, hence to H

r";p

.M /,gives(1.4)

in this case, and estimates from the third line of (1.15), together with Sobolev

imbedding theorems, give (1.6), and furthermore establish that actually, for each

t>0;u.t/ 2 C

.M /,forr

 r>0, estimable from below. Repeating this

argument a ﬁnite number of times, we obtain u.t/ 2 C

.M /, with r

>1.At

this point, the regularity result of Proposition 1.2 applies.

We can now establish a global existence theorem for solutions to (1.16).

Proposition 1.4. Suppose F

satisfy (1.17) with p D 1. Then, given f 2 L

.M /,

the equation (1.16) has a unique solution

(1.24) u 2 C



Œ0; 1/; L

.M /



\ C



.0; 1/  M



;

provided, when u takes values in R

.u/ D



.u/;:::;F

.u/



, that

(1.25)

;1 i; k  K:

Proof. We have u 2 C.Œ0; T ; L

/ \ C

..0; T /  M/,since(1.22) holds with

q D 2. To get global existence, it therefore sufﬁces to bound ku.t/k

;weprove

this is nonincreasing. Indeed, for t>0,

(1.26)

ku.t/k

D 2



u.t/;

.u.t//



 2kru.t/k

 2



u.t/;

.u.t//



1. Semilinear parabolic equations 319

Now by (1.25) there exist smooth G

such that F

D @G

=@u

, and hence the

right side of (1.26) is equal to

(1.27) 2

.u/du D 0:

The proof is complete.

The hypothesis (1.25) implies that the  D 0 analogue of (1.16) is a symmetric

hyperbolic system, as will be seen in the next chapter.

The condition p D 1 for (1.17) is rather restrictive. In the case of a scalar equa-

tion, we can eliminate this restriction, at least for bounded initial data, obtaining

the following important existence theorem.

Proposition 1.5. If (1.16) is scalar and f 2 L

.M /, then there is a unique

solution

u 2 L

.Œ0; 1/  M/\ C

..0; 1/  M/;

such that, as t & 0; u.t/ ! f in L

.M / for all p<1.

Proof. Suppose kf k

 M . Alter F

.u/ on jujM C 1=2, obtaining

.u/,

constant on u M 1 and on u  M C1. Then Proposition 1.4 yields a global

solution u to the modiﬁed PDE. This u solves

(1.28)

D u C

.t; x/@

u;a

.t; x/ D

.u.t; x//;

so the maximum principle for linear parabolic equations applies; ku.t/k

nonincreasing. Thus ku.t/k

 M for all t, and hence u solves the original

PDE.

The solution operator produced from Proposition 1.5 has an important

-contractive property, which will be useful for passing to the  D 0 limit

in  6 of the next chapter. We present an elegant demonstration from [Ho].

Proposition 1.6. Let u

be solutions to the equation (1.16) in the scalar case,

with initial data u

.0/ D f

2 L

.M /. Then, for each t>0,

(1.29) ku

.t/  u

.t/k

.M /

kf

 f

.M /

Proof. Set v D u

 u

.Thenv solves

(1.30)

D v C



; u



;

with

; u

/ D



C .1  s/u



ds;

320 15. Nonlinear Parabolic Equations

so F

/ F

/ D ˆ

; u

/.u

u

/.SetG

.t; x/ D ˆ

; u

/.Now,for

given T>0,letw solve the backward evolution equation

(1.31)

Dw C

.t; x/@

w; w.T / D w

2 C

.M /:

Then w.t/ is well deﬁned for t  T , and the maximum principle yields

(1.32) kw.t/k

kw

; for t  T:

Note that kv.T /k

is the sup of .v.T /; w

/ over kw

 1.Now,fort 2

.0; T /,wehave

(1.33)

.v; w/ D .v; w/ C



v/; w



 .v; w/ C

.v; G

w/ D 0:

Since



v.0/; w.0/



kv.0/k

kw.0/k

,thisproves(1.29).

We next produce global weak solutions to (1.16), for K  K systems, with the

symmetry hypothesis (1.25), in case (1.17) holds with p D 2. As before, we take

M D T

. We will use a version of what is sometimes called a Galerkin method

to produce a sequence of approximations, converging to a solution to (1.16).

Give ">0, deﬁne the projection P

on L

.M / by

f.x/ D

jkj1="

f.k/e

ikx

;

where, for k 2 Z

;

f.k/form the Fourier coefﬁcients of f . Consider the initial-

value problem

(1.34)

D P

P

C P

/; u

.0/ D P

We take f 2 L

.M /. For each " 2 .0; 1, ODE theory gives a unique short-time

solution, satisfying u

.t/ D P

.t/.Furthermore,

(1.35)

.t/k

D 2.P

P

; u

/ C 2



/; u



1. Semilinear parabolic equations 321

The ﬁrst term on the right is 2krP

.t/k

 0. The last term is equal to

(1.36)



/; P



D2



/; @



D2





D 0;

where G

is as in (1.27). We deduce that

(1.37) ku

.t/k

kf k

Hence, for each ">0,(1.34) is solvable for all t>0,and

(1.38) fu

W " 2 .0; 1g is bounded in L

.M //:

Note that further use of (1.35)–(1.37)gives

(1.39) 2

krP

.t/k

dt DkP

f k

ku

.T /k

;

for any T 2 .0; 1/. Hence, for each bounded interval I D Œ0; T ,since

D u

(1.40) fu

g is bounded in L

.I; H

.M //:

Given that jF

.u/jC hui

, it follows from (1.38)that

(1.41)

/g is bounded in L

.M //

 L

n=2ı

.M //;

for each ı>0. Now using the evolution equation (1.34)for@u

=@t and (1.40)–

(1.41), we conclude that

(1.42)

is bounded in L

.I; H

n=21ı

.M //;

hence

(1.43) fu

g bounded in H

.I; H

n=21ı

.M //:

Now we can interpolate between (1.40)and(1.43) to obtain

(1.44) fu

g bounded in H

.I; H

1s.n=2C1Cı/

.M //;

322 15. Nonlinear Parabolic Equations

for each s 2 Œ0; 1.Nowifwepicks>0very small and apply Rellich’s theorem,

we deduce that

(1.45) fu

W 0<" 1g is compact in L

.I; H

1

.M //;

for all >0.

The rest of the argument is easy. Given T<1, we can pick a sequence

D u

! 0, such that

(1.46) u

! u in L

.Œ0; T ; H

1

.M //; in norm.

We can arrange that this hold for all T<1, by a diagonal argument. We can also

assume that u

is weakly convergent in each space speciﬁed in (1.38)and(1.40),

and that @u

=@t is weakly convergent in the space given in (1.42). From (1.46)

we deduce

(1.47) F

/ ! F

.u/ in L

.Œ0; T ; L

.M //; in norm,

as k !1, hence

(1.48) @

/ ! @

.u/ in L

.Œ0; T ; H

1;1

.M //:

Using H

1;1

.M /  H

n=21ı

.M /, we see that each term in (1.34)converges

as "

! 0. We have proved the following:

Proposition 1.7. If jF

.u/jC hui

and jrF

.u/jC hui, then, for each f 2

.M /,aK  K system of the form (1.16), satisfying the symmetry hypothesis

(1.25), possesses a global weak solution

(1.49)

u 2 L

.M // \ L

loc

.M //

\ Lip

loc



2

.M / C H

n=21ı

.M /



When reading the discussion of the Navier–Stokes equations in Chap. 17, one

will note a similar argument establishing a classical result of Hopf on global weak

solutions to that system.

Exercises

1. Verify the estimates on operator norms of e

t

W Y ! X listed in (1.15).

2. Show that, given f 2 C

.M /; M D T

, the solution to

D u CF.t;u; ru/; u.0/ D f;

continues to exist as long as ku.t/k

does not blow up, provided this equation is

scalar and F

 0.(Hint:DeriveaPDEforu

D @u=@x

and apply the maximum

principle.)

Exercises 323

3. Generalize the treatment of PDE (1.16), done above in the case M D T

,tothe

following situation for a general compact Riemannian manifold M :

D u C div F.u/;

where

(a) u is scalar and F.u/ D F.x;u/ 2 T

M ,forx 2 M; u 2 R,

(b) u is a vector ﬁeld and F.u/ D F.x;u/ 2˝

M ,forx 2 M; u 2 T

M .

Consider other generalizations.

4. More generally, extend the treatment of (1.16)to

D u C DF.u/; u.0/ D f;

where D is a ﬁrst-order differential operator on the compact Riemannian manifold

M ,andF satisﬁes the estimates (1.17). What additional properties must D have for

Proposition 1.4 to extend?

5. For given ; " > 0; k 2 Z

, consider the .4k/-order operator

L D   "

Show that ke

L.Y;X/

satisﬁes estimates of the form (1.15), where, in the right col-

umn, one replaces t



by t

=2k

,andC depends on  and ".

6. Consider the PDE

(1.50)

D u "

u C

.u/; u.0/ D f:

Suppose F

satisﬁes (1.17), that is,

(1.51) jF

.u/jC hui

; jrF

.u/jC hui

p1

Show that there is a unique local solution

u 2 C.Œ0; T ; L

.M // \ C

..0; T  M/

given f 2 L

.M /, provided

q  p and q>

n.p  1/

4k  1

7. Suppose that (1.51) holds with p D 2 and that dim M D n<8k 2. Suppose

also that the symmetry hypothesis (1.25) holds (for a K  K system), so F

.u/ D

=@u

.Givenf 2 L

.M /, show that (1.50) has a unique global solution u 2

C.Œ0;1/; L

.M // \ C

..0; 1/  M/,andku.t/k

kf k

,fort>0.

8. Let u D u

be the solution to (1.50) under the hypotheses of Exercise 7.Wetake>0

ﬁxed and let " & 0. Obtain bounds on fu

W " 2 .0; 1gwhich imply that a subsequence

converges to a weak solution u

of the " D 0 case of (1.50), thus providing another

proof of Proposition 1.7.(Hint: Start with the following analogue of (1.39):

2

kru

.t/k

dt C 2"

k

.t/k

dt Dkf k

ku

.T /k

324 15. Nonlinear Parabolic Equations

9. Let u be a smooth solution for 0<t<Tof the system (1.16), under the hypothesis

(1.25) of Proposition 1.4, namely,

(1.52)

D u C

.u/; @

D @

; u.0/ D f:

Thus F

D @

. Show that

u.t/k

D2kuk

 2



.u/



2kuk





ku.t/k



C kuk

;

where

(1.53) ˆ.M / D sup

jujM

.u/kD sup

jujM

.u/k:

Integrating this and using the estimate on 

u./k

d that follows from

(1.26)–(1.27), deduce that, for 0<t<T,

(1.54) kr

u.t/k

C 

ku./k

d kr

f k

C 

2

‰.u;t/

kf k

;

where ‰.u;t/ D ˆ.M /, with M D supfku./k

W 0    tg.

10. In the context of Exercise 9, suppose that the space dimension is n D 1. Note that in

this case, ku.t/k

 C kr

u.t/k

ku.t/k

C C ku.t/k

: Show that under the

hypothesis

ˆ.M /M

2

! 0; as M !C1;

we have a bound on ku.t/k

as t % T , and hence a global existence result for (1.52).

Compare with [Smo], p. 427.

11. Solve the system

(1.55)

D u

C u.u

C v

/; v

D v

C v.u

C v

u.0; x/ D A cos kx; v.0; x/ D A sin kx;

with A>1. Here, x 2 S

D R=2Z. Show that the maximal t-interval of existence

in R

is Œ0; C.A/=k

/. This example is given in [LSU] and attributed to E. Heinz.

12. Consider the multidimensional “Burger’s equation”

(1.56) u

u D u; u.0; x/ D f.x/;

for u.t; x/ W R

 T

! R

. Show that, for each t>0,

sup

x2T

.t; x/j sup

x2T

.x/j;1 j  n:

Deduce that (1.56) has a global solution. (Hint: Show that

ku.t/k

 C

j;`

j˛jCjˇjk

k.D

/.D

kuk

kC1

 2kruk

;

and use Proposition 3.6 of Chap. 13 to estimate k.D

/.D

Note that the case n D 1 is also treated by Proposition 1.5.

2. Applications to harmonic maps 325

2. Applications to harmonic maps

Let M and N be compact Riemannian manifolds. Using Nash’s result, proved

in  5 of Chap.14, we take N to be isometrically imbedded in some Euclidean

space; N  R

. A harmonic map u W M ! N is a critical point for the energy

functional

(2.1) E.u/ D

jru.x/j

dV .x/;

among all such maps. In the integrand, we use the natural square norm on T



M ˝

u.x/

N  T



M ˝R

. The quantity (2.1) clearly depends only on the metrics on

M and N , not on the choice of isometric imbedding of N into Euclidean space.

If u

is a smooth family of maps from M to N ,then

(2.2)

E.u

sD0

D

v.x/u.x/ dV;

where u D u

,andv.x/ D .@=@s/u

.x/ 2 T

u.x/

N . One can vary u

so that v is

any map M ! R

such that v.x/ 2 T

u.x/

N , so the stationary condition is that

(2.3) u.x/ ? T

u.x/

N; for all x 2 M:

We can rewrite the stationary condition (2.3) by a process similar to that used in

(11.12)–(11.14)in Chap. 1. Suppose that, near a point z 2 N  R

;Nis given by

(2.4) f

.y/ D 0; 1  `  L;

where L D k  dim N , with rf

.y/ linearly independent in R

, for each y

near z.Ifu W M ! N is smooth and u.x/ is close to z,thenwehave

(2.5)



D 0; 1  `  L; 1  j  m;

where .x

;:::;x

/ is a local coordinate system on M . Hence

(2.6)



u



D

;;j;k









Since fr

.y/ W 1  `  Lg is a basis of the orthogonal complement in R

of T

N , it follows that, for smooth u W M ! N , the normal component of u

depends only on the ﬁrst-order derivatives of u, and is quadratic in ru;thatis,we

have a formula

(2.7) .u/

D .u/.ru; ru/:

326 15. Nonlinear Parabolic Equations

Thus the stationary condition (2.3)foru is equivalent to

(2.8) u  .u/.ru; ru/ D 0:

Denote the left side of (2.8)by.u/; it follows from (2.7) that, given

u 2 C

.M; N /;  .u/ is tangent to N at u.x/.

J. Eells and J. Sampson [ES] proved the following result.

Theorem 2.1. Suppose N has negative sectional curvature everywhere. Then,

given v 2 C

.M; N /, there exists a harmonic map w 2 C

.M; N / which is

homotopic to v.

As in [ES], the existence of w will be established via solving the PDE

(2.9)

D u  .u/.ru; ru/; u.0/ D v:

It will be shown that under the hypothesis of negative sectional curvature on

N , there is a smooth solution to (2.9)forallt  0 and that, for a sequence

!1; u.t

/ tends to the desired w. In outline, our treatment follows that pre-

sented in [J2], with some simpliﬁcations arising from taking N to be imbedded in

(as in [Str]), and also some simpliﬁcations in the use of parabolic theory.

The local solvability of (2.9) follows directly from Proposition 1.2.Since.u/

is tangent to N for u 2 C

.M; N /, it follows that u.t/ W M ! N for each t in

the interval Œ0; T / on which the solution to (2.9) exists. To get global existence for

(2.9), it sufﬁces to estimate ku.t/k

In order to estimate r

u, we use a differential inequality for the energy density

(2.10) e.t; x/ D

u.t; x/j

In fact, there is the identity

(2.11)

 e Dj



du  Ric

/; du  e

.du  e

; du  e

/du  e

; du e

;

where fe

g is an orthonormal frame at T

M and we sum over repeated indices.

The operator

is obtained from the second covariant derivative:

u.x/ W˝

M ! T

u.x/

See the exercises for a derivation of (2.11).

2. Applications to harmonic maps 327

Given that N has negative sectional curvature, (2.11) implies the inequality

(2.12)

 e  ce:

If f.t;x/ D e

ct

e.t; x/,wehave@f = @ t  f  0, and the maximum principle

yields f.t;x/ kf.0;/k

, hence

(2.13) e.t; x/  e

krvk

This C

-estimate implies the global existence of a solution to (2.9), by

Proposition 1.2.

For the rest of Theorem 2.1, we need further bounds on u, including an im-

provement of (2.13). For the total energy

(2.14) E.t/ D

e.t; x/ dV.x/ D

jruj

dV .x/;

we claim there is the identity

(2.15) E

.t/ D

dV .x/:

Indeed, one easily obtains E

.t/ D

;uidV.x/. Then replace u by

C .u/.ru; ru/.Sinceu

is tangent to N and .u/.ru; ru/ is normal to N ,

(2.15) follows. The desired improvement of (2.13) will be a consequence of the

following estimate:

Lemma 2.2. Let e.t; x/  0 satisfy the differential inequality (2.12). Assume that

E.t/ D

e.t; x/ dV.x/  E

is bounded. Then there is a uniform estimate

(2.16) e.t; x/  e

;t 1;

where K depends only on the geometry of M .

Proof. Writing @e=@t  e D ce  g; g.t; x/  0,wehave,for0  s  1,

(2.17)

e.t C s; x/ D e

s.Cc/

e.t; x/ 

.s/.Cc/

g.; x/ d

 e

s.Cc/

e.t; x/: