Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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358 15. Nonlinear Parabolic Equations
Thus ˇ.t/ Dkw.t/k
V
satisfies the integral inequality
(5.35) ˇ.t/ A
Z
t
0
e
c.t/
ˇ./ d C B n
ı
t
1
;ˇ.0/D 0;
where A and B depend on kf k
V
. The conclusion (5.32) follows from Gronwall’s
inequality.
As one simple example of useful Banach spaces, we consider
(5.36) V D BC
1
.R
n
/; W D BC
0
.R
n
/;
where, as in (4.14), BC
k
.R
n
/ denotes the space of functions whose derivatives
of order k are bounded and continuous on R
n
and extend continuously to the
compactification
c
R
n
via the sphere at infinity (approached radially). This is a sub-
space of the space BC
k
.R
n
/, consisting of functions whose derivatives of order
k are bounded and continuous on R
n
. We want the functions to take values
in R
`
, but we suppress that in the notation. Suppose L is a constant-coefficient,
second-order, elliptic, self-adjoint operator. If (5.1)isan` ` system, let us hy-
pothesize the invariance property (4.4), so, with an appropriate norm on R
`
,we
have that e
tL
is a contraction semigroup on V (and on W ). Furthermore, the other
estimates in (5.28) hold in this case, with D ı D 1=2.
To investigate the estimate (5.30), we have
(5.37)
kF
t
f k
BC
1
DkF
t
f k
L
1
CkD.F
t
f/k
L
1
D sup
x
F
t
X
f.x/
R
`
C sup
x
DF
t
.f / ı Df . x /
R
`
:
Thus (5.30) holds, provided
(5.38) kF
t
X
.y/k
R
`
e
ct
kyk
R
`
;0 t T;
and
(5.39) kDF
t
X
.y/k
L.R
`
/
e
ct
;0 t T:
As shown in 6 of Chap.1, DF
t
X
.y/ D G
t
.y/ satisfies the linear ODE
(5.40)
d
dt
G
t
.y/ D DX
F
t
X
.y/
G
t
.y/; G
0
.y/ D I:
Consequently, it is clear that (5.38)and(5.39) hold as long as X is a C
1
-vector
field on R
`
, satisfying
(5.41) kX.y/k
R
`
c; kDX.y /k
L.R
`
/
c:
5. A nonlinear Trotter product formula 359
These conditions are also enough to yield the boundedness of the maps X W V !
V and Y W V V ! L.W /,in(5.31), but in order to have boundedness of
Y W V V ! L.V /, we need C
1
-bounds on Y.f;g/, hence C
1
-bounds on DX .
In other words, we need X 2 BC
2
.R
`
/. We have the following result.
Proposition 5.2. Let u 2 C
Œ0; T ; BC
1
.R
n
/
solve (5.1), and let v.t/ be defined
by (5.4)–(5.5). Assume that L is a constant-coefficient,second-order, elliptic oper-
ator, generating a contraction semigroup on BC
0
.R
n
/ and that X is a vector field
on R
`
with coefficients in BC
2
.R
`
/. Then, for any bounded interval t 2 Œ0; T ,
(5.42) ku.t/ v.t/k
BC
1
C
kf k
BC
1
n
1=2
:
As another example of Banach spaces to which Proposition 5.1 applies,
consider
(5.43) V D H
k
.R
n
/; W D H
k2
.R
n
/; k >
n
2
;0<<1:
Assume k 2 Z
C
.Then(5.28) holds, with ı D . We have the Moser estimate
(5.44) kX.f /k
H
k
C
k
kf k
L
1
1 Ckf k
H
k
;
where
(5.45) C
k
./ D C
0
k
sup
˚
X
./
.f / Wjf j; jjk
:
Thus (5.31) is seen to hold as long as X 2 BC
k
.R
`
/. To see whether (5.30) holds,
we estimate .d=dt/kF
t
f k
2
H
k
, exploiting (5.44) to obtain
(5.46)
d
dt
kF
t
f k
2
H
k
D 2
X.F
t
f/;F
t
f
H
k
C
k
kF
t
f k
L
1
kF
t
f k
H
k
CkF
t
f k
2
H
k
:
Now for kF
t
f k
H
k
1, the right side is 2C
k
kF
t
f k
L
1
kF
t
k
2
H
k
.IfX 2
BC
k
.R
`
/, there is a bound on 2C
k
kF
t
f k
L
1
strong enough to yield (5.30).
We have the following result:
Proposition 5.3. Assume k>n=2is an integer. Let u 2 C
Œ0; T ; H
k
.R
n
/
solve (5.1), and let v.t/ be defined by (5.4)–(5.5). Assume that L is a constant-
coefficient, second-order, elliptic operator, generating a contraction semigroup on
L
2
.R
n
/, and that X is a vector field on R
`
with coefficients in BC
k
.R
`
/. Then,
for any bounded interval t 2 Œ0; T ,
(5.47) ku.t/ v.t/k
H
k
C
kf k
H
k
n
;
360 15. Nonlinear Parabolic Equations
for any <1. Furthermore, for any ">0,
(5.48) ku.t/ v.t/k
H
k"
C
"
kf k
H
k
n
1
:
It remains to establish (5.48). Indeed, if we set
e
W D H
k2
.R
n
/, we easily get
kR.t/k
e
W
Cn
1
,viauseofke
tL
I k
L.V;
e
W/
ct, instead of the last estimate
of (5.28). Then we can use
e
V D H
k"
.R
n
/, replacing the second estimate of
(5.28)byke
tL
k
L.
e
W;
e
V/
C
"
t
.1"=2/
, and parallel the analysis in (5.33)–(5.35)
to obtain (5.48).
It is desirable to have product formulas for which the existence of solutions to
(5.1)isaconclusion rather than a hypothesis. Suppose that v,givenby(5.4)–(5.5),
is compared, not with the solution u to (5.1), but to the functionev, constructed by
the same process as v, but using intervals of half the length. Thus, for an integer
or half-integer k,define
(5.49) ev
k
D
e
.1=2n/L
F
1=2n
2k
.f /;
and set
(5.50) ev.t/ D e
sL
F
s
ev
k
; for t D
k
n
C s; 0 s
1
2n
:
Parallel to (5.6), we have
(5.51)
@ev
@t
D Lev C X.ev/C
e
R.t/;
where, for t D k=n C s; 0 s < 1=2n,
(5.52)
e
R.t/ D .e
sL
I/X.F
s
ev
k
/ C
X.F
s
ev
k
/ X.e
sL
F
s
ev
k
/
:
Consequently, ew D v ev satisfies the PDE
(5.53)
@ew
@t
D Lew C
e
A.t; x/ew C R.t/
e
R.t/; ew.0/ D 0;
where, parallel to (5.11),
(5.54)
e
A.t; x/ D Y
ev.t;x/;v.t;x/
:
Pick Banach spaces V and W as above, and assume f 2 V . As long as the
hypotheses (5.28)–(5.31) hold, we again have
(5.55) kR.t/
e
R.t/k
W
Cn
ı
;0 t T:
We also have bounds on kv.t/k
V
and kevk
V
, independent of n, hence bounds on
e
A.t; x/, so the analysis in (5.33)–(5.35) extends to yield
5. A nonlinear Trotter product formula 361
(5.56) kv.t/ ev.t/k
V
Cn
ı
;0 t T:
Consequently, if we take n D 2
j
and denote v,definedby(5.4)–(5.5), by u
.j /
,
so ev is logically denoted u
.j C1/
,wehave
(5.57)
˚
u
.j /
W j 2 Z
C
is Cauchy in C
Œ0; T ; V
;
and the limit is seen to satisfy (5.1).
There are some unsatisfactory aspects of using the smoothing of e
tL
that fol-
lows when L is elliptic. For example, Propositions 5.1–5.3 do not apply to the
Fitzhugh–Nagumo system (4.2), since the operator L given by (4.3) is not elliptic.
We now derive a convergence result that does not make use of such a hypothesis;
the conclusion will be weaker, in that we get convergence in a weaker norm. We
will establish the following variant of Proposition 5.1:
Proposition 5.4. Let V and W be Banach spaces of (`-tuples of) functions for
which e
tL
satisfies the estimates
(5.58) ke
tL
k
L.V /
e
ct
; ke
tL
k
L.W /
e
ct
; ke
tL
I k
L.V;W /
Ct
ı
;
for 0<t T , with some ı>0.LetX be a vector field on R
`
, generating a
flow F
t
X
, whose action on functions via F
t
f.x/ D F
t
X
f.x/
satisfies (5.29)–
(5.31). Take f 2 V .Then(5.1) has a solution u 2 C
Œ0; T ; W
, and the function
v 2 C
Œ0; T ; V
given by (5.4)–(5.5) satisfies
(5.59) kv.t/ u.t /k
W
Cn
ı
;0 t T:
Proof. If v and ev are defined by (5.4)–(5.5) and by (5.49)–(5.50), we will show
that
(5.60) sup
0tT
kv.t/k
V
B;
with B independent of n,andthat
(5.61) kv.t/ ev.t/k
W
Cn
ı
;0 t T:
In fact, the hypotheses (5.58) together with (5.29)–(5.30) immediately yields
(5.60). If we also have (5.31), then there is the estimate
(5.62) kR.t/k
W
Cn
ı
; k
e
R.t/k
W
Cn
ı
;
established just as before. Again, ew D v ev solves the PDE (5.53), and hence,
parallel to (5.33)–(5.34), we have
362 15. Nonlinear Parabolic Equations
(5.63) kew.t/k
W
A
Z
t
0
e
c.t/
kew./k
W
d C C
0
n
ı
; kew.0/k
W
D 0:
Thus Gronwall’s inequality yields (5.61), and the proposition follows.
Note that in Proposition 5.4, we can weaken the hypothesis (5.31)to
(5.64) X W V ! V and Y W V V ! L.W / are bounded;
omitting mention of L.V /. Let us also note that the limit function u 2
C
Œ0; T ; W
also satisfies
(5.65) u 2 L
1
Œ0; T ; V
;
provided V is reflexive.
Proposition 5.4 essentially applies to the Fitzhugh–Nagumo system (4.2),
which we recall:
(5.66)
@v
@t
D D
@
2
v
@x
2
C f.v/ w;
@w
@t
D ".v w/:
As we did in 4, we modify the vector field X.v; w/ D
f.v/ w;".v w/
outside some compact set to keep its components and sufficiently many of their
derivatives bounded.
Exercises
1. Investigate Strang’s splitting method:
u.t/ D lim
n!1
F
t=2n
e
.t=n/L
F
t=2n
n
.f /:
Obtain faster convergence than that given by (5.48) for the splitting method (5.3).
2. Write a computer program to solve numerically the Fitzhugh–Nagumo system (4.2),
using the splitting method. Take M D S
1
.Use(4.32) to specify the constants ; a,and
". Alternatively, take D 10. Try various values of D. Use the FFT to solve the linear
PDE @v=@t D D@
2
x
v, and use a reasonable difference scheme to integrate the planar
vector field X.
6. The Stefan problem
The Stefan problem models the melting of ice. We consider the problem in one
space dimension. We assume that the point separating ice from water at time t is
given by x D s.t/, with water, at a temperature u.t; x/ 0, on the left, and ice, at
6. The Stefan problem 363
temperature 0, on the right. Let us also assume that the region x 0 is occupied
by a solid maintained at temperature 0. In appropriate units, u and s satisfy the
equations
(6.1)
u
t
D u
xx
; u.0; x/ D f.x/;
u.t; 0/ D 0; u.t; s.t// D 0;
where 0 x s.t/ and
(6.2) Ps Da u
x
.t; s.t//; s.0/ D 1:
We suppose f is given, in C
1
.I /; I D Œ0; 1, such that f.x/ 0 and f.0/ D
f.1/ D 0.In(6.2), a is a positive constant.
It is convenient to change variables, setting v.t; x/ Du.t; s.t/x/,for0 x 1.
The equations then become
(6.3)
v
t
D s.t/
2
v
xx
C
Ps
s
xv
x
;v.0;x/D f.x/;
v.t; 0/ D 0; v.t; 1/ D 0;
and
(6.4) Ps D
a
s
v
x
.t; 1/:
Note that (6.4) is equivalent to .d=dt/s
2
D2av
x
.t; 1/,soifweset.t/ D
s.t/
2
, we can rewrite the system as
v
t
D .t/
1
v
xx
C
1
2
P
xv
x
;v.0;x/D f.x/; v.t;0/ D v.t; 1/ D 0;(6.5)
P
.t/ D2a v
x
.t; 1/; .0/ D 1:(6.6)
Note that the system (6.5)–(6.6) is equivalent to the system of integral equations
v.t/ D e
˛.t;0/
f C
Z
t
0
ˇ./e
˛.t;/
xv
x
./
d;(6.7)
.t/ D 1 2a
Z
t
0
v
x
.; 1/ d ;(6.8)
where
(6.9) ˛.t; / D
Z
t
d
s./
2
D
Z
t
d
./
;ˇ./D
Ps./
s./
D
1
2
P
./
./
:
364 15. Nonlinear Parabolic Equations
Here, e
t
is the solution operator to the heat equation on R
C
I , with Dirichlet
boundary conditions at x D 0; 1.
We will construct a short-time solution as a limit of approximations as follows.
Start with
0
.t/ D 1 2af
0
.1/t.Solve(6.5)forv
1
.t; x/, with D
1
.Thenset
1
.t/ D 1 2a
R
t
0
@
x
v
1
.; 1/ d .Nowsolve(6.5)forv
2
.t; x/, with D
1
.Then
set
2
.t/ D 1 2a
R
t
0
@
x
v
2
.; 1/ d , and continue. Thus, when you have
j
.t/,
solve for v
j C1
.t; x/ the equation
(6.10)
@
@t
v
j C1
D
j
.t/
1
@
2
x
v
j C1
C
1
2
P
j
j
x@
x
v
j C1
;v
j C1
.0; x/ D f.x/;
v
j C1
.t; 0/ D v
j C1
.t; 1/ D 0:
Then set
(6.11)
j C1
.t/ D 1 2a
Z
t
0
@
x
v
j C1
.; 1/ d :
Lemma 6.1. Suppose
j
.t/ satisfies
(6.12)
j
.0/ D 1;
P
j
.0/ D2af
0
.1/;
P
j
0:
Then
j C1
also has these properties.
Proof. The first two properties are obvious from (6.11), which implies
(6.13)
P
j C1
.t/ D2a @
x
v
j C1
.t; 1/:
Furthermore, the maximum principle applied to (6.10) yields
(6.14) v
j C1
.t; x/ 0:
Since v
j C1
.t; 1/ D 0,wemusthave@
x
v
j C1
.t; 1/ 0.
The PDE (6.10)forv
j C1
is equivalent to
(6.15) v
j C1
.t/ D e
˛
j
.t;0/
f C
Z
t
0
ˇ
j
./e
˛
j
.t;/
x@
x
v
j C1
./
d;
where
(6.16) ˛
j
.t; / D
Z
t
d
j
./
;ˇ
j
./ D
1
2
P
j
./
j
./
:
One way to analyze e
t
on functions on I is to construct S
1
, the “double” of
I , and use the identity
6. The Stefan problem 365
(6.17) e
t
g D e
t
Og
;
where Og is the extension of g 2 L
2
.I / to Og 2 L
2
.S
1
/, which is odd with
respect to the natural involution on S
1
(i.e., the reflection across @I ), and G is
the restriction of G 2 L
2
.S
1
/ to I . It is useful to note that
(6.18) O W C
r
b
.I / ! C
r
.S
1
/; for 0 r<2;
where C
r
b
.I / is the subspace of u 2 C
r
.I / such that u.0/ D u.1/ D 0.Ifr D
1 C ; 0 < < 1,thenC
r
.I / D C
1;
.I /.Furthermore,
(6.19) O W C
1;1
b
.I / ! C
1;1
.S
1
/:
It is useful to note that
(6.20) e
˛
.@
x
g/ D @
x
e
˛
N
g if g 2 C
1
.I /;
where e
˛
N
is the solution operator to the heat equation on R
C
I , with Neumann
boundary conditions, as can be seen by taking the even extension of g to S
1
.
Hence (6.15) can be written as
(6.21)
v
j C1
.t/ D e
˛
j
.t;0/
f
C
Z
t
0
ˇ
j
./
@
x
e
˛
j
.t;/
N
M
x
e
˛
j
.t;/
v
j C1
./ d ;
where M
x
is multiplication by x. In analogy with (6.17), we have
(6.22) e
t
N
g D e
t
Eg
;
where Eg is the even extension of g to S
1
. In place of (6.18)–(6.19), we have
(6.23)
E W C
r
.I / ! C
r
.S
1
/; for 0 r<1;
E W C
0;1
.I / ! C
0;1
.S
1
/:
We now look for estimates on v
j C1
and
j C1
. The simplest is the uniform
estimate
(6.24) kv
j C1
.t/k
L
1
kf k
L
1
;
which follows from the maximum principle. Other estimates can be derived using
(6.15)and(6.21) together with such estimates as
(6.25) ke
t
gk
C
r
Ct
r=2
kgk
L
1
; ke
t
N
gk
C
r
Ct
r=2
kgk
L
1
;
366 15. Nonlinear Parabolic Equations
valid for any f 2 L
1
.I /,anyr>0;t2 .0; T , with C D C.r;T/. Hence,
using (6.21), we obtain, for 0<<1, an estimate
(6.26) kv
j C1
.t/k
C
ke
˛
j
f k
C
C A
j
.t/kf k
L
1
;
where
(6.27) A
j
.t/ D A
Z
t
0
ˇ
j
./˛
j
.t; /
.1C/=2
d:
Now, by (6.16), ˛
j
.t; /
j
.t/
1
.t /, granted that
P
j
0,so
(6.28)
A
j
.t/ A
j
.t/
.1C/=2
Z
t
0
P
j
./
j
./
.t /
.1C/=2
d
B
j
.t/
.1C/=2
h
sup
0t
P
j
./
i
t
.1/=2
:
Now we can apply (6.21) again to obtain, for 0<r<1; 0<<1; C r ¤ 1,
(6.29) kv
j C1
.t/k
C
Cr
ke
˛
j
f k
C
Cr
C A
jr
.t/ sup
0t
kv
j C1
./k
C
;
using
(6.30)
ke
t
gk
C
Cr
Ct
r=2
kgk
C
;r>0;2 Œ0; 2/; g 2 C
r
b
.I /;
ke
t
N
gk
C
Cr
Ct
r=2
kgk
C
;r>0;2 Œ0; 1/; g 2 C
r
.I /;
for t 2 .0; T .If C r D 1, it is necessary to replace C
1
by the Zygmund space
C
1
. Combining this with (6.26), we obtain, for
(6.31) N
jr
.t/ D sup
0t
kv
j
./k
C
r
;
the estimates
(6.32)
N
j C1;Cr
.t/ C
0
kf k
C
Cr
C C
1
A
jr
.t/kf k
C
C C
2
A
jr
.t/A
j
.t/kf k
L
1
:
Recall that
(6.33)
P
j
./ 2akv
j
./k
C
1
:
Hence
(6.34)
j
.t/ 1 C 2atN
j1
.t/;
6. The Stefan problem 367
where N
j1
.t/ is the case r D 1 of (6.31). Therefore, by (6.28),
(6.35) A
j
.t/ 2aB
h
1 C a.1 C /N
j1
.t/t
i
N
j1
.t/t
.1/=2
:
Consequently, taking r D 2 .1=2; 1/,so2 2 .1; 2/,wehave
(6.36) N
j C1;2
.t/ P
N
j1
.t/t; N
j1
.t/t
.1/=2
;
where P.X;Y/ is a polynomial of degree 4, with coefficients depending on such
quantities as kf k
C
2
, but not on j . A fortiori, we have
(6.37) N
j C1;1
.t/ P
N
j1
.t/t; N
j1
.t/t
.1/=2
:
Such an estimate automatically implies a uniform bound
(6.38) N
j1
.t/ K; for t 2 Œ0; T ;
for some T>0, chosen sufficiently small. Appealing again to (6.36), we conclude
furthermore that there are uniform bounds
(6.39) N
j;2
.t/ K
; for t 2 Œ0; T ; 2 < 2:
That is to say,
(6.40) fv
j
W j 2 Z
C
g is bounded in C
Œ0; T ; C
r
.I /
;r<2:
Taking r D 1, we conclude that
(6.41) f
j
W j 2 Z
C
g is bounded in C
1
Œ0; T
:
Of course, we know that each
j
.t/ is monotone increasing, with
j
.0/ D 1.
From (6.40)and(6.18), we have
(6.42) fOv
j
W j 2 Z
C
g bounded in C
Œ0; T ; C
r
.S
1
/
;r<2:
Also, fE .xv
j
/ W j 2 Z
C
g is bounded in C
Œ0; T ; C
0;1
.S
1
/
, so we deduce from
(6.10)that
(6.43) f@
t
.Ov
j
/ W j 2 Z
C
g is bounded in C
Œ0; T ; C
.2r/
.S
1
/
;r<2:
Interpolation with (6.42), together with Ascoli’s theorem, gives
(6.44) fOv
j
W2 Z
C
g compact in C
Œ0; T ; C
r2
.S
1
/
;