Well-posedness for the heat flow of polyharmonic maps with rough initial data

0.18 MB·English

by  Tao Huang Changyou Wang

Well-posedness for the heat flow of polyharmonic maps with
rough initial data
Tao Huang∗ Changyou Wang∗
0
1
0
2
Abstract

n
a Weestablishbothlocalandglobalwell-posednessoftheheatflowofpolyharmonic
J maps from Rn to a compact Riemannian manifold without boundary for initial data

5 with small BMO norms.
2

]
P
1 Introduction
A
.
h For k ≥ 1, let N be a k-dimensional compact Riemannian manifold without boundary,
at isometrically embedded in some Euclidean space Rl. For n ≥ 2 and m ≥ 1, we consider
m the m-th order energy functional
[
1 E (u) = 1 |∇mu|2 = 21 Rn|∆m2 u|2 if m is even
8v m 2ZRn (12 RRn|∇∆m2−1u|2 if m is odd
8
4 for any u∈ Wm,2(Rn,N), where ∆ is the LRaplace operator on Rn and
4
.
1 Wm,2(Rn,N) = v ∈Wm,2(Rn,Rl) : v(x) ∈ N for a.e. x ∈ Ω .
0
0 n o
1 Recall that a map u∈ Wm,2(Rn,N) is called a polyharmonic map if u is the critical point
v: of Em. The Euler-Lagrange equation of polyharmonic maps is (see Gastel-Scheven [9]):
i
X
m−2
m−1
r (−1)m∆mu = F(u) :=(−1)mdivm ∇m−k−1(Π(u))∇k+1u
a k
!
k=0 (cid:18) (cid:19)
X (1.1)
m−1
m
− (−1)k divk ∇m−k(Π(u))∇mu
k
Xk=0 (cid:18) (cid:19) (cid:16) (cid:17)
where Π : N → N is the nearest point projection from the δ-neighborhood of N to N,
δ
which is smooth provide δ = δ(N) > 0 is sufficiently small. It is readily seen that (1.1)
becomes the equation of harmonic maps for m = 1, and of extrinsic biharmonic maps for
m = 2.
Motivated by the study of heat flow of harmonic and biharmonic maps, we consider
the heat flow of polyharmonic maps, i.e. u: Rn×R → N solves
+
∗Department of Mathematics, University of Kentucky,Lexington, KY 40506
1
u +(−1)m∆mu=F(u) in Rn×(0,+∞) (1.2)
t
u =u on Rn, (1.3)
t=0 0
where u0 :Rn → N is a given map. (cid:12)(cid:12)
The heat flow of harmonic maps, (1.2) for m = 1, has been extensively studied. For
smooth initial data, the existence of global smooth heat flow of harmonic maps has been
established by (i) Eells-Sampson [7] under the assumption that the sectional curvature
K ≤ 0, and (ii) Hildebrandt-Kaul-Widman [11] under the assumption that the image
N
of u is contained in a geodesic ball B in N with radius R < π . In general,
0 R
2 max|KN|
rBR
the short time smooth heat flow of harmonic maps may develop singularity at finite time,
see Coron-Ghidaglia [4], Chen-Ding [2], and Chang-Ding-Ye [3]. However, Chen-Struwe
[6] (see also Chen-Lin [5] and Lin-Wang [19]) proved the existence of partially smooth,
global weak solutions to (1.2)-(1.3) for smooth initial data u . For rough initial data u ,
0 0
the second author recently proved in [25] the well-posedness for the heat flow of harmonic
maps provided the BMO norm of u is small.
0
Whenm = 2,(1.2)becomestheheatflowofextrinsicbiharmonicmaps,whichwasfirst
studied by Lamm in [15, 16, 17]. In particular, it was proven in [15, 16, 17] that if n = 4
and ku0kW2,2(R4) is sufficiently small, then there exists a unique global smooth solution.
For an arbitrary u ∈ Wm,2(R2m), it was later independently proved by Wang [23] (for
0
m = 2) and Gastel [8] (for m ≥ 2) that there exists a global weak solution to (1.2)-(1.3)
that is smooth away from finitely many singular times. Very recently, the second author
established in [24] the well-posedness for the heat flow of biharmonic maps for u with
0
small BMO norm.
We would like to mention that there have been some works on the regularity of poly-
harmonic maps for m ≥ 3 in the critical dimensions n = 2m. We refer the readers to
Gastel-Scheven [9], Lamm-Wang [18], Goldstein-Strzelecki-Zatorska-Goldstein[10], Moser
[20], and Angelsberg-Pumberger [1].
In this paper, we are interested in the well-posedness of the heat flow of polyharmonic
maps with rough initial data. In particular, we aim to extend the techniques from [25, 24]
to establish the well-posedness of the heat flow of polyharmonic maps (1.2) and (1.3) for
m ≥ 3 with u having small BMO norm.
0
We remark that the techniques employed by Wang [25, 24] were motivated by the
earlier work by Koch and Tataru [14] on the global well-posedness of the incompressible
Navier-Stokes equation, and the recent work by Koch-Lamm [13] on geometric flows with
rough initial data.
We first recall the BMO spaces. For x ∈ Rn and r > 0, let B (x) ⊂ Rn be the ball
r
with center x and radius r. For f :Rn → R, let f be the average of f over B (x).
x,r r
Definition 1.1 For f : Rn → R and R > 0, define
BMO (Rn)= f : Rn → R| [f] := sup r−n |f −f | < +∞ .
R BMO(Rn) x,r
( x∈Rn,0<r≤R ZBr(x) )
When R = +∞, we simply write BMO(Rn) for BMO (Rn).
∞
2
For 0 < T ≤ ∞, define the functional space X by
T
XT := (f :Rn×[0,T] → Rl | kfkXT := 0<sut≤pTkfkL∞(Rn)+[f]XT), (1.4)
where
m
[f]XT = Xk=1{0<sut≤pT t2kmk∇kfkL∞(Rn)+x∈Rn,0s<upr≤T21m(r−nZPr(x,r2m)|∇kf|2km)2km} (1.5)
whereP (x,r2m)= B (x)×[0,r2m]. It is clear that (X ,k·k ) is a Banach space. When
r r T XT
T = +∞, we simply write X, k·k , and [·] for X , k·k , and [·] respectively.
X X ∞ X∞ X∞
The main theorem is
Theorem 1.2 There exists an ε > 0 such that for any R > 0 if u : Rn → N has
0 0
[u0]BMOR(Rn) ≤ ε0, then there exists a unique global solution u : Rn × [0,R2m] → N to
(1.2) and (1.3) with small semi-norm [u] .
XR2m
As a direct consequence, we have
Corollary 1.3 There exists an ε0 > 0 such that if u0 : Rn → N has [u0]BMO(Rn) ≤ ε0,
then there exists a unique global solution u : Rn×R → N to (1.2) and (1.3) with small
+
semi-norm [u] .
X
We follow the arguments in [25, 24] very closely. The paper is written as follows. In
section 2, we present some basic estimates on the polyharmonic heat kernel. In section
3, we present some crucial estimates on the polyharmonic heat equation. In section 4, we
prove Theorem 1.2.
2 The polyharmonic heat kernel
In this section, we will prove some basic properties on the polyharmonic heat kernel.
The fundamental solution of the polyharmonic heat equation:
b (x,t)+(−1)m∆mb(x,t) = 0 in Rn×R (2.1)
t +
is given by
x
b(x,t) =t−2nmg , (2.2)
1
(cid:18)t2m(cid:19)
where
g(x) = (2π)−n2 eix·ξ−|ξ|2mdξ, x ∈ Rn. (2.3)
Rn
Z
It is easy to see that g is smooth, radial, and
Proposition 2.1 For any L ≥ 0,k ≥ 0, there exists C = C(k,L)> 0 such that
|∇kg(x)| ≤ C(1+|x|)−L, ∀x ∈ Rn. (2.4)
3
Proof. For k ≥ 0 and L ≥ 0, since
∇k eix·ξ = (iξ)k(ix)−L∇L eix·ξ ,
x ξ
(cid:16) (cid:17) (cid:16) (cid:17)
we have, by integration by parts,
|∇kg(x)| = (ix)−Leix·ξ∇L (iξ)ke−|ξ|2m dξ
ξ
≤ (cid:12)(cid:12)(cid:12)CZ(Rkn,L)(1+|x|)−L.(cid:16) (cid:17) (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
This completes the proof. 2
As adirectconsequenceof (2.4), wehave thefollowing propertiesfor thepolyharmonic
heat kernel b
Lemma 2.2 For any k,L ≥ 0, there exist C > 0 depending on n,L and C ,C > 0
1 2 3
depending on n,k,L such that for any x ∈ Rn and t > 0, it holds:
|x| −L
|b(x,t)| ≤ C1t−2nm 1+ 1 , (2.5)
(cid:18) t2m(cid:19)
|∇kb(x,t)| ≤ C2 t−21m n+k−L t21m +|x| −L, (2.6)
(cid:16) (cid:17) (cid:16) (cid:17)
k∇kb(x,t)kL1(Rn) ≤ C3t−2km. (2.7)
At the end of this section, we recall that the solution to the Dirichlet problem of
inhomogeneous polyharmonic heat equation
u (x,t)+(−1)m∆mu(x,t) =f(x,t) in Rn×R , (2.8)
t +
u(x,0) =u (x) on Rn (2.9)
0
is given by the following Duhamel formula:
u = Gu +Sf, (2.10)
0
where
Gu (x,t) := b(x−y,t)u (y)dy, (x,t) ∈ Rn×R , (2.11)
0 0 +
Rn
Z
and
t
Sf(x,t) := b(x−y,t−s)f(y,s)dyds, (x,t) ∈ Rn×R . (2.12)
+
Z0 ZRn
4
3 Basic estimates for the polyharmonic heat equation
In this section, we will provide some crucial estimates for the solution of the polyharmonic
heat equation with initial data in BMO spaces.
Lemma 3.1 For 0 < R ≤ +∞, if u ∈ BMO (Rn), then uˆ := Gu satisfies
0 R 0 0
m
sup r−n r2k−2m|∇kuˆ |2 ≤ C[u ]2 , (3.1)
k=1x∈Rn,0<r≤R ZPr(x,r2m) 0 0 BMOR(Rn)
X
and
m
Xk=10<st≤uRp2mt2km (cid:13)∇kuˆ0(t)(cid:13)L∞(Rn) ≤ C[u0]BMOR(Rn). (3.2)
If, in addition, u0 ∈ L∞(Rn) then (cid:13)(cid:13) (cid:13)(cid:13)
m−1
k=1 x∈Rns,u0<pr≤Rr−nZPr(x,r2m)|∇kuˆ0|2km ≤ Cku0kL2km∞−(R2n)·[u0]2BMOR(Rn), (3.3)
X
The proof of Lemma 3.1 is similar to [24] Lemma 3.1. For completeness, we sketch it
here. Let S denote the class of Schwartz functions, the following characterization of BMO
spaces, due to Carleson, is well-known (see, Stein [21]).
Lemma 3.2 For 0 < R ≤ +∞, let Φ ∈ S be such that Φ = 0 and denote for t > 0,
Rn
Φ (x) = t−nΦ(x), x ∈ Rn. If f ∈ BMO (Rn), then
t t R R
r dxdt
sup r−n |Φ ∗f|2(x,t) ≤ C[u ] (3.4)
x∈Rn,0<r≤R Z0 ZBr(x) t t 0 BMOR(Rn)
for some C = C(n)> 0.
Proof of Lemma 3.1. Let g be given by (2.3) and Φi = ∇ig for i = 1,··· ,m. Then Φi ∈S
and Φi = 0 for i = 1,··· ,m. Direct calculations show
Rn
R Φi(x) = t−n(∇ig) x = ti∇i t−ng(x) = ti∇ig (x),
t t t t
(cid:16) (cid:17) (cid:16) (cid:17)
where g (x) = t−ng(x). Hence we have
t t
Φi ∗u (x) = ti∇i(g ∗u )(x).
t 0 t 0
Since the polyharmonic heat kernel b(x,t) = g (x), we have
1
t2m
Φi ∗u (x) = ti∇i[(b(·,t2m)∗u )(x)] = ti∇i(Gu )(x,t2m).
t 0 0 0
Thus Lemma 3.1 implies that for i =1,··· ,m,
r dxdt
C[u ]2 ≥ sup r−n |Φi ∗u |2
0 BMOR(Rn) x∈Rn,0<r≤R Z0 ZBr(x) t 0 t
r
= sup r−n t2i−1|∇iGu |2(x,t2m)dxdt
0
x∈Rn,0<r≤R Z0 ZBr(x)
= 1 sup r−n t2i2−m2m|∇iGu0|2(x,t)dxdt.
2m x∈Rn,0<r≤R ZPr(x,r2m)
5
This clearly implies (3.1), since for i = 1,··· ,m, t2i2−m2m ≥ r2i−2m when 0< t ≤ r2m.
Since uˆ solves the polyharmonic heat equation:
0
(∂ +(−1)m∆m)uˆ = 0 on Rn×(0,+∞),
t 0
the standard theory implies that for any x ∈ Rn and r > 0,
m m
rmk |∇kuˆ0|2(x,r2m)≤ C r−n r2k−2m|∇kuˆ0|2.
k=1 k=1 ZPr(x,r2m)
X X
Taking supremum over x ∈Rn and 0 < t = r2m ≤R2m yield (3.2).
For (3.3), observe that u ∈ L∞(Rn) implies Φi ∗u ∈ L∞(Rn) for i = 1,··· ,m−1,
0 t 0
and
kΦit∗u0kL∞(Rn) ≤ kΦikL1(Rn)ku0kL∞(Rn) ≤ k∇igkL1(Rn)ku0kL∞(Rn) ≤ Cku0kL∞(Rn).
Hence
sup r−n |∇iGu0|2mi dxdt
x∈Rn,0<r≤R ZPr(x,r2m)
dxdt
= x∈Rns,u0<pr≤Rr−nZBr(x)×[0,r]|Φit∗u0|2mi t
2mi −2 dxdt
≤ (cid:18)st>up0 kΦit∗u0kL∞(Rn)(cid:19)L∞(Rn)·x∈Rns,u0<pr≤Rr−nZBr(x)×[0,r]|Φit∗u0|2 t
≤ Cku k2mi −2 ·[u ]2
0 L∞(Rn) 0 BMOR(Rn)
This implies (3.3). 2
Now we prove an important estimate on the distance of uˆ to the manifold N in term
0
of the BMO norms of u . More precisely,
0
Lemma 3.3 For any δ > 0, there exists K = K (δ,N) > 0 such that for 0 < R ≤ +∞,
0 0
if u ∈ BMO (Rn) then
0 R
R
dist(uˆ0(x,t),N) ≤ K0[u0]BMOR(Rn)+δ, ∀x ∈ Rn, 0 ≤ t ≤ (K )2m. (3.5)
0
Proof. For any x ∈ Rn, t > 0 and K > 0, denote
cKx,t = |B 1(0)| u0(x−t21mz)dz.
K ZBK(0)
Let g be given by (2.3). Then, by a change of variables, we have
1
uˆ0(x,t) = g(y)u0(x−t2my)dy.
Rn
Z
6
Applying (2.5) (with L = n+1) from Lemma 2.2, we have
uˆ0(x,t)−cKx,t ≤ g(y) u0(x−t21my)−cKx,t dy
Rn
Z (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)≤ (cid:12)+ g(y) u(cid:12)0(x−t21my)−cKx,t dy
(ZBK(0) ZRn\BK(0)) (cid:12) (cid:12)
(cid:12) (cid:12)
≤ u0(x−t21my)−cKx,t (cid:12)dy (cid:12) (3.6)
ZBK(0)(cid:12) (cid:12)
(cid:12) (cid:12) 1
+Cku0(cid:12)kL∞(Rn)ZRn\BK(0) |(cid:12)y|n+1 dy
≤Kn[u ] +δ,
0 BMO 1 (Rn)
Kt2m
provided we choose a sufficiently large K = K (δ,N) > 0 so that
0
1
Cku0kL∞(Rn)ZRn\BK(0) |y|n+1 dy ≤ δ.
On the other hand, since u (Rn) ⊂ N, we have
0
dist(cKx,t,N) ≤ |B 1(0)| cKx,t−u0(x−t21my) dy ≤ [u0]BMO 1 (Rn). (3.7)
K ZBK(0)(cid:12) (cid:12) Kt2m
(cid:12) (cid:12)
Putting (3.6) and (3.7) together yie(cid:12)lds (3.5) holds for t(cid:12)≤ (R)2m. 2
K
4 Boundedness of the operator S
In this section, we introduce several function spaces and establish the boundedness of the
operator S between these spaces.
For 0 < T < ∞, the spaces Yk, for k = 0,··· ,m − 1, are the sets consisting of all
T
functions f : Rn×[0,T] → R such that
2m−k
2m
kfkYk := sup t2m2m−kkfkL∞(Rn)+ sup R−n |f|2m2m−k . (4.1)
T 0<t≤T x∈Rn,0<R≤T21m ZPR(x) !
Notice that (Yk,k ·k ) is a Banach space for k = 0,··· ,m −1. When T = +∞, we
T Yk
T
simply denote (Yk,k·k ) for (Yk,k·k ).
Yk ∞ Y∞k
Let the operator S be defined by (2.12). Then we have
Lemma 4.1 For any 0 < T ≤ +∞ and k = 0,··· ,m−1, if f ∈ Yk, then S(∇αf)∈ X
T T
and
kS(∇αf)k ≤ Ckfk , (4.2)
XT YTk
where α= (α ,···α ) is any multi-index of order k.
1 n
7
Proof. We need to show the point wise estimate
m
Ri|∇iS(∇αf)|(x,R2m) ≤ CkfkYk, ∀x ∈Rn, 0< R ≤ T21m, (4.3)
T
i=0
X
1
and the integral estimate for 0< R ≤ T2m:
m
R−2imn ∇iS(∇αf) L2mi (PR(x,R2m)) ≤ CkfkYTk (4.4)
i=1
X (cid:13) (cid:13)
By suitable scaling, we may ass(cid:13)ume T ≥ 1.(cid:13)Since both estimates are translation and scale
invariant, it suffices to show (4.3) and (4.4) hold for x = 0 and R = 1.
For i = 0,··· ,m and α = (α ,··· ,α ) with order k, we have
1 n
1
∇iS(∇αf) (0,1) = ∇i+αb(y,1−s)f(y,s)dyds
(cid:12)Z0 ZRn (cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12) 1 + 12 + 21 (cid:12)(cid:12)(cid:12)∇i+kb(y,1−s) |f(y,s)|dyds
(Z12 ZRn Z0 ZB2 Z0 ZRn\B2)(cid:12) (cid:12)
(cid:12) (cid:12)
=I +I +I .
1 2 3 (cid:12) (cid:12)
Applying Lemma 2.2, we can estimate I , I and I as follows.
1 2 3
1
|I1|≤ sup kf(s)kL∞(Rn) ∇i+kb(·,1−s) ds
21≤s≤1 ! Z21 (cid:13) (cid:13)L1(Rn) !
1 (cid:13) (cid:13)
≤CkfkYk 2 s−i2+mkds (by (2.7(cid:13))) (cid:13)
1
Z0
≤Ckfk (since i+k ≤2m−1).
Yk
1
|I2| ≤ sup k∇i+kb(·,1−s)kL∞(Rn) |f(y,s)|dyds
0≤s≤21 ! ZB2×[0,21] !
≤C |f(y,s)|dyds
ZB2×[0,21]
≤Ckfk .
Yk
1
1
|I | ≤ 2 ∇i+kb(y,1−s) |f(y,s)|dyds
3
Z0 ZRn\B2(cid:12) (cid:12)
1 (cid:12) (cid:12)
≤C 2 (cid:12) |y|−(n+1)|f(y,s(cid:12))|dyds (by (2.6) for L = n+1)
Z0 ZRn\B2
∞
≤ kn−1k−(n+1) sup |f(y,s)|dyds
k=2 ! x∈RnZP1(x,1) !
X
∞
≤C k−2 kfk ≤ kfk .
Yk Yk
! 1 1
k=2
X
8
Now we want to show (4.4) by the energy method. Denote w = S(∇αf). Then w
solves
(∂ +(−1)m∆m)w = ∇αf in Rn×(0,+∞); w| = 0. (4.5)
t t=0
Let η ∈ C∞(B ) be a cut-off function of B . Multiplying (4.5) by η4w and integrating
0 2 1
over Rn×[0,1], we obtain
|w|2η4+2 ∇mw·∇m(wη4)= 2 ∇αf ·wη4. (4.6)
ZR×{1} ZRn×[0,1] ZRn×[0,1]
By the H¨older inequality, we have
∇mw·∇m(wη4)
ZRn×[0,1]
m−1
= |∇m(wη2)|2+ ∇mw ∇β(wη2)·∇m−β(η2)
ZRn×[0,1] ZRn×[0,1] β=0 
X
  (4.7)
m−1
− ∇m(wη2) ∇β(w)·∇m−β(η2)
ZRn×[0,1] β=0 
X
 m−1 
1
≥ |∇m(wη2)|2−C |∇βw|2
2ZRn×[0,1] β=0ZB2×[0,1]
X
∇αf ·wη4 = (−1)k f ·∇α[(wη2)η2]
ZRn×[0,1] ZRn×[0,1]
k
≤C |f||∇β(wη2)|
β=0ZRn×[0,1]
X
k−1 1 (4.8)
≤C sup t2m2m−kkfkL∞(Rn)· sup t2βmk∇βwkL∞(Rn)· t−1+k2−mβdt
β=00<t≤1 0<t≤1 Z0
X
+Ckfk ·k∇k(wη2)k
2m 2m
L2m−k(B2×[0,1]) L k (Rn×[0,1])
≤Ckfk2 +Ckfk ·k∇k(wη2)k
Y1k Y1k L2km(Rn×[0,1])
To estimate the last term, we need the Nirenberg interpolation inequality: for k ≤ m−1,
k∇k(wη2)k2km ≤ Ckwη2k2km−2 k∇m(wη2)k2 ,
L2km(Rn) L∞(Rn) L2(Rn)
which, after integrating with respect to t ∈ [0,1], implies
k∇k(wη2)k ≤ C sup kwk1−mk k∇m(wη2)kmk , (4.9)
L2km(Rn×[0,1]) 0≤t≤1 L∞(Rn) L2(Rn×[0,1])
Putting (4.9), (4.7) and (4.8) into (4.6), we have
9
|∇m(wη2)|2
ZRn×[0,1]
m−1
≤C |∇βw|2 +Ckfk2 +Ckfk ·k∇k(wη2)k
β=0ZB2×[0,1] Y1k Y1k L2km(Rn×[0,1])
X
m−1 1
≤C t−mβ dt· sup (tmβ k∇βw(t)k2L∞(Rn))
β=0(cid:20)Z0 0<t≤1 (cid:21) (4.10)
X
+Ckfk2 +Ckfk sup kwk1−mk k∇m(wη2)kmk
Y1k Y1k0≤t≤1 L∞(Rn) L2(Rn×[0,1])
≤Ckfk2 + 1 |∇m(wη2)|2+Ckfkq ·kwk(1−mk)q
Y1k 2ZRn×[0,1] Y1k L∞(Rn)
1
≤ |∇m(wη2)|2 +Ckfk2 ,
2 ZRn×[0,1] Y1k
where q = 2m . Therefore, we obtain
2m−k
|∇mw|2 ≤ |∇m(wη2)|2 ≤Ckfk2 . (4.11)
Yk
ZP1(0,1) ZRn×[0,1] 1
For i = 1,··· ,m−1, applying Nirenberg’s interpolation inequality gives
|∇iw|2mi ≤ |∇i(wη2)|2mi
ZP1(0,1) ZRn×[0,1]
≤ sup kwk2mi −2 k∇m(wη2)k2
L∞(Rn) L2(Rn×[0,1])
0≤t≤1 (4.12)
≤Ckfk2mi −2· |∇m(wη2)|2
Yk
1 ZRn×[0,1]
2m
≤Ckfk i
Yk
1
(4.11) and (4.12) imply (4.4). This completes the proof. 2
5 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. The idea is based on the fixed point
theorem in a small ball inside X .
R2m
Sincetheimage of amapu ∈ X may notbecontained inN, wefirstneedtoextend
R2m
Π to Rl, denoted as Π, such that Π∈ C∞(Rl) and Π≡ Π in N .
δN
Let
e e e
m−2
m−1
F(u) :=(−1)mdivm ∇m−k−1(Π(u))∇k+1u
k
!
k=0 (cid:18) (cid:19)
X (5.1)
e m−1 e
m
− (−1)k divk ∇m−k(Π(u))∇mu .
k
Xk=0 (cid:18) (cid:19) (cid:16) (cid:17)
e
10