Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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198 J. Escher, M. Kohlmann and B. Kolev

aﬃne connection ∇ on Diﬀ

∞

)by

∇

[ξ

,ξ

]+B(ξ

,ξ

), (2.2)

where ξ

and ξ

are the right-invariant vector ﬁelds on Diﬀ

∞

)withvaluesu, v

at the identity. It can be shown that a smooth curve t → g(t)inDiﬀ

∞

)isa

geodesic if and only if u =(R

−1

)

∗

˙g solves the Euler equation

= −B(u, u); (2.3)

here, u is the Eulerian velocity (cf. [2]). Hence the Euler equation (2.3) corre-

sponds to the geodesic ﬂow of the aﬃne connection ∇ on the diﬀeomorphism

group Diﬀ

∞

). Paradigmatic examples are the following: In [8], the authors

show that the Euler equation for the right-invariant L

-metric on Diﬀ

∞

)is

given by the inviscid Burgers equation. Equipping on the other hand C

∞

)with

the H

-metric, one obtains the Camassa-Holm equation. Similar correspondences

for the general H

-metrics are explained in [9].

Conversely, starting with an equation of type u

= −B(u, u) with a bilinear

operator B, one associates an aﬃne connection ∇ on Diﬀ

∞

(S) by formula (2.2).

It is however by no means clear that this connection corresponds to a Riemannian

structure on Diﬀ

∞

(S). It is worthwhile to mention that the connection ∇ cor-

responding to the family of b-equations is compatible with some metric only for

b = 2: In [18] the authors explain that for any b =2,theb-equation (1.1) cannot be

realized as an Euler equation on Diﬀ

∞

) for any regular inertia operator. This

motivates the notion of non-metric Euler equations. An analogous result holds true

for the μ-b-equations from which we conclude that the μDP equation belongs to

the class of non-metric Euler equations. Although we have no metric for the μDP

equation, we will obtain some geometric information by using the connection ∇,

deﬁned in the following way.

Let X(t)=(ϕ(t),ξ(t)) be a vector ﬁeld along the curve ϕ(t) ∈ Diﬀ

∞

Furthermore let

B(v, w):=

−1

(v(Aw)

+ w(Av)

+3(Av)w

+3(Aw)v

Lemma 2.2 shows that

B(u, u)=A

−1

(u(Au)

+3(Au)u

)=−u

if u is a solution to the μDP equation. Next, the covariant derivative of X(t)in

the present case is given by

(t)=



ϕ(t),ξ

[u(t),ξ(t)] + B(u(t),ξ(t))



where u = ϕ

◦ϕ

−1

.Weseethatu is a solution of the μDP if and only if its local

ﬂow ϕ is a geodesic for the connection ∇ deﬁned by B via (2.2).

Although we are mainly interested in the smooth category, we will ﬁrst discuss

ﬂows ϕ(t)onDiﬀ

) for technical purposes. Regarding Diﬀ

)asasmooth

The Periodic μDP Equation 199

Banach manifold modelled over C

), the following result has to be understood

locally, i.e., in any local chart of Diﬀ

Proposition 2.3. Given n ≥ 3, the function u ∈ C(J, C

)) ∩C

(J, C

n−1

)) is

asolutionof (1.2) if and only if (ϕ, ξ) ∈ C

(J, Diﬀ

) ×C

)) is a solution of



= ξ,

= −P

(ξ),

(2.4)

where P

:= R

◦ P ◦ R

−1

and P (f ):=3A

−1

+(Af )f

Proof. The function u and the corresponding ﬂow ϕ ∈ Diﬀ

) satisfy the relation

= u ◦ ϕ. Setting ϕ

= ξ, the chain rule implies that

=(u

+ uu

) ◦ ϕ.

Applying Lemma 2.2, we see that u is a solution of the μDP equation (1.2) if and

only if

+ uu

= −A

−1

(u(Au)

−A(uu

)+3(Au)u

)

= −A

−1

(−uu

xxx

+ u

+ uu

xxx

+2u

+3(Au)u

)

= −3A

−1

+(Au)u

)

= −P (u).

Recall that

μ(uu



dx =



∂

)dx =

(1) − u

(0)) = 0,

since u is continuous on S

.Withu = ξ ◦ϕ

−1

the desired result follows. 

3. Short time existence of geodesics

We now deﬁne the vector ﬁeld

F (ϕ, ξ):=(ξ,−P

(ξ))

such that (ϕ

,ξ

)=F (ϕ, ξ). We know that

F :Diﬀ

) × C

) → C

) × C

since P is of order zero. We aim to prove smoothness of the map F .Itisworth

to mention that this will not follow from the smoothness of P since neither the

composition nor the inversion are smooth maps on Diﬀ

). The following lemma

will be crucial for our purposes.

Lemma 3.1. Assume that p is a polynomial diﬀerential operator of order r with

coeﬃcients depending only on μ, i.e.,

p(u)=

I=(α

,...,α

∈N∪{0}, |I|<K

(μ(u)) u



)

···(u

(r)

)

200 J. Escher, M. Kohlmann and B. Kolev

Then the action of p

:= R

◦ p ◦ R

−1

(u)=





u(y)ϕ

(y)dy



(u; ϕ

,...,ϕ

(r)

where q

are polynomial diﬀerential operators of order r with coeﬃcients being

rational functions of the derivatives of ϕ up to the order r. Moreover, the denom-

inator terms only depend on ϕ

Proof. It is suﬃcient to consider a monomial

m(u)=a(μ(u))u



)

···(u

(r)

)

We have

(u)=a(μ(u ◦ ϕ

−1

))u

[(u ◦ϕ

−1

)



◦ ϕ]

···[(u ◦ ϕ

−1

)

(r)

◦ ϕ]

where ◦ denotes again the composition with respect to the spatial variable. First,

we observe that

μ(u ◦ ϕ

−1



u(ϕ

−1

(x)) dx =



u(y)ϕ

(y)dy,

where we have omitted the time dependence of u and ϕ. Recall that ϕ(S

)=S

> 0andthatμ(u ◦ ϕ

−1

) is a constant with respect to the spatial variable

x ∈ S

. Let us introduce the notation

=(u ◦ ϕ

−1

)

(k)

◦ ϕ, k =1, 2,...,r.

Then, by the chain rule,

=(∂

(u ◦ ϕ

−1

)) ◦ ϕ =

◦ ϕ

−1

◦ ϕ

−1

◦ ϕ =

and

k+1

=(∂

(u ◦ ϕ

−1

)

(k)

) ◦ ϕ =(∂

◦ ϕ

−1

)) ◦ ϕ =

∂

so that our theorem follows by induction. 

Recall that in the Banach algebras C

), n ≥ 1, addition and multiplication

as well as the mean value operation μ and the derivative

are smooth maps. We

therefore conclude that if the coeﬃcients a

are smooth functions for any multi-

index I and u and ϕ are at least r times continuously diﬀerentiable, then p

(u)

depends smoothly on (ϕ, u).

Proposition 3.2. The vector ﬁeld

F :Diﬀ

) × C

) → C

) × C

)

is smooth for any n ≥ 3.

Proof. We write F =(F

). Since F

:(ϕ, ξ) → ξ is smooth, it remains to check

that F

:(ϕ, ξ) →−P

(ξ) is smooth. For this purpose, we consider the map

P :Diﬀ

) ×C

) → Diﬀ

) ×C

)

The Periodic μDP Equation 201

deﬁned by

P (ϕ, ξ)=(ϕ, (R

◦ P ◦ R

−1

)(ξ)).

Observe that we have the decomposition

P =

−1

◦

Q with

A(ϕ, ξ)=(ϕ, (R

◦ A ◦ R

−1

)(ξ))

and

Q(ϕ, ξ)=(ϕ, (R

◦ Q ◦ R

−1

)(ξ)),

where Q(f):=3(f

+(Af )f

). We now apply Lemma 3.1 to deduce that

Q :Diﬀ

) × C

) → Diﬀ

) × C

n−2

)

are smooth. To show that

−1

:Diﬀ

) × C

n−2

) → Diﬀ

) × C

)is

smooth, we compute the derivative D

A at an arbitrary point (ϕ, ξ). We have the

following directional derivatives of the components

and

=id,D

=0,D

= R

◦ A ◦ R

−1

It remains to compute (D

(ϕ, ξ))(ψ)=

dε

(ϕ + εψ, ξ)

ε=0

.Inaﬁrststep,we

calculate

∂

(ξ ◦ (ϕ + εψ)

−1

)=∂

.

+ εψ



◦ (ϕ + εψ)

−1



(ϕ

+ εψ

)

− ξ

+ εψ

(ϕ

+ εψ

)



◦ (ϕ + εψ)

−1

from which we get

dε



∂

(ξ ◦ (ϕ + εψ)

−1

) ◦ (ϕ + εψ)



dε



(ϕ

+ εψ

)

− ξ

+ εψ

(ϕ

+ εψ

)



= −2

(ϕ

+ εψ

)

−

(ϕ

+ εψ

)

(ϕ

+ εψ

)

(ϕ

+ εψ

)

and ﬁnally

dε



∂

(ξ ◦ (ϕ + εψ)

−1

) ◦ (ϕ + εψ)



ε=0

= −2

−

Secondly, we observe that

dε

μ(ξ ◦ (ϕ + εψ)

−1

)

ε=0

dε



ξ(y)(ϕ

+ εψ

)(y)dy

ε=0



ξ(y)ψ

(y)dy,

since ϕ + εψ ∈ Diﬀ

) for small ε>0. Hence

(ϕ, ξ))(ψ)=



ξ(y)ψ

(y)dy +2

− 3

202 J. Escher, M. Kohlmann and B. Kolev

and

A(ϕ, ξ)=



id 0

(ϕ, ξ) R

◦ A ◦R

−1



It is easy to check that D

A(ϕ, ξ) is an invertible bounded linear operator C

)×

) → C

) ×C

n−2

). By the open mapping theorem, D

A is a topological

isomorphism and, by the inverse mapping theorem,

−1

is smooth. 

Since F is smooth, we can apply the Banach space version of the Picard-

Lindel¨of Theorem (also known as Cauchy-Lipschitz Theorem) as explained in [27],

Chapter XIV-3. This yields the following theorem about the existence and unique-

ness of integral curves for the vector ﬁeld F.

Theorem 3.3. Given n ≥ 3, there is an open interval J

centered at zero and

an open ball B(0,δ

) ⊂ C

) such that for any u

∈ B(0,δ

) there exists a

unique solution (ϕ, ξ) ∈ C

∞

, Diﬀ

)×C

)) of (2.4) with initial conditions

ϕ(0) = id and ξ(0) = u

. Moreover, the ﬂow (ϕ, ξ) depends smoothly on (t, u

From the above theorem, we get a unique short-time solution u = ξ ◦ ϕ

−1

in C

)oftheμDP equation with continuous dependence on (t, u

). We now

aim to obtain a similar result for smooth initial data u

. But since C

∞

)isa

Fr´echet space, classical results like the Picard-Lindel¨of Theorem or the local inverse

theorem for Banach spaces are no longer valid in C

∞

). In the proof of our main

theorem, we will make use of a Banach space approximation of the Fr´echet space

∞

). First, we shall establish that any solution u of the μDP equation does

neither lose nor gain spatial regularity as t increases or decreases from zero. For

this purpose, the following conservation law is quite useful. In its formulation we

use the notation m

(x):=(Au)(0,x)=μ(u

) −(u

)

Lemma 3.4. Let u be a C

)-solution of the μDP equation on (−T,T) and let ϕ

be the corresponding ﬂow. Then

(Au)(t, ϕ(t, x))ϕ

(t, x)=m

for all t ∈ (−T,T).

Proof. We compute

[(μ(u) −u

◦ ϕ)ϕ

]

=[μ(u

) −u

xxt

◦ ϕ − (u

xxx

◦ ϕ)ϕ

] ϕ

+3ϕ

(μ(u) − u

◦ ϕ)

=[μ(u

) −u

xxt

◦ ϕ − (u

xxx

◦ ϕ)(u ◦ ϕ)] ϕ

+3ϕ

(u ◦ ϕ)

(μ(u) − u

◦ ϕ)

=[(μ(u

) − u

xxt

− u

xxx

u) ◦ϕ] ϕ

+3ϕ

◦ϕ)ϕ

(μ(u) − u

◦ ϕ)

=[(μ(u

) − u

xxt

− u

xxx

u) ◦ϕ] ϕ

+3ϕ

(μ(u) − u

)] ◦ϕ

=[(3u

−3μ(u)u

) ◦ ϕ] ϕ

− 3ϕ

−μ(u)u

) ◦ϕ

=0.

Since ϕ(0) = id and ϕ

(0) = 1, the proof is completed. 

The Periodic μDP Equation 203

Lemma 3.5. Let (ϕ, ξ) ∈ C

∞

, Diﬀ

) × C

)) be a solution of (2.4) with

initial data (id,u

), according to Theorem 3.3.Then,forallt ∈ J

(t)=ϕ

(t)





μ(u)ϕ

(s)ds − m



(s)

−2



(3.1)

and

(t)=ξ

(t)

+ ϕ

(t)[μ(u)ϕ

(t) − m

(t)

−2

]. (3.2)

Proof. We have





xxt

− ϕ

Since ϕ

= u ◦ ϕ,

= ϕ

= ∂

(u ◦ ϕ)=(u

◦ ϕ)ϕ

and

xxt

= ϕ

txx

= ∂

(u ◦ ϕ)

= ∂

[(u

◦ ϕ)ϕ

]

=(u

◦ ϕ)ϕ

+(u

◦ ϕ)ϕ

Hence





=(u

◦ ϕ)ϕ

According to the previous lemma, we know that

◦ ϕ = μ(u) − m

−3

Integrating





= μ(u)ϕ

− m

−2

over [0,t] leads to equation (3.1) and taking the time derivative of (3.1) yields

(3.2). 

Remark 3.6. Since the μDP equation is equivalent to the quasi-linear evolution

equation

+ uu

+3μ(u)∂

−1

u =0,

we see that μ(u

) = 0 and hence μ(u)=μ(u

)sothatμ(u) can in fact be written

in front of the ﬁrst integral sign in equation (3.1).

Corollary 3.7. Let (ϕ, ξ) be as in Lemma 3.5.Ifu

∈ C

) then we have

(ϕ(t),ξ(t)) ∈ Diﬀ

) ×C

) for all t ∈ J

Proof. We proceed by induction on n.Forn = 3 the result is immediate from our

assumption on (ϕ(t),ξ(t)). Let us assume that (ϕ(t),ξ(t)) ∈ Diﬀ

) × C

)

for some n ≥ 3. Then Lemma 3.5 shows that, if u

∈ C

n+1

), then (ϕ(t),ξ(t)) ∈

Diﬀ

n+1

) × C

n+1

), ﬁnishing the proof. 

204 J. Escher, M. Kohlmann and B. Kolev

Corollary 3.8. Let (ϕ, ξ) be as in Lemma 3.5. If there exists a nonzero t ∈ J

such

that ϕ(t) ∈ Diﬀ

) or ξ(t) ∈ C

) then ξ(0) = u

∈ C

Proof. Again, we use a recursive argument. For n = 3, there is nothing to do. For

some n ≥ 3, suppose that u

∈ C

). By the previous corollary, (ϕ(t),ξ(t)) ∈

Diﬀ

) × C

) for all t ∈ J

. Assume that there is 0 = t

∈ J

such that

ϕ(t

) ∈ Diﬀ

n+1

)orξ(t

) ∈ C

n+1

). Since ϕ

> 0, Lemma 3.5 immediately

implies that also u

∈ C

n+1

). 

Now we discuss Banach space approximations of Fr´echet spaces.

Deﬁnition 3.9. Let X be a Fr´echet space. A Banach space approximation of X is

a sequence {(X

, ||·||

); n ∈ N

} of Banach spaces such that

⊃ X

⊃···⊃X, X =

∞

n=0

and {||·||

; n ∈ N

} is a sequence of norms inducing the topology on X with

||x||

≤||x||

≤···

for any x ∈ X.

We have the following result. For a proof, we refer to [17].

Lemma 3.10. Let X and Y be Fr´echet spaces with Banach space approximations

{(X

, ||·||

); n ∈ N

} and {(Y

, ||·||

); n ∈ N

}.LetΦ

: U

→ V

beasmoothmap

between the open subsets U

⊂ X

and V

⊂ Y

.Let

U := U

∩ X and V := V

∩Y,

as well as

:= U

∩ X

and V

:= V

∩Y

for any n ≥ 0. Furthermore, we assume that, for each n ≥ 0, the following prop-

erties are satisﬁed:

(1) Φ

) ⊂ V

(2) the restriction Φ

:= Φ

: U

→ V

is a smooth map.

Then Φ

(U) ⊂ V and the map Φ:=Φ

: U → V is smooth.

Now we come to our main theorem which we ﬁrst formulate in the geometric

picture.

Theorem 3.11. There exists an open interval J centered at zero and δ>0 such

that for all u

∈ C

∞

) with ||u

)

<δ,thereexistsauniquesolution(ϕ, ξ) ∈

∞

(J, Diﬀ

∞

)×C

∞

)) of (2.4) such that ϕ(0) = id and ξ(0) = u

. Moreover,

the ﬂow (ϕ, ξ) depends smoothly on (t, u

) ∈ J ×C

∞

The Periodic μDP Equation 205

Proof. Theorem 3.3 for n = 3 shows that there is an open interval J centered at

zero and an open ball U

= B(0,δ) ⊂ C

)sothatforanyu

∈ U

there exists a

unique solution (ϕ, ξ) ∈ C

∞

(J, Diﬀ

)×C

)) of (2.4) with initial data (id,u

)

and a smooth ﬂow

: J × U

→ Diﬀ

) ×C

Let

:= U

∩ C

)andU

∞

:= U

∩ C

∞

By Corollary 3.7, we have

(J × U

) ⊂ Diﬀ

) × C

)

for any n ≥ 3andthemap

:= Φ

J×U

: J × U

→ Diﬀ

) × C

)

is smooth. Lemma 3.10 implies that

(J ×U

∞

) ⊂ Diﬀ

∞

) ×C

∞

completing the proof of the short-time existence for smooth initial data u

.More-

over, the mapping

∞

:= Φ

J×U

∞

: J × U

∞

→ Diﬀ

∞

) × C

∞

)

is smooth, proving the smooth dependence on time and on the initial condition.



Under the assumptions of Theorem 3.11, the map

Diﬀ

∞

) ×C

∞

) → C

∞

), (ϕ, ξ) → ξ ◦ ϕ

−1

= u

is smooth. Thus we obtain the result stated in Theorem 1.1.

4. The exponential map

For a Banach manifold M equipped with a symmetric linear connection, the ex-

ponential map is deﬁned as the time one of the geodesic ﬂow, i.e., if t → γ(t)is

the (unique) geodesic in M starting at p = γ(0) with velocity γ

(0) = u ∈ T

then exp

(u)=γ(1). Roughly speaking, the map exp

(·) is a projection from

M to the manifold M. Since the derivative of exp

at zero is the identity, the

exponential map is a smooth diﬀeomorphism from a neighbourhood of zero of

M to a neighbourhood of p ∈ M. However, this fails for Fr´echet manifolds

like Diﬀ

∞

) in general. We know that the Riemannian exponential map for the

-metric on Diﬀ

∞

) is not a local C

-diﬀeomorphism near the origin, cf. [9].

For the Camassa-Holm equation and more general for the H

-metrics, k ≥ 1, the

Riemannian exponential map in fact is a smooth local diﬀeomorphism. This result

was generalized to the family of b-equations, see [17], and in this section we obtain

a similar result for the μDP equation.

The basic idea of the proof of Theorem 1.2 is to consider a perturbed problem:

Let (ϕ

,ξ

) denote the local expression of an integral curve of (2.4) in T Diﬀ

)

206 J. Escher, M. Kohlmann and B. Kolev

with initial data (id,u+ εw), where u, w ∈ C

). Let

ψ(t):=

∂ϕ

(t)

∂ε

ε=0

By the homogeneity of the geodesics,

(t)=exp(t(u + εw)),

so that

ψ(t)=D (exp(tu)) tw =: L

(t, u)w,

where L

(t, u) is a bounded linear operator on C

Lemma 4.1. Suppose that u ∈ C

n+1

).Then,fort =0,

(t, u)(C

)\C

n+1

)) ⊂ C

)\C

n+1

Proof. First, we write down equation (3.1) for ϕ

(t),

(t)=ϕ

(t)

μ(u + εw)



(s)ds − m



(s)

−2

and take the derivative with respect to ε,

∂ϕ

∂ε

(t)=

∂ϕ

∂ε

(t)

μ(u + εw)



(s)ds − m



(s)

−2

+ ϕ

(t)

μ(w)



(s)ds + μ(u + εw)



∂ϕ

∂ε

(s)ds

− ϕ

(t)

∂m

∂ε



(s)

−2

ds + m



∂

∂ε

(s)

−2

Notice that

∂m

∂ε

= μ(w) − w

= Aw

and that m

→ m

= Au as ε → 0. Hence

(t)=ψ

(t)

μ(u)



(s)ds − m



(s)

−2

+ ϕ

(t)

μ(w)



(s)ds + μ(u)



(s)ds

−ϕ

(t)

(μ(w) − w

)



(x)

−2

ds −2m



(s)

−3

(s)ds

= a(t)ψ

(t)+b(t)



c(s)ψ

(s)ds + d(t)+e(t)w

with a(t),b(t),c(t),d(t),e(t) ∈ C

n−1

)ande(t) =0fort = 0. Finally, if

w ∈ C

)\C

n+1

then

ψ(t)=L

(t, u)w ∈ C

)\C

n+1

). 

The Periodic μDP Equation 207

Let us now turn to the proof of Theorem 1.2. Since C

)isaBanachspace

and Diﬀ

) is a Banach manifold modelled over C

), we know that the expo-

nential map is a smooth diﬀeomorphism near zero, i.e., there are neighbourhoods

of zero in C

)andV

of id in Diﬀ

) such that

exp

:= exp |

: U

→ V

is a smooth diﬀeomorphism. For n ≥ 3, we now deﬁne

:= U

∩ C

)andV

= V

∩ Diﬀ

Let exp

:= exp

. Since exp

is a restriction of exp

, it is clearly injective. We

now use Corollary 3.7 and Corollary 3.8 to deduce that exp

is also surjective,

more precisely, exp

)=V

.Ifthegeodesicϕ with ϕ(1) = exp(u)startsat

id ∈ Diﬀ

) with velocity vector u belonging to C

), then ϕ(t) ∈ Diﬀ

)

for any t and hence exp

) ⊂ V

.Conversely,ifv ∈ V

is given, then there

is u ∈ U

with exp

(u)=v. Corollary 3.8 immediately implies that u ∈ C

);

hence u ∈ U

and exp

(u)=v.Notethatexp

is a bijection from U

to V

Furthermore, exp

is a smooth map and diﬀeomorphic U

→ V

. We now show

that exp

is a smooth diﬀeomorphism; precisely, we show that exp

−1

: V

→ U

smooth by virtue of the inverse mapping theorem. For each u ∈ C

), D exp

(u)

is a bounded linear operator C

) → C

). Notice that

D exp

(u)=D exp

(u)|

)

from which we conclude that D exp

(u) is injective. Let us prove the surjectiv-

ity of D exp

(u), n ≥ 3, by induction. For n = 3, this follows from the fact

that exp

: U

→ V

is diﬀeomorphic and hence a submersion. Assume that

D exp

(u) is surjective for some n ≥ 3andthatu ∈ C

n+1

). We have to show

that this implies the surjectivity of D exp

n+1

(u). But this is a direct consequence

of D exp

(u)=L

(1,u) and the previous lemma: Let f ∈ C

n+1

). We have to

ﬁnd g ∈ C

n+1

) with the property D exp

n+1

(u)g = f.Byourassumption,there

is g ∈ C

) such that D exp

(u)g = f . Suppose that g/∈ C

n+1

). But then

f = L

(1,u)g/∈ C

n+1

) in contradiction to the choice of f .Thusg ∈ C

n+1

)

and D exp

n+1

(u)g = f. Now we can apply the open mapping theorem to deduce

that for any n ≥ 3andanyu ∈ C

)themap

D exp

(u):C

) → C

)

is a topological isomorphism. By the inverse function theorem, exp

: U

→ V

a smooth diﬀeomorphism. If we deﬁne

∞

:= U

∩C

∞

)andV

∞

:= V

∩Diﬀ

∞

Lemma 3.10 yields that

exp

∞

:= exp

∞

: U

∞

→ V

∞

as well as

exp

−1

∞

: V

∞

→ U

∞

are smooth maps. Thus exp

∞

is a smooth diﬀeomorphism between U

∞

and V

∞