Table Of ContentTHE GRADIENT FLOW OF O’HARA’S KNOT ENERGIES
SIMONBLATT
Abstract. JunO’HarainventedafamilyofknotenergiesEj,p,j,p∈(0,∞),
in [O’H92]. We study the negative gradient flow of the sum of one of the
6 energies Eα=Eα,1,α∈(2,3),andapositivemultipleofthelength.
1 Showing that the gradients of these knot energies can be written as the
0 normal part of a quasilinear operator, we derive short time existence results
2 fortheseflows. Wethenprovelongtimeexistenceandconvergence tocritical
points.
n
a
J
2
1
1. Introduction
]
P Is there an optimal way to tie a knot in Euclidean space? And if so, how nice
A are these optimal shapes? Is there a natural way to transform a given knot into
this optimal shape?
.
h To give a precise meaning to such questions a variety of energies for immersions
t
a havebeeninventedandstudiedduringthelasttwentyfiveyearswhicharesubsumed
m under the term knot energies.
[ In this article we deal with the third of the questions above. But first of all, let
us gather some known answers to the first two questions.
1
The first family of geometric knot energies goes back to O’Hara. In [O’H92],
v
0 O’Hara suggested for j,p (0, ) the energy
∈ ∞
4
8 Ej,p(c)= 1 1 p c′(x) c′(y)dxdy
2 c(x) c(y)j − d (x,y)j | |·| |
0 (RZ/lZZ)2(cid:18)| − | c (cid:19)
.
1 of a regularclosedcurve c C0,1(R/lZ,Rn). Here, d (x,y) denotes the distance of
0 ∈ c
the points x and y along the curve c, i.e., the length of the shorter arc connecting
6
1 these two points.
: O’Hara observed that these energies are knot energies if and only if jp > 2
v
[O’H92, Theorem 1.9] in the sense that then both pull-tight of a knot and selfin-
i
X tersections are punished. Furthermore, he showed that minimizers of the energies
r exist within every knot class if jp > 2. So: Yes, there is an optimal way to tie a
a
knot – actually even several ways to do so.
Abrams et al proved in [ACF+03] that for p 1 and jp 1<2p these energies
≥ −
areminimizedbycirclesandthattheseenergiesareinfinite foreveryclosedregular
curve if jp 1 2p.1
− ≥
Thereis areasonwhy forthe restofourquestionswewillonlyconsiderthe case
p = 1: For p = 1, we expect that the first variation of Ej,p leads to a degenerate
6
elliptic operator of fractional order – even after breaking the symmetry of the
equation coming from the invariance under re-parameterizations. We will only
Date:January 13,2016.
2010 Mathematics Subject Classification. 53C44,35S10.
1Infactonecaneven showthat forjp−1≥2ptheenergy isonlyfiniteforopencurves that
arepartofastraightline-inwhichcasetheenergyis0(cf. [BR15,Remark1.3]).
1
2 SIMONBLATT
consider the non-degenerate case p=1 and look at the one-parameter family
1 1
(1.1) Eα(c):=Eα,1(c)= c′(x) c′(y)dxdy.
c(x) c(y)α − d (x,y)α | || |
RZ/LZZ(cid:18)| − | c (cid:19)
We leave the case p=1 for a later study.
6
Themostprominentmemberofthisfamily isE2 whichisalsoknownasMöbius
energy due to the fact that it is invariant under Möbius transformations. While
for α (2,3) the Euler-Lagrange equation is a non-degenerate elliptic sub-critical
∈
operator it is a critical equation for the case of the Möbius energy E2.
In [FHW94], Freedman, He, and Wang showed that even E2 can be minimized
within every prime knot class. Whether or not the same is true for composite
knot classes is an open problem, though there are some clues that this might not
be the case in very such knot class. Furthermore, they derived a formula for the
L2-gradient of the Möbius energy which was extended by Reiter in [Rei12] to the
energiesEα for α [2,3). They showedthatthe firstvariationofthese functionals
∈
can be given by
Eα(c+εh) Eα(c)
Eα(c):= lim − = Hα(c)(x),h(x) c′(x)dx
h
∇ ε→0 h h i·| |
R/ZlZ
where
(1.2) Hα(c)(x)
l
2
c(x+w) c(x) κ(x)
:=p.v. P⊥ 2α − (α 2)
c′(x) c(x+w) c(x)2+α − − d (x+w,x)α
Z (cid:26) | − | c
−l
2
κ(x)
2 c′(x+w)dw.
− c(x+w) c(x)α | |
| − | (cid:27)
Here, P⊥(u) = u u, c′ c′ denotes the orthogonal projection onto the normal
c′ −h |c′|i|c′|
l
part, and p.v 2 =lim denotes Cauchy’s principal value.
−l ε↓0 [−l/2,l/2]\[−ε,ε]
2
In the case that c is parameterized by arc length this reduces to
R R
(1.3) Hα(c)(x)
l
2
c(s+w) c(s) c′′(s)
:=p.v. P⊥ 2α − (α 2)
c′(x) c(s+w) c(s)2+α − − wα
Z (cid:26) | − | | |
−l
2
c′′(s)
2 dw.
− c(s+w) c(s)α
| − | (cid:27)
Using the Möbius invariance of E2, Freedman, He, and Wang showedthat local
minimizers of the Möbius energy are of class C1,1 [FHW94] – and thus gave a
firstanswertothequestionaboutthenicenessoftheoptimalshapes. Zheng-XuHe
combinedthiswithasophisticatedbootstrappingargumenttofindthatminimizers
oftheMöbiusenergyareofclassC∞ [He00]. Reitercouldprovethatcriticalpoints
c of Eα, α (2,3) with κ Lα are smooth embedded curves [Rei12], a result we
∈ ∈
extended to critical points of finite energy in [BR13] and to the Möbius energy in
[BRS15].
Let us now turn to the last of the three questions we started this article with:
Is there a natural way to transform a given knot into its optimal shape? Since Eα
is notscalinginvariantfor α (2,3),the L2-gradientflowofEα alonecannothave
∈
GRADIENT FLOW OF KNOT ENERGIES 3
a nice asymptotic behavior. We want to avoidthat the curve would get larger and
largerinordertodecreasetheenergy. Here,thelengthL(c)ofthe curvecwillhelp
us.
To transform a given knotted curve into a nice representative, we will look at
the L2-gradient flow of E = Eα+λL for α (2,3) and λ > 0 instead. This leads
∈
to the evolution equation
(1.4) ∂ c= Hα(c)+λκ .
t c
−
Wewillseethattherighthandsideofthisequationcanbewrittenasthenormal
part of a quasilinear elliptic but non-local operator of order α+1 [3,4).
∈
The main result of this article is the following theorem. Roughly speaking, it
tells us that, given an initial regular embedded curve of class C∞, there exists a
unique solution to the above evolution equations. This solution is immortal and
converges to a critical point. More precisely we have:
Theorem 1.1. Let α (2,3) and c C∞(R/Z,Rn) be an injective regular curve.
0
∈ ∈
Then there is a unique smooth solution
c C∞([0, ) R/Z)
∈ ∞ ×
to (1.4)withinitialdatac(0)=c thataftersuitablere-parameterizations converges
0
smoothly to a critical point of Eα+λL.
LinandSchwetlickshowedsimilarresultsfortheelasticenergyplussomepositive
multiple of the Möbius energy and the length [LS10]. They succeeded in treating
the term in the L2-gradient coming from the Möbius energy as a lower order per-
turbationofthegradientoftheelasticenergyofcurves. Thisallowedthemtocarry
over the analysis due to Dziuk, Kuwert and Schätzle of the latter flow [DKS02].
They proved long time existence for their flow and sub-convergence to a critical
point up to re-parameterizationsand translations.
The situation is quite different in the case we treat in this article. We have to
understandthegradientofO’Hara’senergiesinamuchmoredetailedwayandhave
to use sharper estimates than in the work of Lin and Schwetlick. Furthermore, in
contrastto Lin and Schwetlick we show that the complete flow, without going to a
subsequence and applying suitable translations,convergesto a criticalpoint of our
energy.
We want to conclude this introduction with an outline of the proof of Theo-
rem 1.1. In Section 2, we prove shorttime existence results of these flow for initial
data in little Hölder spaces and smooth dependence on the initial data. To do
that, weshow thatthe gradientofEα is the normalpartofanabstractquasilinear
differential operator of fractional order (cf. Theorem 2.3). Combining Banach’s
fixed-point theorem with a maximal regularity result for the linearized equation,
we get existence for a short amount of time. In order to keep this article as easily
accessible as possible, we give a detailed prove of the necessarymaximal regularity
result.
Themostimportantingredienttothe proofoflongtime existenceinSection 3.4
is a strengthening of the classification of curves of finite energy Eα in [Bla12a]
using fractional Sobolev spaces. For s (0,1), p [1, ) and k N the space
0
Wk+s,p(R/Z,Rn) consists of all function∈s f Wk,p∈(R/R∞,Rn) for w∈hich
∈
1/p
f(x) f(y)p
|fk|Ws,p := | x −y 1+sp| dxdy
RZ/ZRZ/Z | − |
is finite. This space is equipped with the norm f := f + f(k) .
k kWk+s,p k kWk,p | |Ws,p
For a thorough discussion of the subject of fractional Sobolev space we point the
4 SIMONBLATT
readertothemonographofTriebel[Tri83]. Chapter7of[AF03]andtheverynicely
written and easy accessible introduction to the subject [DNPV12].
We know that a curve parameterized by arc length has finite energy Eα if and
only ifit isbi-Lipschitz andbelongsto the spaceWα+21,2. InTheorem3.2we show
that even
γ′ CEα
| |Wα−21,2 ≤
and hence the Wα+21,2-norm of the flow is uniformly bounded in time.
We then derive the evolution equation of
(1.5) k = ∂kκ2 c′(x)dx.
E | s | | |
RZ/Z
whereκ denotes the curvatureofthe curvec and∂ := ∂x is the derivativewith
s |c′(x)|
respect to the arc length parameters. Note that in contrastto previous workslike
theworkduetoDziuk, Kuwert,andSchätzleonelasticfloworthe workdue toLin
and Schwetlick on the gradient flow of the elastic plus a positive multiple of the
Möbius energy, we do not consider the normal derivatives of the curvature but the
full derivatives with respect to arc length.
Using Gagliardo-Nirenberg-Sobolev inequalities for Besov spaces, which quite
naturallyappearduringthecalculation,togetherwithcommutatorestimateswecan
thenshowthatalsothe k areboundeduniformlyintime. Bystandardarguments
E
this will lead to long time existence and smooth subconvergence to a critical point
after suitable translations and re-parameterizationof the curves.
To get the full statement, we study the behavior of solutions near such critical
points using a Łojasievicz-Simon gradient estimate. This allows us to show that
flowsstartingcloseenoughtoacriticalpointandremainingabovethiscriticalpoint
in the sense of the energy, exist for all time and converge to critical points. More
precisely we have:
Theorem 1.2 (Longtimeexistenceabovecriticalpoints). Let c C∞(R/Z,Rn)
M
be a critical point of the energy E = Eα +λL, α [2,3), let k ∈N, δ > 0, and
∈ ∈
β >α. Then there is a constant ε>0 such that the following is true:
Suppose that (c ) is a maximal solution of the gradient flow of the energy
t t∈[0,T)
E for λ>0 with smooth initial data satisfying
λ
c c ε
k 0− MkCβ ≤
and
E(c ) E(c )
t M
≥
wheneverthereisadiffeomorphism φ :R/Z R/Zsuchthat c φ c δ.
t → k t◦ t− MkCβ ≤
Thentheflow(c ) existsforalltimesandconverges,aftersuitablere-parameterizations,
t t
smoothly to a critical point c of E satisfying
∞ λ
E(c )=E(c ).
∞ M
Thisshowsthatinthe situationofTheorem3.1the completesolutionconverges
to a critical point of Eα + λL – even without applying any translations or re-
parameterizations.
2. Short Time Existence
This section is devoted to an almost self-containedproof of shorttime existence
for equation (1.4). We will show that for all c C∞(R/Z) a solution exists for
∈
some time.
For any space X C1(R/Z,Rn) we will denote by X the (open) subspace
ir
⊂
consisting of all injective (embedded) and regular curves in X.
GRADIENT FLOW OF KNOT ENERGIES 5
Theorem 2.1 (Short time existence for smooth data). Let c C∞(R/Z). Then
0 ∈ ir
there exists some T =T(c )>0 and a unique solution
0
c C∞([0,T) R/Z,Rn)
∈ ×
of (1.4).
We canstrengthenthis resultabove. Infact, wecanreducethe regularityofthe
initial curve even below the level where our evolution equation makes sense. We
still can prove existence and in a sense also uniqueness of a family of curves with
normal velocity given by the gradient of Eα+λL. For this purpose we will work
in little Hölder spaces hβ, β / N, which are the completion of C∞ with respect to
∈
the Cβ-norm.
Theorem 2.2 (Short time existence for non-smooth data). For α (2,3) let
∈
c hβ (R/Z,Rn) for some β > α, β / N. Then there is a constant T > 0 and a
0 ∈ i,r ∈
re-parameterization φ Cβ(R/Z,R/Z) such that there is a solution
∈
c C([0,T),hβ (R/Z,Rn)) C1((0,T),C∞(R/Z,Rn))
∈ i,r ∩
of the initial value problem
∂⊥c= H(c)+λκ t [0,T],
t − c ∀ ∈
(c(0)=c0 φ.
◦
This solution is unique in the sense that for each other solution
c˜ C([0,T˜),hβ (R/Z,Rn)) C1((0,T˜),C∞(R/Z,Rn))
∈ i,r ∩
and all t (0,min(T,T˜)] there is a smooth diffeomorphism φ C∞(R/Z,R/Z)
t
∈ ∈
such that
c(t, )=c˜(t,φ ()).
t
· ·
As in the special case of the Möbius energy dealt with in [Bla12b], these results
are based on the fact that the functional Hα possesses a quasilinear structure – a
statement that will be proven in the next subsection. After that we build a short
timeexistencetheoryforthelinearizationoftheseequationsfromscratchandprove
a maximal regularity result for this equation.
Toagetasolutionoftheevolutionequation(1.4),wefirsthavetobreakthesym-
metry that comes from the invariance of the equation under re-parameterizations.
We do this by writing the time dependent family of curves c as a normal graph
over some fixed smooth curve c . Applying Banach’s fixed-point theorem as done
0
in [Ang90] for nonlinear and quasilinear semiflows, we get short time existence for
the evolutionequationofthe normalgraphsandandcontinuousdependence ofthe
solutionontheinitialdata. Astandardre-parameterizationthengivesTheorem2.1
while Theorem 2.2 is obtained via an approximation argument.
2.1. Quasilinear Structure of the Gradient. By exchangingeveryappearance
of c(u+w) c(u) andd (u+w,u) by their firstorder Taylorexpansion c′(u) w
c
| − | | || |
and c′(u+w) by c′(u) in the formula for H˜α, we are led to the conjecture that
| | | |
the leading order term of α is the normal part of α Qα(c) where
H |c′|α+1
c(u+w) c(u) wc′(u) dw
Qα(c):=p.v. 2 − − c′′(u) .
w2 − wα
Z (cid:18) (cid:19)| |
[−l,l]
ThisheuristiccanbemaderigoroususingTaylor’sexpansionsofthe errorterms
and estimates for multilinear Hilbertransforms (cf. Lemma B.2) leading to the
next theorem. It will be essential later on that the remainder term is an analytic
operator between certain function spaces – which we denote by Cω.
6 SIMONBLATT
Theorem 2.3 (Quasilinear structure). For α (2,3) there is a mapping
∈
Fα Cω(Cα+β(R/Z,Rn),Cβ(R/Z,Rn))
∈ i,r
β>0
\
such that
α
Hαc= P⊥(Qαc)+Fαc
c′ α+1 c′
| |
for all c Hα+1(R/Z,Rn).
∈ i,r
Proof of Theorem 2.3. It is enough to show that there is a mapping
F˜α Cω(Cα+β(R/Z,Rn),Cβ(R/Z,Rn))
∈ i,r
β>0
\
such that
α
H˜αc= Qαc+F˜αc
c′ α+1
| |
for all c Hα+1(R/Z,Rn), where
∈
(2.1)
c(x+w) c(x) wc′(s) c′′(x)
H˜αc= lim 2α − − (α 2)
εց0 Z (cid:26) |c(x+w)−c(x)|2+α − − |c′(x)|2dc(x+w,x)α
w∈Iǫ,l
c′′(x)
2 c′(x+w)dw.
− c′ 2 c(x+w) c(x)α | |
| | | − | (cid:27)
The theorem then follows easily using Hαc=P⊥H˜α =H˜α Hα, c′ c′ .
c′ −h |c′|i|c′|
We decompose
α
(2.2) H˜αc= Qαc+2αRαc (α 2)Rαc 2Rαc+αRαc
c′ α+1 1 − − 2 − 3 4
| |
where
(Rαc)(x):= (c(x+w) c(x) wc′(x))
1 − −
RZ/Z
1 1
c′(x+w)dw,
c(x+w) c(x)α+2 − c′(x)α+2 wα+2 | |
(cid:18)| − | | | | | (cid:19)
c′′(x) 1 1
(Rαc)(x):= c′(x+w)dw,
2 c′(x)2 d (x+w,x)α − c′(x)αwα) | |
RZ/Z | | (cid:18) c | | (cid:19)
c′′(x) 1 1
(Rαc)(x):= c′(x+w)dw,
3 c′(x)2 c(x+w) c(x)α − c′(x)αwα) | |
RZ/Z | | (cid:18)| − | | | (cid:19)
1 c(x+w) c(x) wc′(x) c′(x+w) c′(x)
(Rαc)(x):= 2 − − c′′(x) | |−| |dw.
4 c′ α+2 w2 − wα
| | RZ/Z (cid:18) (cid:19) | |
Using Taylor’s expansion up to first order, we get
1
c′(u+τw)
d (x+w,w)=wc′(x)+w2 ,c′′(u+τw) dτ
c c′(u+τw)
Z (cid:28)| | (cid:29)
0
=wc′(x)(1+wX˜ (x,w))
c
where
1
1 c′(u+τw)
X˜ (x,w):= ,c′′(u+τw) dτ,
c c′ c′(u+τw)
| |Z (cid:28)| | (cid:29)
0
GRADIENT FLOW OF KNOT ENERGIES 7
and
2
1
c(x+w) c(x)2 = wc′(u)+w2 c′′(u+τw)dτ =w2 c′(x)(1+wX (x,w))
| − | (cid:12) (cid:12) | | c
(cid:12) Z (cid:12)
(cid:12) 0 (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
1 1 2
1
X (x,w):= c′(u) c′′(u+τw)dτ +w c′′(u+τw)dτ .
c c′ 2
| | Z Z
0 0
Together with the Taylor expansion
1
1+x−σ =1 σx+σ(σ+1)x2 (1 τ)1+τx−σ−2dτ
| | − − | |
Z
0
for σ >0 and x> 1, this leads to
−
1 1 1
c(x+w) c(x)σ − c′(x)σ wσ = c′(x)σ wσ (1+Xc(x,w))−σ2 −1
| − | | | | | | | | |
(cid:0) (cid:1)
1 σ
= wX (x,w)
c′(x)σ wσ − 2 c
| | | |
1
σ σ
+ +1 w2Xc(u,w)2 (1+τwXc(x,w))−σ2−2dτ
2 2
!
(cid:16) (cid:17) Z0
and
1 1
d(x+w,x)σ − c′(x)σ wσ
| | | |
1
1
= σwX˜ (x,w)+σ(σ+1)w2X˜ (u,w)2 (1+τwX˜)−σ−2dτ .
c′(u)σ wσ − c c
| | | | Z0
Furthermore, we will use the identities
1
c(x+w) c(x) wc′(x)=w2 (1 τ)c′′(x+τw)dτ
− − −
Z
0
and
1
c′(x+τw)
c′(x+w) c′(x) =w ,c′′(x+τw) dτ.
| |−| | c′(x+τw
Z (cid:28)| | (cid:29)
0
Plugging these formulas into the expressions for the terms Rα,Rα,Rα,Rα and
1 2 3 4
factoring out, we see that they can be written as the sum of integrals as in the
following Lemma 2.4. Hence, Lemma 2.4 completes the proof. (cid:3)
Lemma 2.4. For ˜l l , ˜l l , and ˜l l and a multilinear operator M let
1 1 2 2 3 3
≤ ≤ ≤
I := M(c′′(x+τ w),...,c′′(x+τ w),c′(x+τ ),...,c′(x+τ w),
1 l1 l1+1 l1+l2
Z
[0,1]˜l1+˜l2+˜l3
c′(x+τ w) c′(x+τ w)
l1+l2+1 ,..., l1+l2+l3
c′(x+τ w) c′(x+τ )
| l1+l2+1 | | l1+l2+l3 |(cid:19)
dτ dτ dτ dτ dτ dτ ,
1··· ˜l1 l1+1··· l1+˜l2 l1+l2+1··· l1+l2+˜l3
8 SIMONBLATT
i.e. we integrate over some of the τ but not over all. Then for α˜ (0,1) the
i
∈
functionals
1/2
I
T˜ (c)(x):= dw,
1 wwα˜
Z | |
−1/2
1/2
I( 1(1+τwX˜(x,w))−σ)dτ
T˜ (c)(x):= 0 dw,
2 wα˜
Z R | |
−1/2
1/2
I
T (c)(x):= dw,
1 wwα˜
Z | |
−1/2
and
1/2
I( 1(1+τwX(x,w))−σ)dτ
T (c)(x):= 0 dw
2 wα˜
Z R | |
−1/2
are analytic from Cβ+α˜+2 to Cβ for all β >0.
Proof. The statement of the lemma for T˜ and T follows immediately from the
1 1
boundedness of the multilinear Hilberttransform in Hölderspaces as stated in Re-
mark B.3 combined with the Lemmata A.2 and A.3.
Using a similar argument, one deduces that c 1+τX is analytic, hence we
c
→
get that for a given c there is a neighborhood U such that
0
Dm(1+τX (,w) Cm!
k c c · kL(Cβ+2,Cβ) ≤
for allc U where C does not depend on w and τ. Using that v v σ is analytic
∈ →| |
away from 0, we deduce that
Dm(1+τX (,w)σ Cm!
k c c · kL(Cβ+2,Cβ) ≤
using Lemma A.2. Hence, the integrandsin the definitions of T andT˜ satisfy the
2 2
assumptions of Lemma A.3. Hence, T and T˜ are even analytic operators from
2 2
Cβ+2(R/Z,Rn) to Cβ(R/Z,Rn).
(cid:3)
2.2. Short Time Existence. Using the quasilinear form of α, we derive short
H
time existence resultsfor the gradientflowofO’Hara’senergiesinthis section. For
this task, we will work with families of curves that are normal graphs over a fixed
smooth curve c and whose normal part belongs to a small neighborhood of 0 in
0
hβ, β >α.
To describe these neighborhoods, note that there is a strictly positive, lower
semi-continuous function r :C2 (R/Z,Rn) (0, ) such that
i,r → ∞
c+ N C1(R/Z,Rn)⊥ : N <r(c)
{ ∈ c0 k kC1 }
only contains regular embedded curves for all c C1 (R/Z,Rn) and
∈ i,r
(2.3) r(c) 1/2 inf c′(x).
≤ x∈R/Z| |
Here, Cβ(R/Z,Rn)⊥ denotes the space of all vectorfields N Cβ(R/Z,Rn) which
are normal to c, i.ec. for which c′(u),N(u) =0 for all u R∈/Z. Letting
h i ∈
(c):= N hβ(R/Z,Rn)⊥ : N <r(c)
Vr,β { ∈ c k kC1 }
GRADIENT FLOW OF KNOT ENERGIES 9
we have for all c hβ (R/Z,Rn)
∈ i,r
(2.4) c+ (c) hβ (R/Z,Rn).
Vr,β ⊂ i,r
Let N (c). Equation (2.3) guarantees that P⊥ is an isomorphism
∈ Vr,β (c+N)′(u)
from the normal space along c at u to the normal space along c+N. Otherwise
there would be a v =0 in the normal space of c at u such that
6
(c+N)′(u) (c+N)′(u)
0=P⊥ (v)=v v,
(c+N)′(u) − (c+N)′(u) (c+N)′(u)
(cid:28) | |(cid:29)| |
which would contradict
(c+N)′(u) (c+N)′(u) N′(u)
v v, v v, v /2>0.
− (c+N)′(u) (c+N)′(u) ≥| |− (c+N)′(u) | ≥| |
(cid:12) (cid:28) | |(cid:29)| |(cid:12) (cid:12)(cid:28) | |(cid:29) (cid:12)
F(cid:12)(cid:12)or c C1((0,T),C1 (R/Z,Rn)) we d(cid:12)(cid:12)enote by(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) ∈ i,r (cid:12) (cid:12) (cid:12)
∂⊥c=P⊥(∂ c)
t c′ t
the normal velocity of the family of curves.
We prove the following strengthened version of the short time existence result
mentioned at the beginning of Section 2.
Theorem 2.5 (Short time existence for normal graphs). Let c C∞(R/Z,Rn)
0
be an embedded regular curve, α (2,3), and β > α, β / N. ∈Then for every
∈ ∈
N (c )thereisaconstantT =T(N )>0andaneighborhoodU ofN
0 r,β 0 0 r,β 0
∈V ⊂V
such that for every N˜ U there is a uniquesolution N C([0,T),hβ(R/Z)⊥)
0 ∈ N˜0 ∈ c0 ∩
C1((0,T),C∞(R/Z)⊥) of
c0
∂⊥(c +N = Hα(c +N)+λκ t [0,T],
(2.5) t 0 − 0 c0+N ∈
((cid:8)N(0)=N˜0.(cid:1)
Furthermore, the flow (N˜ ,t) N (t) is in C1((U (0,T)),C∞(R/Z)).
0 7→ N˜0 ×
The proofof Theorem 2.5 consists of two steps. First we show that (2.5) canbe
transformedintoanabstractquasilinearsystemofparabolictype. Thesecondstep
is to establish short time existence results for the resulting equation.
Thesecondstepcanbedoneusinggeneralresultsaboutanalyticsemigroups,reg-
ularity of pseudo-differential operators with rough symbols [Bou88], and the short
time existence results for quasilinear equations in [Ang90] or [Ama93]. Further-
more, we need continuous dependence of the solution on the data and smoothing
effects in order to derive the long time existence results in Section 4.
For the convenience of the reader,we go a different way here and present a self-
containedproofofthe shorttime existence that only reliesona characterizationof
the little Hölder spaces as trace spaces. In Subsection 2.2.1, we deduce a maximal
regularity result for solutions of linear equations of type ∂ u+a(t)Qu+b(t)u=f
t
in little Hölder spaces using heat kernel estimates. Following ideas from [Ang90],
we then prove short time existence and differentiable dependence on the data for
the quasilinear equation.
2.2.1. The Linear Equation. We willderivea prioriestimatesandexistenceresults
for linear equations of the type
∂ u+aQsu+bu=f in R/Z (0,T)
t
×
(u(0)=u0
using little Hölder spaces, where b(t) L(hβ(R/Z,Rn),Cβ(R/Z,Rn)) and a(t)
hβ(R/Z,(0, )). ∈ ∈
∞
10 SIMONBLATT
For θ (0,1), β >0, and T >0 we will consider solutions that lie in the space
∈
Xθ,β := g C((0,T),hβ+s(R/Z,Rn)) C1((0,T),hβ(R/Z,Rn):
T ∈ ∩
(cid:26)
sup t1−θ( ∂ g(t) + g(t) )<
k t kCβ k kCs+β ∞
t∈(0,T) (cid:27)
and equip this space with the norm
g := sup t1−θ( ∂ g(t) + g(t) ).
k kXTθ,β t∈(0,T) k t kCβ k kCs+β
The right hand side f of our equation should then belong to the space
Yθ,β := g C((0,T),hβ(R/Z,Rn)): sup t1−θ g(t) <
T ( ∈ t∈(0,T) k kCβ ∞)
equipped with the norm
g := sup t1−θ g(t) .
k kYTθ,β t∈(0,T) k kCβ
Fromthetracemethodinthetheoryofinterpolationspaces(cf. [Lun95,Section
1.2.2]), the following relation of the space X to the little Hölder space hβ+sθ is
θ,β
well known if β+sθ is not an integer:
If u Xθ,β then u(t) converges in hβ+sθ to a function u(0) as t 0 with
∈ T ց
(2.6) u(0) C u .
k kCβ+sθ ≤ k kXθT,β
Ontheotherhand,foreveryu hβ+sθ thereisau Xθ,β suchthatu(t)converges
0 ∈ ∈ T
in hβ+sθ to u(0) for t 0 and
→
(2.7) u u .
k kXθT,β ≤k 0kCβ+sθ
Given u we will see that the solution of the initial value problem
0
∂ u+( ∆)s/2u=0 on Rn
t
−
(u=u0 at t=0.
satisfies (2.7). The well-known embedding Xθ,β Cθ((0,T),Cβ(R/Z,Rn)) will
T ⊂
also be essential in the proof.
Theaimofthissubsectionistoprovethefollowingtheoremaboutthesolvability
of our linear equation:
Theorem 2.6. Let T >0, β >0, θ (0,1) with β+sθ / N, and
∈ ∈
a C1([0,T],hβ(R/Z,[1/Λ, ))), b C0((0,T),L(hβ(R/Z,Rn),hβ(R/Z,Rn)))
∈ ∞ ∈
with
a + sup t1−θ b(t) <
k kC1([0,T],Cβ) k kL(hβ,hβ) ∞
t∈(0,T)
Then the mapping J : u (u(0),∂ u + aQs−1u + bu) defines an isomorphism
t
7→
between Xθ,β and hβ+θs(R/Z,Rn) Yθ,β.
T × T
This will be enough to prove short time existence of a solution for some quasi-
linear equations later on using Banach’s fixed-point theorem.
Equation (2.6) already guarantees that J is a bounded linear operator. So we
only have to prove that it is onto for which we will use some a priori results (also
calledmaximalregularityresults in this context) togetherwith the method of con-
tinuity.
