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

Подождите немного. Документ загружается.

188 B. Ducomet and

S. Neˇcasov´a

Then we will get from 8.12,



√



(T )

+ μD(u

(Q)



√

− ˜u

(Q)

≤ 0. (8.13)

This gives us the following bounds:

• u

is bounded in L

(0,T,W

1,p

(T )),

•

√

and (u)

are bounded in L

∞

(0,T,L

(T )),

•

√

− ˜u

)and

− ˜u) are bounded in L

(0,T; L

(T )).

Then we can extract subsequences such that

• u

→ u weakly in L

(0,T,W

1,p

(Ω)),

• [u

]

→ [u]

weakly

∗

in L

(0,T,W

1,∞

(Ω)), div[u]

=0.

Since Ω is regular then u =0on∂Ω.

Observing that {

}

ε>0

solves the transport equation (5.1), we can use Propo-

sition 8.1 and (6.1) to deduce that



→  in C([0,T] ×T), (8.14)

where

inf

x∈T



0,δ

≤ inf

x∈T



(t, x) ≤ sup

x∈T



(t, x) ≤ sup

x∈T



0,δ

In particular,

inf

x∈T



0,δ

≤ inf

x∈T

(t, x) ≤ sup

x∈T

(t, x) ≤ sup

x∈T



0,δ

. (8.15)

Consequently, employing once more the energy inequality (5.14), we conclude that

→ u in L

∞

(0,T; L

(T ; R

)).

Clearly, the limit density  satisﬁes the approximate equation of continuity

∂

 +div

([u]

)=dΔρ in (0,T) × R

, (8.16)

provided  has been extended to be 

outside Ω.

Moreover,

• (

√

−

√

u) → 0inL

(0,T,L

(Ω)) strongly, and

• H

− H

u → 0inL

(0,T,L

(Ω)) strongly.

8.2.2. Passing to the limit in the rigid velocity. The rigid velocity is deﬁned as

(x,t)=u

ε,G

(t)+ω

(t) × r

(x, t),

with

ε,G

(t)=



dx, ω

(t)= J

−1



× u

dx .

Then it follows that

• u

ε,G

(t) is bounded in L

∞

(0,T),

• ω

(t) is bounded in L

∞

(0,T).

It implies that

→

u in the weak

∗

sense in L

∞

(0,T,L

∞

(Ω)).

On the Motion of Several Rigid Bodies 189

Taking gradients of the rigid velocity

implies that

→

u in L

(0,T,W

1,∞

(Ω)) weak

∗

Applying the compactness results to the transport equation on H

gives us

→ H

a.e. in C(0,T,L

(Ω)) strongly for p ∈ [1, ∞[,

satisfying the transport equation

u ·∇H

=0,H

(0,x)=H

(0).

This implies the strong convergence of r

in C(0,T,L

(Ω)), for all p ≥ 1and

we can also pass to the limit in the expression ˜u

and ω

8.2.3. Strong convergence of u

. To prove the strong convergence of a subsequence

of u

in L

(Q)wewrite



−u|

dxdt ≤

min





|ρ(u

−u

)|dxdt +



|2ρu(u −u

|dxdt



(8.17)

Since the second term in (8.17) converges to 0, then



− u|

dxdt ≤

min





|ρ

− ρu

|dxdt +



|(−ρ

+ ρ)u

dxdt



(8.18)

From (8.14) and boundedness of u

in L

∞

(0,T,L

) we get that convergence of the

second term of (8.18) converges to 0, and



− u|

dxdt

≤

min





|ρ



∪

i=1

(t)[



] −ρu ·P



∪

i=1

(t)[



])|dxdt



|ρ



−P



∪

i=1

(t)[



])



|dxdt



|ρu · (P



∪

i=1

(t)[



[u] − u)|dxdt + l



≤

min



|ρ



∪

i=1

(t)[



] − ρu ·P



∪

i=1

(t)[



[u](u)|

(Q)

+ C(u

−P



∪

i=1

(t)[



]

(Q)

+ c(u −P



∪

i=1

(t)[



[u])

(Q)

+ l



, (8.19)

where l

→ 0whenε → 0.

It implies the strong convergence of u

→ u in L

(Q). Now from previous

estimates and Sobolev imbedding we get

• H

→ H

u weakly in L

(0,T,L

3−p

(Ω)),

• H

→ H

˜u weakly in L

(Q),

190 B. Ducomet and

S. Neˇcasov´a

• H

− H

→ H

u − H

u weakly in L

(0,T,L

3−p

(Ω)),

• H

− H

u → 0 strongly in L

(Q).

Thus

Hu = H

u, div (uH)=div(

uH),

and ﬁnally we obtain

u = H

The proof of the inequality (8.19) goes exactly in the same way as in Section 6.3.

Finally it remains to show the strong convergence of the gradient of velocity and

to pass to the limit, see Section 6.4. Then we pass to the limit with d similarly as

in Feireisl and Novotn´y [18]. The last step is passing to the limit with δ which must

proceed as with the ε limit together with using the transport theory developed by

DiPerna-Lions [10].

Acknowledgment

The work of

S.N. was supported by Grant IAA100190804 of GA ASCR in the

framework of the general research programme of the Academy of Sciences of the

Czech Republic, Institutional Research Plan AV0Z10190503. The ﬁrst part of the

paper was done during her stay in Cea, Arpajon. The ﬁnal version was supported

by the Grant P201/11/1304 of Grant Agency of Czech Republic. She would like to

thank Prof. Ducomet and his colleagues for their hospitality during her stay there.

We would like to thank the referee for his stimulating remarks and suggestions.

References

[1] W. Borchers. Zur Stabilit¨at und Faktorisierungsmethode f¨ur die Navier-Stokes-Glei-

chungen inkompressibler viskoser Fl¨ussigkeiten. Habilitation Thesis, Univ. of Pader-

born, 1992.

[2] C. Bost, G.H. Cottet and E. Maitre. Numerical analysis of a penalization method

for the three-dimensional motion of a rigid body in an incompressible viscous ﬂuid.

Preprint, 2009.

[3] H. Brenner. The Stokes resistance of an arbitrary particle II. Chem. Engng. Sci. 19,

599–624, 1959.

[4] M. Bul´ıˇcek, E. Feireisl and J. M´alek. Navier-Stokes-Fourier system for incompressible

ﬂuids with temperature dependent material coeﬃcients. Nonlinear Analysis: Real

World Applications, 10, 2, 992–1015, 2009.

[5] C. Conca, J. San Martin and M. Tucsnak. Existence of solutions for the equations

modelling the motion of a rigid body in a viscous ﬂuid. Commun. Partial Diﬀerential

Equation, 25, 1019–1042, 2000.

[6] B. Desjardins and M.J. Esteban. Existence of weak solutions for the motion of rigid

bodies in a viscous ﬂuid. Arch. Rational Mech. Anal., 146, 59–71, 1999.

[7] B. Desjardins and M.J. Esteban. On weak solutions for ﬂuid-rigid structure interac-

tion: Compressible and incompressible models. Commun. Partial Diﬀerential Equa-

tions, 25, 1399–1413, 2000.

On the Motion of Several Rigid Bodies 191

[8] B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba. Global in time weak solutions

for compressible barotropic self-graviting ﬂuids, DCDS, A, 11, 1, 113–130.

[9] B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba: Existence global pour un ﬂuide

barotropique autogravitant, C. R. Acad. Paris, 332, 627-632, 2001.

[10] R.J. DiPerna and P.-L. Lions. Ordinary diﬀerential equations, transport theory and

Sobolev spaces. Invent. Math., 98, 511–547, 1989.

[11] R. Farwig. An L

-analysis of viscous ﬂuid ﬂow past a rotating obstacle. Tˆohoku

Math. J. , 58 , 129–147, 2005

[12] R. Farwig. Estimates of lower order derivatives of viscous ﬂuid ﬂow past a rotating

obstacle. Banach Center Publications Warsaw, 70, 73–82, 2005.

[13] R. Farwig and T. Hishida. Stationary Navier-Stokes ﬂow around a rotating obstacle.

Funkcialaj Ekvacioj, 3, 371–403, 2007.

[14] R. Farwig, T. Hishida, and D. M¨uller. L

-theory of a singular “winding” integral

operator arising from ﬂuid dynamics. Paciﬁc J. Math. , 215 , 297–312, 2004.

[15] E. Feireisl and J. M´alek. On the Navier- Stokes equations with temperature depen-

dent transport coeﬃcients. Diﬀer. Equ. Nonl. Mech., Art. Id 90616, 2006.

[16] E. Feireisl. On the motion of rigid bodies in a viscous compressible ﬂuid. Arch.

Rational Mech. Anal., 167, 281–308, 2003.

[17] E. Feireisl, M. Hillairet and

S. Neˇcasov´a. On the motion of several rigid bodies in an

incompressible non-Newtonian ﬂuid. Nonlinearity, 21, 1349–1366, 2008.

[18] E. Feireisl and A. Novotn´y. Singular limits in thermodynamics of viscous ﬂuids.

Birkh¨auser, Basel, 2008.

[19] E. Feireisl, J. Neustupa, J. Stebel. Convergence of a Brinkman-type penalization for

compressible ﬂuid ﬂows. Preprint of the Necas Centrum, 2010.

[20] J. Frehse, J. M´alek and M. R˚uˇziˇcka. Large data existence result for unsteady ﬂows

of inhomogeneous heat-conducting incompressible ﬂuids. Preprint of the Necas Cen-

trum, 2008.

[21] J. Frehse and M. R˚uˇziˇcka. Non-homogeneous generalized Newtonian ﬂuids. Math.

Z., 260, 35–375, 2008.

[22] G.P. Galdi. On the steady self-propelled motion of a body in a viscous incompressible

ﬂuid. Arch. Rat. Mech. Anal., 148, 53–88, 1999.

[23] G.P. Galdi. On the motion of a rigid body in a viscous ﬂuid: A mathematical analysis

with applications. Handbook of Mathematical Fluid Dynamics, Vol. I, Elsevier Sci.,

Amsterdam, 2002.

[24] D. G´erard-Varet and M. Hillairet. Regularity issues in the problem of ﬂuid structure

interaction. Archive for Rational Mechanical Analysis. In press.

[25] M.D. Gunzburger, H.C. Lee, and A. Seregin. Global existence of weak solutions

for viscous incompressible ﬂow around a moving rigid body in three dimensions. J.

Math. Fluid Mech., 2, 219–266, 2000.

[26] T.I. Hesla. Collision of smooth bodies in a viscous ﬂuid: A mathematical investiga-

tion. 2005. PhD Thesis – Minnesota.

[27] M. Hillairet. Lack of collision between solid bodies in a 2D incompressible viscous

ﬂow. Comm. Partial Diﬀerential Equations, 32, 7-9, 1345–1371, 2007. Preprint –

ENS Lyon.

[28] M. Hillairet and T. Takahashi. Collisions in three dimensional ﬂuid structure inter-

action problems SIAM J. Math. Anal., 40, 6, 2451–2477, 2009

192 B. Ducomet and

S. Neˇcasov´a

[29] K.-H. Hoﬀmann and V.N. Starovoitov. On a motion of a solid body in a viscous

ﬂuid. Two dimensional case. Adv. Math. Sci. Appl., 9, 633–648, 1999.

[30] K.H. Hoﬀmann and V.N. Starovoitov. Zur Bewegung einer Kugel in einer z¨ahen

Fl¨ussigkeit. (German) [On the motion of a sphere in a viscous ﬂuid] Doc. Math., 5 ,

15–21, 2000.

[31] G. Kirchhoﬀ.

Uber die Bewegung eines Rotationsk¨orpers in einer Fl¨ussigkeit. (Ger-

man) [On the motion of a rotating body in a ﬂuid] Crelle J. 71 , 237–281, 1869.

[32] H. Koch and V.A. Solonnikov. L

estimates for a solutions to the nonstationary

Stokes equations. J. Math. Sci., 106, 3042–3072, 2001.

[33] J. M´alek,J.Neˇcas, M. Rokyta, and M. R˚uˇziˇcka. Weak and measure-valued solutions

to evolutionary PDE’s. Chapman and Hall, London, 1996.

[34] J.A. San Martin, V. Starovoitov, and M. Tucsnak. Global weak solutions for the two

dimensional motion of several rigid bodies in an incompressible viscous ﬂuid. Arch.

Rational Mech. Anal., 161, 93–112, 2002.

[35] D. Serre. Chute libre d’un solide dans un ﬂuide visqueux incompressible. Existence.

Jap. J. Appl. Math., 4, 99–110, 1987.

[36] H. Sohr. The Navier-Stokes equations: An elementary functional analytic approach.

Birkh¨auser Verlag, Basel, 2001.

[37] V.N. Starovoitov. Nonuniqueness of a solution to the problem on motion of a rigid

body in a viscous incompressible ﬂuid. J. Math. Sci., 130, 4893–4898, 2005.

[38] V.N. Starovoitov. Behavior of a rigid body in an incompressible viscous ﬂuid near

boundary. In International Series of Numerical Mathematics, 147, 313–327, 2003.

[39] N.V. Judakov. The solvability of the problem of the motion of a rigid body in a

viscous incompressible ﬂuid. (Russian) Dinamika Sploˇsn. Sredy Vyp., 18, 249–253,

1974.

[40] L. Tartar. Compensated compactness and applications to partial diﬀerential equa-

tions. Heriot-Watt Symposium, Vol. IV, Res. Notes in Math. 39, Pitman, Boston,

Mass., 136–212, 1979.

[41] W. Thomson (Lord Kelvin). Mathematical and Physical Papers, Vol. 4. Cambridge

University Press, 1982.

[42] H.F. Weinberger. On the steady fall of a body in a Navier-Stokes ﬂuid. Proc. Symp.

Pure Mathematics 23, 421–440, 1973.

[43] H.F. Weinberger. Variational properties of steady fall in Stokes ﬂow. J. Fluid Mech.

52, 321–344, 1972.

[44] J. Wolf. Existence of weak solutions to the equations of non-stationary motion of

non-newtonian ﬂuids with shear rate dependent viscosity. J. Math. Fluid Dynamics,

8, 1–35, 2006.

Bernard Ducomet

CEA, DAM, DIF, F-91297 Arpajon, France

e-mail: bernard.ducomet@cea.fr

S´arka Neˇcasov´a

Institute of Mathematics of the Academy of Sciences of the Czech Republic

Zitn´a 25, CZ-11567 Praha 1, Czech Republic

e-mail: matus@math.cas.cz

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 193–209

 2011 Springer Basel AG

Geometric Aspects of the Periodic

μ-Degasperis-Procesi Equation

Joachim Escher, Martin Kohlmann and Boris Kolev

Dedicated to Herbert Amann on the oc casion of his 70th birthday

Abstract. We consider the periodic μDP equation (a modiﬁed version of

the Degasperis-Procesi equation) as the geodesic ﬂow of a right-invariant

aﬃne connection ∇ on the Fr´echet Lie group Diﬀ

∞

)ofallsmoothand

orientation-preserving diﬀeomorphisms of the circle S

= R/Z.OntheLie

algebra C

∞

)ofDiﬀ

∞

), this connection is canonically given by the sum

of the Lie bracket and a bilinear operator. For smooth initial data, we show

the short time existence of a smooth solution of μDP which depends smoothly

on time and on the initial data. Furthermore, we prove that the exponential

map deﬁned by ∇ is a smooth local diﬀeomorphism of a neighbourhood of

zero in C

∞

) onto a neighbourhood of the unit element in Diﬀ

∞

). Our

results follow from a general approach on non-metric Euler equations on Lie

groups, a Banach space approximation of the Fr´echet space C

∞

), and a

sharp spatial regularity result for the geodesic ﬂow.

Mathematics Subject Classiﬁcation (2000). Primary 53D25; Secondary 37K65.

Keywords. Degasperis–Procesi equation, Euler equation, geodesic ﬂow.

1. Introduction

In recent years, several nonlinear equations arising as approximations to the gov-

erning model equations for water waves attracted a considerable amount of atten-

tion in the ﬂuid dynamics research community (cf. [22]). The Korteweg-de Vries

(KdV) equation is a well-known model for wave-motion on shallow water with

small amplitudes over a ﬂat bottom. This equation is completely integrable, al-

lows for a Lax pair formulation and the corresponding Cauchy problem was the

subject of many studies. However, it was observed in [3] that solutions of the KdV

equation do not break as physical water waves do: the ﬂow is globally well posed

for square integrable initial data (see also [23, 24] for further results).

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

The Camassa-Holm (CH) equation

+3uu

=2u

+ uu

xxx

+ u

txx

was introduced to model the shallow-water medium-amplitude regime (see [4]).

Closely related to the CH equation is the Degasperis-Procesi (DP) equation

+4uu

=3u

+ uu

xxx

+ u

txx

which was discovered in a search for integrable equations similar to the CH equa-

tion (see [12]). Both equations are higher-order approximations in a small ampli-

tude expansion of the incompressible Euler equations for the unidirectional motion

of waves at a free surface under the inﬂuence of gravity (cf. [10]). They have a bi-

Hamiltonian structure, are completely integrable and allow for wave breaking and

peaked solitons [6, 13, 16, 21]. The Cauchy problem for the periodic CH equation

in spaces of classical solutions has been studied extensively (see, e.g., [5, 33]); in

[7] and [11] the authors explain that this equation is also well posed in spaces

which include peakons, showing in this way that peakons are indeed meaningful

solutions of CH. Well-posedness for the periodic DP equation and various features

of solutions of the DP on the circle are discussed in [20]. Both, the CH equation

and the DP equation, are embedded into the family of b-equations

= −(m

u + bmu

),m:= u − u

, (1.1)

where u(t, x) is a function of a spatial variable x ∈ S

and a temporal variable t ∈

R. Note that the family (1.1) can be derived as the family of asymptotically equiv-

alent shallow water wave equations that emerges at quadratic-order accuracy for

any b = −1 by an appropriate Kodama transformation [14]. For b = 2, we recover

the CH equation and for b = 3, we get the DP equation. Note that the b-equation is

integrable only if b =2orb = 3. For further results and references we refer to [19].

Since the pioneering works [1, 15], geometric interpretations of evolution

equations led to several interesting results in the applied analysis literature. A

detailed discussion of the CH equation in this framework was given by [26]. The

geometrical aspects of some metric Euler equations are explained in [8, 9, 25,

32]. Studying the b-equations as a geodesic ﬂow on the diﬀeomorphism group

Diﬀ

∞

), it was shown recently in [17] that for smooth initial data u

,thereis

a unique short-time solution u(t, x) of (1.1), depending smoothly on (t, u

). The

crucial idea is to deﬁne an aﬃne (not necessarily Riemannian) connection ∇ on

Diﬀ

∞

), given at the identity by the sum of the Lie bracket and a bilinear sym-

metric operator B,sothatB(u, u)=−u

. Most importantly, this approach also

works for b-equations of non-metric type and it motivates the study of geometric

quantities like curvature or an exponential map for the family (1.1). In particu-

lar, the authors of [17] proved that the exponential map for ∇ is a smooth local

diﬀeomorphism near zero in C

∞

). Recently it has been shown in [18] that the b-

equation can be realized as a metric Euler equation only if b = 2. In all other cases

b = 2 there is no Riemannian metric on Diﬀ

∞

) such that the corresponding

The Periodic μDP Equation 195

geodesic ﬂow is re-expressed by the b-equation. Geometric aspects of some novel

nonlinear PDEs related to CH and DP are discussed in [31].

In this paper, we study the μDP equation

μ(u

) −u

txx

+3μ(u)u

−3u

−uu

xxx

=0, (1.2)

where μ denotes the projection μ(u)=



u dx and u(t, x) is a spatially periodic

real-valued function of a temporal variable t ∈ R and a space variable x ∈ S

.The

μDP equation belongs to the family of μ-b-equations which follows from (1.1) by

replacing m = μ(u) − u

. The study of μ-variants of (1.1) is motivated by the

following key observation: Letting m = −∂

u, equation (1.1) for b = 2 becomes

the Hunter-Saxton (HS) equation

+ uu

xxx

+ u

txx

=0,

which possesses various interesting geometric properties, see, e.g., [29, 30], whereas

the choice m =(1−∂

)u leads to the CH equation as explained above. In the search

for integrable equations that are given by a perturbation of −∂

,theμ-b-equation

has been introduced and it could be shown that it behaves quite similarly to the

b-equation; see [31] where the authors discuss local and global well-posedness as

well as ﬁnite time blow-up and peakons. Our study of the μDP equation is inspired

by the results in [17]. In fact using the approach of [17] we shall conceptualise a

geometric picture of the μDP equation.

Our study is mostly performed in the C

∞

-category. Elements of C

∞

)are

sometimes also called smooth for brevity.

We will reformulate the μDP equation in terms of a geodesic ﬂow on Diﬀ

∞

)

to obtain the following main result: Given a smooth initial data u

(x), for which

||u

)

is small, there is a unique smooth solution u(t, x) of (1.2) which depends

smoothly on (t, u

). More precisely, we have

Theorem 1.1. There exists an open interval J centered at zero and δ>0 such

that for each u

∈ C

∞

) with ||u

)

<δ, there exists a unique solution

u ∈ C

∞

(J, C

∞

)) of the μDP equation such that u(0) = u

. Moreover, the

solution u depends smoothly on (t, u

) ∈ J × C

∞

It is known that the Riemannian exponential mapping on general Fr´echet

manifolds fails to be a smooth local diﬀeomorphism from the tangent space back

to the manifold, cf. [8]. Therefore the following result is quite remarkable.

Theorem 1.2. The exponential map exp at the unity element for the μDP equation

on Diﬀ

∞

) is a smooth local diﬀeomorphism from a neighbourhood of zero in

∞

) onto a neighbourhood of id in Diﬀ

∞

Our paper is organized as follows: In Section 2, we rewrite (1.2) in terms of

alocalﬂowϕ ∈ Diﬀ

), n ≥ 3, and explain the geometric setting. The resulting

equation is an ordinary diﬀerential equation and in Section 3, we apply the Theo-

rem of Picard-Lindel¨of to obtain a solution of class C

) with smooth dependence

on t and u

(x). In addition, we show that this solution in Diﬀ

) ×C

)does

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

neither lose nor gain spatial regularity as t varies through the associated interval of

existence. We then approximate the Fr´echet Lie group Diﬀ

∞

) by the topologi-

cal groups Diﬀ

)andtheFr´echet space C

∞

) by the Banach spaces C

)

to obtain an analogous existence result for the geodesic equation on Diﬀ

∞

Finally, in Section 4, we make again use of a Banach space approximation to prove

that the exponential map for the μDP is a smooth local diﬀeomorphism nero zero

as a map C

∞

) → Diﬀ

∞

2. Geometric reformulation of the μDP equation

We write Diﬀ

∞

) for the smooth orientation-preserving diﬀeomorphisms of the

unit circle S

= R/Z and Vect

∞

) for the space of smooth vector ﬁelds on

. Clearly, Diﬀ

∞

) is a Lie group and it is easy to see that its Lie algebra

is Vect

∞

): If t → ϕ(t) is a smooth path in Diﬀ

∞

)withϕ(0) = id, then

(0,x) ∈ T

for all x ∈ S

and thus the Lie algebra element ϕ

(0, ·)isa

smooth vector ﬁeld on S

. Furthermore, since T S

 S

× R is trivial, we can

identify the Lie algebra Vect

∞

)withC

∞

). Note that [u, v]=u

v − v

u is

the corresponding Lie bracket. In the following, we will also use that Diﬀ

∞

)

has a smooth manifold structure modelled over the Fr´echet space C

∞

). In

particular, Diﬀ

∞

)isaFr´echet Lie group and thus it is parallelizable, i.e.,

T Diﬀ

∞

)  Diﬀ

∞

) ×C

∞

). Let Diﬀ

) denote the group of orientation-

preserving diﬀeomorphisms of S

which are of class C

). Similarly, Diﬀ

)

has a smooth manifold structure modelled over the Banach space C

). Note

that Diﬀ

) is only a topological group but not a Banach Lie group, since the

composition and inversion maps are continuous but not smooth. Furthermore, the

trivialization T Diﬀ

)  Diﬀ

) ×C

) is only topological and not smooth.

In this section, we write (1.2) as an ordinary diﬀerential equation on the

tangent bundle Diﬀ

) × C

), where n ≥ 3. In a ﬁrst step, we rewrite (1.2)

using the operator A := μ − ∂

.Hereμ denotes the linear map given by f →



f(t, x)dx for any function f (t, x) depending on time t and space x ∈ S

.Observe

that μ(∂

f)=0fork ≥ 1iff and its derivatives are continuous functions on S

Furthermore, μ(f ) is still depending on the time variable t. The following lemma

establishes the invertibility of A as an operator acting on C

)forn ≥ 2.

Lemma 2.1. Given n ≥ 2, the operator A = μ − ∂

maps C

) isomorphically

onto C

n−2

). The inverse is given by

−1

f)(x)=



−

x +





f(a)da +



x −





f(b)db da

−



f(b)db da +



f(c)dc db da.

Proof. Clearly, μ(A

−1

f)=μ(f)and(A

−1

= μ(f) −f so that A(A

−1

f)=f.

To verify that A is surjective, we observe that ∂

−1

f)(0) = ∂

−1

f)(1) for

The Periodic μDP Equation 197

all k ∈{0,...,n}.ToseethatA is injective, assume that Au =0foru ∈ C

)

and n ≥ 2. Then there are constants c, d ∈ R such that u =

μ(u)x

+ cx + d.By

periodicity we ﬁrst conclude that c =0andμ(u)=0.Henced has to vanish as

well. 

Lemma 2.2. Assume that u ∈ C((−T,T), C

)) ∩ C

((−T,T), C

n−1

)) is a

solution of (1.2) for some n ≥ 3 with T>0. Then the μDP equation can be

written as

= −A

−1

(u(Au)

+3(Au)u

). (2.1)

Proof. Writing (1.2) in the form

μ(u

) − u

txx

= uu

xxx

−3u

(μ(u) − u

weseethatitisequivalentto

= −u(Au)

− 3(Au)u

Thus u is a solution of (1.2) if and only if (2.1) holds true. 

As explained in [27, 28], the vector ﬁeld u(t, x) admits a unique local ﬂow ϕ

of class C

), i.e.,

(t, x)=u(t, ϕ(t, x)),ϕ(0,x)=x

for all x ∈ S

and all t in some open interval J ⊂ R. We will use the short-hand

notation ϕ

= u ◦ϕ for ϕ

(t, x)=u(t, ϕ(t, x)); i.e., ◦ denotes the composition with

respect to the spatial variable. Particularly, we have that u = ϕ

◦ϕ

−1

.Moreover,

given (ϕ, ξ) ∈ C

(J, Diﬀ

) × C

)), then ϕ

−1

(t)isaC

)-diﬀeomorphism

for all t ∈ J and ξ ◦ ϕ

−1

∈ C

(J, C

)).

In this paper, we are mainly interested in smooth diﬀeomorphisms on S

For the reader’s convenience we brieﬂy recall the basic geometric setting. Let us

consider the Fr´echet manifold Diﬀ

∞

) and a continuous non-degenerate inner

product ·, · on C

∞

), i.e., u →u, u is continuous (and hence smooth) and

u, v =0forallv ∈ C

∞

) forces u = 0. To deﬁne a weak right-invariant

Riemannian metric on Diﬀ

∞

(S), we extend the inner product ·, · to any tangent

space by right-translations, i.e., for all g ∈ Diﬀ

∞

)andallu, v ∈ T

Diﬀ

∞

we set

u, v



−1

)

∗

u, (R

−1

)

∗

where e denotes the identity. Observe that any open set in the topology induced by

this inner product is open in the Fr´echet space topology of C

∞

) but the converse

is not true. We therefore call ·, · a weak Riemannian metric on Diﬀ

∞

(S), cf. [8].

We next deﬁne a bilinear operator B : Vect

∞

) ×Vect

∞

) → Vect

∞

)by

B(u, v)=

((ad

)

∗

(v)+(ad

)

∗

(u)),

where (ad

)

∗

is the adjoint (with respect to ·, ·) of the natural action of the Lie

algebra on itself given by ad

: v → [u, v]. Observe that B deﬁnes a right-invariant