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

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328 15. Nonlinear Parabolic Equations

Since e

s.Cc/

is uniformly bounded from L

.M / to L

.M / for s 2 Œ1=2; 1,the

bound (2.16)fort 2 Œ1=2; 1/ follows from the hypothesized L

-bound on e.t/.

We remark that a more elaborate argument, which can be found on pp. 84–86

of [J2], yields an explicit bound K depending on the injectivity radius of M and

the ﬁrst (nonzero) eigenvalue of the Laplace operator on M .

Since Lemma 2.2 applies to e.t; x/ Djruj

when u solves (2.9), we see that

solutions to (2.9) satisfy

(2.18) ku.t/k

 K

kvk

; for all t  0:

Hence, by the regularity estimate in Proposition 1.2, there are uniform bounds

(2.19) ku.t/k

 K

kvk

;t 1;

for each `<1. Of course there are consequently also uniform Sobolev bounds.

Now, by (2.15), E.t/ is positive and monotone decreasing as t %1. Thus

the quantity

.t; x/j

dV.x/ is an integrable function of t, so there exists a

sequence t

!1such that

(2.20) ku

; /k

! 0:

From (2.19)andthePDE(2.9), we have bounds

.t; /k

 C

;

and interpolation with (2.20) then gives, for any ` 2 Z

(2.21) ku

; /k

! 0:

Therefore, by the PDE (2.9), one has for u

.x/ D u.t

;x/,

(2.22) u

 .u

/.ru

; ru

/ ! 0 in H

.M /;

as well as a uniform bound from (2.19). It easily follows that a subsequence

converges in a strong norm to an element w 2 C

.M; N / solving (2.8)and

homotopic to v, which completes the proof of Theorem 2.1.

We next show that there is an energy-minimizing harmonic map w W M ! N

within each homotopy class when N has negative sectional curvature.

Proposition 2.3. Under the hypotheses of Theorem 2.1, if we are given v 2

.M; N /, then there is a smooth map w W M ! N that is harmonic, and

homotopic to v, and such that E.w/  E.Qv/ for any Qv 2 C

.M; N / homo-

topic to v.

2. Applications to harmonic maps 329

Proof. If ˛ is the inﬁmum of the energies of smooth maps homotopic to v,pick



, homotopic to v, such that E.v



/ & ˛. Then solve (2.9), for u



, with initial

data u



.0/ D v



. We have some sequence u



j

/ ! w



2 C

.M; N /,har-

monic. The proof of Theorem 2.1 gives E.w



/  E.v



/, hence E.w



/ ! ˛.

Also, via (2.16)and(2.19), we have uniform C

-bounds on w



,forall`. Thus



g has a limit point w with the desired properties.

We record a local existence result for parabolic equations with a structure like

that of (2.9), with initial data less smooth than C

. Thus we look at equations of

the form (1.1), with

(2.23) F.x;D

u/ D B.u/.ru; ru/;

a quadratic form in ru. In this case, we take

(2.24) X D H

1;p

;YD L

;qD

;p>n;

and verify the conditions (1.3)–(1.6), using the Sobolev imbedding result

s;p

 L

np=.nsp/

;p<

This yields the following:

Proposition 2.4. If (2.23) is a quadratic form in ru, then the PDE

(2.25)

D u C B.u/.ru; ru/; u.0/ D f;

has a solution in C.Œ0; T ; H

1;p

/ \ C

..0; T /  M/,providedf 2 H

1;p

.M /,

p>n.

The smoothness is established by the same sort of arguments as described

before. Of course, the proof of Proposition 2.4 yields persistence of solutions as

long as ku.t/k

1;p

is bounded for some p>n.

We mention further results on harmonic maps. First, in the setting of

Theorem 2.1,thatis,whenN has negative sectional curvature, any harmonic

map is energy minimizing in its homotopy class, a fact that makes Proposition 2.3

superﬂuous. An elegant proof of this fact can be found in [Sch]. It is followed by a

proof of a uniqueness result of P. Hartman, which says that under the hypotheses

of Theorem 2.1, any two homotopic harmonic maps coincide, unless both have

rank  1.

Theorem 2.1 does not extend to arbitrary N . For example, it was established

by Eells and Wood that if v 2 C

/ has degree 1,thenv is not homotopic

to a harmonic map. Among positive results not contained in Theorem 2.1,we

mention a result of Lemaire and Sacks–Uhlenbeck that if 

.N / D 0 and dim

330 15. Nonlinear Parabolic Equations

M D 2,thenanyv 2 C

.M; N / is homotopic to a smooth harmonic map. If

dim M  3, there are nonsmooth harmonic maps, and there has been considerable

work on the nature of possible singularities. Details on matters mentioned in this

paragraph, and further references, can be found in [Hild, J1, Str, Str2]. We also

refer to [Ham] for extensions of Theorem 2.1 to cases where M and N have

boundary.

In case M and N are compact Riemann surfaces of genus  2 (endowed with

metrics of negative curvature, as done in  2 of Chap. 14), harmonic maps of de-

gree 1 are unique and are diffeomorphisms, as shown by R. Schoen and S.-T. Yau.

They measure well the degree to which M and N may fail to be conformally

equivalent, and they provide an excellent analytical tool for the study of Teich-

muller theory, replacing the more classical use of “quasi-conformal maps.” This

material is treated in [Tro].

We mention some other important geometrical results attacked via parabolic

equations. R. Hamilton [Ham2] obtained topological information on 3-manifolds

with positive Ricci curvature and in [Ham3] provided another approach to the

uniformization theorem for surfaces, an approach that works for the sphere as well

as for surfaces of higher genus; see also [Chow]. S. Donaldson [Don] constructed

Hermitian–Einstein metrics on stable bundles over compact algebraic surfaces;

see [Siu] for an exposition. Some facets of the Yamabe problem were treated via

the “Yamabe ﬂow” in [Ye].

Hamilton’s Ricci ﬂow equation

D2 Ric.g/

is a degenerate parabolic equation, but D. DeTurk [DeT] produced a strongly

parabolic modiﬁcation, which ﬁts into the framework of  7 of this chapter, giv-

ing short time solutions. Solutions typically develop singularities, and there has

been a lot of work on their behavior. Work of G. Perelman, [Per1]–[Per3], was

a tremendous breakthrough, greatly reﬁning understanding of the Ricci ﬂow and

using this to prove the Poincar´e Conjecture and Thurston’s Geometrization Con-

jecture, for compact 3-dimensional manifolds. This work has generated a large

additional body of work, quite a bit of it devoted to giving more digestible pre-

sentations of Perelman’s work. We refer to [CZ]and[MT] for such presentations,

and other references.

Exercises

For Exercises 1–3, choose local coordinates x near a point p 2 M and local coordi-

nates y near q D u.p/ 2 N . Then the energy density is given by

(2.26) e.t; x/ D





.x/h



.u.t; x//;

where u.x/ D



.x/; ; u

.x/



in the y-coordinate system, n D dim N . Here,

and h



deﬁne the metrics on M and N , respectively, and we use the summation

Exercises 331

convention, here and below. Assume the coordinate systems are normal at p and q,

respectively.

1. Using these coordinate systems, show that the PDE (2.9) takes the form

(2.27)



D g



 g

k` M





C g

k` N











;

where



and







are the connection coefﬁcients of M and N , respectively.

2. Differentiating (2.27), show that, at p,

(2.28)













˛





˛





3. Using (2.28), show that, at p,

(2.29)



C e D



















































Obtain the identity (2.11) by showing that this is equal, at p,to

Ric















To deﬁne

u.x/,letE ! M denote the pull-back u



TN, with its pulled-back

connection

r.ToDu W TM !;TN we associate Du 2 C

.M; T



˝ E/.Ifr

denotes the product connection on T



M ˝ E,wehave

(2.30)

u Dr

Du 2 C

.M; T



˝ T



˝ E/:

Compare the construction of second covariant derivatives in Chap. 2,  3,and

Appendix C,  2.

If N  R

,letF ! M be the pull-back uT



, with its pulled-back (ﬂat) con-

nection r

.WehaveD u 2 C

.M; T



˝ E/  C

.M; T



˝ F/and

(2.31) r

u Dr

Du 2 C

.M; T



˝ T



˝ F/;

obtained by taking the Hessian of u componentwise.

4. Show that

(2.32)

u.X; Y / D P

u.X; Y /;

where P

W F ! E is orthogonal projection on each ﬁber. Parallel to (2.7), produce

aformula

332 15. Nonlinear Parabolic Equations

(2.33) r

u D

u CG.u/.ru; ru/; orthogonal decomposition.

Relate G.u/ to the second fundamental form of N  R

; show that

(2.34) G.u/.ru; ru/.X; Y / D II



Du.x/X; Du.x/Y



5. Suppose N is a hypersurface of R

,givenbyN Dfx 2 R

W '.x/ D C g, with

r' ¤ 0 on N . Show that .u/.ru; ru/ in (2.7) is given in this case by

(2.35) .u/.ru; ru/ D

u.x/  D



u.x/



 @

u.x/



u.x/





u.x/



Compare with the geodesic equation (11.14) in Chap. 1.

6. If dim M D 2, show that the energy E.u/ givenby(2.1)ofasmoothmapu W M ! N

is invariant under a conformal change in the metric of M , that is, under replacing the

metric tensor g on M by g

D e

g, for some real-valued f 2 C

.M /.

7. Show that any isometry w W M ! N of M onto N is a harmonic map.

8. Show that if dim M D dim N D 2 and w W M ! N is a conformal diffeomorphism,

then it is harmonic. (Hint: Recall Exercise 6.)

9. If u W M ! N is an isometry of M onto a submanifold

M  N that is a minimal

submanifold, show that u is harmonic.

10. If dim M D 2 and f W M ! N , show that

E.f /  Area



f.M/



;

with equality if and only if f is conformal.

3. Semilinear equations on regions with boundary

The initial-value problem

(3.1)

D u C F.t;x;u; ru/; u.0/ D f;

for u D u.t; x/, was studied in  1 for x 2 M , a compact manifold without bound-

ary. Here we extend many of these results to the case where x 2

M , a compact

manifold with boundary. As in  1, we assume F is smooth in its arguments. We

will deal speciﬁcally with the Dirichlet problem:

(3.2) u D 0 on R

 @M:

There is an analogous development for other boundary conditions, such as

Neumann or Robin boundary conditions.

Recall that Propositions 1.1 and 1.1A were phrased on a very general level, so

a number of short-time existence results in this case follow simply by verifying

3. Semilinear equations on regions with boundary 333

the hypotheses (1.3)–(1.6), for appropriate Banach spaces X and Y on M .For

example, somewhat parallel to (1.10), consider X D C

.M/; Y D C.M/,

where, for j  0,weset

(3.3) C

.M/ Dff 2 C

.M/ W f D 0 on @M g:

In Proposition 7.4 of Chap.13, it is shown that e

t

is a strongly continuous semi-

group on C

.M/.Also,(7.52) of Chap. 13 gives

(3.4) ke

t

f k

.M/

 Ct

1=2

kf k

; for 0<t 1;

so we have the following:

Proposition 3.1. If f 2 C

.M/,then(3.1)–(3.2) has a unique solution

(3.5) u 2 C



Œ0; T /; C

.M/



;

for some T>0, estimable from below in terms of kf k

If we specialize to F independent of ru, hence look at

(3.6)

D Lu C F.t;x;u/; u.0/ D f;

we can take X D C

.M/; Y D C.M/, and, by arguments similar to those used

above, we obtain the following result:

Proposition 3.2. If f 2 C

.M/,then(3.6), (3.2) has a unique solution

(3.7) u 2 C



Œ0; T /; C.



for some T>0, estimable from below in terms of kf k

We can obtain further regularity results on solutions to (3.1)and(3.6), with

boundary condition (3.2), making use of regularity results for

(3.8)

D u C g.t; x/; u.t; x/ D 0 for x 2 @M;

established in Exercises 4–10 of Chap. 6,  1. To recall the result, let us set, for

k 2 Z

(3.9) H

.I M/ Dfu 2 L

.I M/ W @

u 2 L

.I; H

2k2j

.M //; 0  j  kg:

334 15. Nonlinear Parabolic Equations

The result is that if (3.8) holds on I  M , with I D Œ0; T

,then

(3.10) g 2 H

.I M/H) u 2 H

kC1

 M/;

for I

D Œ"; T

; " > 0.Takingg D F.t;x;u; ru/ for u in Proposition 3.1 and

g D F.t;x;u/ for u in Proposition 3.2, we have in both cases g 2 H

.I  M/,

whenever T

<T, and hence

(3.11) u 2 H

 M/;

in both cases. One also has higher order regularity. For simplicity, we restrict

attention to the setting of Proposition 3.2.

Proposition 3.3. Assume F is smooth in its arguments. The solution (3.7)of

(3.6), (3.2) has the property

(3.12) u 2 C



.0; T / 



Proof. In this case, we start with the implication

(3.13) u 2 C.I 

M/\ H

 M/ H) F.t;x;u/ 2 H

 M/;

as follows from the chain rule and Moser estimates, as in Proposition 3.9 in

Chap. 13. Applying (3.10)thengivesu 2 H

 M/. More generally,

(3.14) u 2 C.I 

M/\ H

 M/ H) F.t;x;u/ 2 H

 M/:

Repeated applications of this plus (3.10) then yield u 2 H

kC1

 M/for all k,

which implies (3.12).

Exercises

1. Work out results parallel to those presented in this section, when the Dirichlet boundary

condition (3.2) is replaced by Neumann or Robin boundary conditions.

2. Consider the 3-D Burger equation

(3.15) u

u D u; u.0; x/ D f.x/; u.t; x/ D 0; for x 2 @;

where u W R

  ! R

,and is a bounded domain in R

with smooth boundary.

Show that the set-up to prove local existence works, with

X D H

./; Y D L

3=2

./:

(Hint: Show that H

./  L

./  L

3=2

./ and D.

1=4

/  L

./, hence

t

f k

./

 Ct

3=4

kf k

3=2

./

; for 0<t 1:/

4. Reaction-diffusion equations 335

4. Reaction-diffusion equations

Here we study `  ` systems of the form

(4.1)

D Lu C X.u/; u.0/ D f;

where u D u.t; x/ takes values in R

;Xis a real vector ﬁeld on R

,andL is a

second-orderdifferential operator,which we assume to be a negative-semideﬁnite,

self-adjoint operator on L

.M /.WetakeM to be a complete Riemannian man-

ifold, of dimension n, often either R

or compact. The numbers n and ` are

unrelated. We do not assume L is elliptic, though that possibility is not precluded.

Such a system arises when “substances” S



;1   `, whose concentrations

are measured by u



, are simultaneously diffusing and interacting via a mechanism

that changes these quantities. Recall from the introduction to Chap. 11 the relation

between the quantity u



of S



and its ﬂux J



, in case S



is being neither created

nor destroyed. This generalizes to the identity



.t; x/ dV .x/ D

N  J



dS.x/ C





u.t; x/



dV.x/

if X



.u/ is a measure of the rate at which S



is created, due to interactions with

the “environment,” namely, with the other S



. Consequently, by the divergence

theorem,



Ddiv J



C X



.u/:

If we assume that each S



obeys a diffusion law independent of the other sub-

stances, of the form considered in Chap. 11, that is,



Dd



grad u



;

then we obtain the system (4.1), with L D D,whereD is a diagonal `` matrix

with diagonal entries d



 0; we allow the possibility d



D 0, which means S



not diffusing.

An example of the sort of system that arises this way is the Fitzhugh–Nagumo

system:

(4.2)

D D

C f.v/ w;

D ".v w/;

with

f.v/ D v.a  v/.v  1/:

336 15. Nonlinear Parabolic Equations

In this case,

(4.3) L D





Here D is a positive constant. This arose as a model for activity along the axon of

a nerve, with v and w related to the voltage and the ion concentration, respectively.

We will mention other examples later in this section. While we will mention what

various examples model, we will not go into the mechanisms behind the models.

Excellent discussions of all these models and more can be found in [Mur].

One property L in (4.3) has is the following generalization of the maximum

principle:

Invariance property. There is a compact, convex neighborhood K of the origin

in R

such that if f 2 L

.M /, then, for all t  0,

(4.4) f.x/ 2 K for all x H) e

f.x/ 2 K for all x:

Thus, if f; g 2 L

.M / have compact support,

(4.5) ke

f k

 kf k

;

with  independent of t  0. If we deﬁned a norm on R

so that K \ .K/ was

the unit ball, would have  D 1. Note that, for such f and g,wehave

(4.6) j.e

f; g/jDj.f; e

g/jkf k

kgk

;

so ke

f k

 kf k

. Thus e

has a unique extension to a linear map

(4.7) e

W L

.M / ! L

.M /; ke

k

;

in case p D 1, hence, by interpolation, for 1  p  2, and, by duality, for

2  p 1, uniqueness for p D1holding in the class of operators whose

adjoints preserve L

.M /.

As mentioned above, in many examples of reaction-diffusion equations,

L D DL

,whereD is a diagonal `  ` matrix, with constant entries d

 0,

and L

is a scalar operator, generating a diffusion semigroup on L

.M /; in fact,

often M D R and L

D @

=@x

.ForsuchL, any rectangular region of the form

K Dfy 2 R

W a

 y

 b

g has the invariance property (4.4). If some of the

diagonal entries d

coincide, there will be a somewhat larger set of such invariant

regions.

We apply the technique of  1 to obtain solutions to (4.1), rewritten as the

integral equation

(4.8) u.t/ D e

f C

.ts/L



u.s/



ds:

4. Reaction-diffusion equations 337

Proposition 4.1. Let V be a Banach space of functions on M with values in R

such that

(4.9) e

W V ! V is a strongly continuous semigroup, for t  0;

and

(4.10) X W V ! V is Lipschitz, uniformly on bounded sets,

where .Xf /.x/ D X.f .x//.Then(4.8) has a unique solution

u 2 C



Œ0; T ; V



;

where T>0is estimable from below in terms of kf k

The proof is simply a specialization of that used for Proposition 1.1. Note that

(4.10) holds for a variety of spaces, such as V D L

.M; R

/; V D C.M;R

when X is a vector ﬁeld on R

satisfying

(4.11) kX.y/kC

; krX.y/kC

; 8 y 2 R

;

provided M is compact. If M has inﬁnite volume, you also need X.0/ D 0 for

V D L

.M; R

/ to work. Whenever X has this property, and L satisﬁes the

invariance property (4.4), it follows that Proposition 4.1 applies, for initial data

f 2 L

.M; R

/; 1  p<1. If, in addition, e

W C.M/ ! C.M/,wealso

have short-time solutions to (4.8)forf 2 C.M;R

/. For example, if M D R

and L has constant coefﬁcients, then (4.7) implies

(4.12) e

W H

k;p

/ ! H

k;p

/; k > 0:

Also

(4.13) e

W C

/ ! C

for t  0,sinceC

/, the space of continuous functions vanishing at inﬁnity,

is the closure of H

k;2

/ in L

/,fork>n=2.

Another useful example when M D R

is the space

(4.14) BC.R

/ Dff 2 C.R

/ W f extends continuously to

where

is the compactiﬁcation of R

via the sphere at inﬁnity (approached

radially). For k 2 Z

,wesayf 2 BC

/ provided D

f 2 BC .R

/ whenever

j˛jk.

If M D R

and (4.12) holds, then Proposition 4.1 applies with V D

k;p

; R

/ whenever the vector ﬁeld X and all its derivatives of order k