http://www.uta.edu/math/preprint/

Technical Report 2008-10

Wave Breaking Phenomena and Stability of Peakons for the

Degasperis-Procesi

Yue Liu

Wave Breaking Phenomena and Stability

of Peakons for the Degasperis-Procesi

Equation

Yue Liu∗

This article is dedicated to Professor Guangchang Dong for his 80th Birthday

Abstract

We survey some recent results concerning with the Degasperis-Procesiequation, which can be derived as a member of a one-parameter familyof asymptotic shallow-water wave approximations to the Euler equa-tions with the same asymptotic accuracy as that of the Camassa-Holmequation. We will focus on some important results including wavebreaking phenomena, blow-up structure, global weak solutions and theorbital stability of the peaked solitons.

AMS subject classification (2000): 35G25, 35L05, 35Q35, 35Q51, 58D05

Keywords: Camassa-Holm equation; Degasperis-Procesi equation; Peakons;Shallow water waves; Wave breaking; Global weak solutions; Blow-upstructure; Stability of peakons

1 Introduction

Considered herein is the Degasperis-Procesi (DP) equation

(1.1) yt + yxu + 3yux = 0, x ∈ R, t > 0,

with y = u− uxx.Degasperis and Procesi [27] studied a family of third order dispersive

nonlinear equations

(1.2) ut − α2uxxt + γuxxx + c0ux = (c1u2 + c2u

2x + c3uuxx)x.

with six real constants c0, c1, c2, c3, γ, α ∈ R. They [27] found that thereare only three equations from this family were asymptotically integrable upto third order, that is, the Korteweg-de Vries (KdV) equation (α = c2 =

∗Department of Mathematics, University of Texas, Arlington, TX 76019, yliu@uta.edu

1

c3 = 0), the Camassa-Holm (CH) equation (c1 = − 3c32α2 , c2 = c3

2 ), and onenew equation (c1 = −2c3

α2 , c2 = c3), which is called the Degasperis-Procesiequation. By rescaling, shifting the dependent variable, and finally apply-ing a Galilean transformation, those three completely integrable1 equationscan be transformed into the following forms, the Korteweg-de Vries (KdV)equation

ut + uxxx + uux = 0,

the Camassa-Holm (CH) shallow water equation [5, 28, 40],

(1.3) yt + yxu + 2yux = 0, y = u− uxx,

and the Depasperis-Procesi equation of the form (1.1). These three casesare all the completely integrable candidates for (1.2) [5, 25, 27]. Applyinga reciprocal transformation to the Degasperis-Procesi equation, Degasperis,Holm and Hone [25] used the Painleve analysis to show the formal integra-bility of the DP equation as Hamiltonian systems by constructing a Lax pairand a bi-Hamiltonian structure.

Equation (1.1) was also derived as, in dimensionless space-time variables(x, t), an approximation to the incompressible Euler equations for shallowwater under the Kodama transformation [29, 38, 39] and its asymptoticaccuracy is the same as that of the Camassa-Holm (CH) shallow waterequation, where u(t, x) is considered as the fluid velocity at time t in thespatial x-direction with momentum density y. More interestingly, the DPequation is recently observed as a model supporting shock waves [49].

More recently, Constantin and Lannes [18] give a rigorous proof of boththe CH equation and the DP equation are valid approximation to the gov-erning equations for water waves (see also [40] for the formal asymptoticprocedures) and also show the relevance of these two equations as modelsfor the propagation of shallow water waves.

To see this rigorous justification of the derivation, one can consider thewater wave equations for one-dimensional surfaces in nondimensionalizedform

µ∂2xψ + ∂2

zΨ = 0, in Ωt,

∂zΨ = 0, at z = −1,

∂tξ − 1µ(−µ∂xξ∂xΦ + ∂zΨ) = 0, at z = εξ,

∂tΨ + ε2(∂xΨ)2 + ε

2µ(∂zΨ)2 = 0, at z = εξ,

where x → εξ(t, x) parameterizes the elevation of the free surface at timet, Ωt = (x, z); −1 < z < εξ(t, x) is the fluid domain delimited by the

1Integrability is meant in the sense of the infinite-dimensional extension of a classicalcompletely integrable Hamiltonian system: there is a transformation which converts theequation into an infinite sequence of linear ordinary differential equations which can betrivially integrated [52].

2

free surface and the flat bottom z = −1, Ψ(t, ·) is the velocity potentialassociated to the flow, and ε and µ are two dimensionless parameters definedby

ε =a

h, µ =

h2

λ2,

where h is the mean depth, a is the typical amplitude, and λ is the typicalwavelength of the waves.

Define the vertically averaged horizontal component of the velocity by

u(t, x) =1

1 + εξ

∫ εξ

−1∂xΨ(t, x, z)dz.

In the shallow-water scaling (µ ¿ 1), one can derive the Green-Naghdiequations [1, 35] for one-dimensional surfaces and flat bottoms without anyassumption on ε(ε = O(1)). These equations couple the free surface elevationξ to the vertically averaged horizontal component of the velocity u and canbe written as

ξt + ((1 + εξ)u)x = 0ut + ξx + εuux = µ

31

1+εξ

((1 + εξ)3(uxt + εuuxx − εu2

x))x,

where O(µ2) terms have been neglected. In the so-called long-wave regime

µ ¿ 1, ε = O(µ),

the right-going wave should satisfy the KdV equation

ut + ux + ε32uux + µ

16uxxx = 0

with ξ = u + O(ε, µ), or a wider class of equations, referred as the BBMequations [2](sometimes also called the regularized long-wave equations),which provide an approximation of the exact water wave equations of thesame accuracy as the KdV equation.

ut + ux +32εuux + µ(αuxxx + βuxxt) = 0, with α− β =

16.

Consider now the so-called Camassa-Holm scaling, that is

µ ¿ 1, ε = O(√

µ).

With this scaling, one still has ε ¿ 1, the dimensionless parameter is, how-ever, larger here than in the long wave scaling, and the nonlinear effects aretherefore stronger and it is possible that a stronger nonlinearity could allowthe appearance of breaking waves, which is a fundamental phenomenon inthe theory of water waves that is not captured by the BBM equations.

3

Define the horizontal velocity uθ (θ ∈ [0, 1]) at the level line θ of the fluiddomain by

v ≡ uθ(x) = ∂xΨ∣∣z=(1+εξ)θ−1.

Let p ∈ R and λ = 12(θ2 − 1

3), with θ ∈ [0, 1]. Assume

α = p + λ, β = p− 16

+ λ, γ = −23p− 1

6− 3

2λ, δ = −9

2p− 23

24− 3

2λ.

Under the Camassa-Holm scaling, one should have the following class ofequations for v ≡ uθ (θ ∈ [0, 1]), namely

(?) vt + vx +32εvvx + µ(αvxxx + βvxxt) = εµ(γvvxxx + δvxvxx),

where O(ε4, η2) terms have been discarded. The vertically averaged hori-zontal velocity u and the free surface ξ satisfy

u = uθ + µλuθxx + 2µελuθuθ

xx,

ξ = u +ε

4u2 + µ

16uxt − εµ

(16uuxx +

548

u2x

).

By rescaling, shifting the dependent variable, and applying a Galileantransformation, the Camassa-Holm equation

Ut + κUx + 3UUx − Utxx = 2UxUxx + UUxxx

can be obtained from (?) if the following conditions hold

β < 0, α 6= β, β = −2γ, δ = 2γ,

where p = −13 , θ2 = 1

2 . The solution uθ of (?) is transformed to the solutionU of the CH equation by

U(t, x) =1auθ

(x

b+

ν

ct,

t

c

),

with a = 2εκ(1− ν), b2 = − 1

β µ, ν = αβ , and c = b

κ(1− ν).On the other hand, the DP equation

Ut + κUx + 4UUx − Utxx = 3UxUxx + UUxxx

can also be derived if the following conditions hold

β < 0, α 6= β, β = −83γ, δ = 3γ,

where p = − 77216 , θ2 = 23

36 . The solution uθ of (?) is also transformed to thesolution U of the DP equation by

U(t, x) =1auθ

(x

b+

ν

ct,

t

c

),

4

with a = 83εκ(1 − ν), b2 = − 1

β µ, ν = αβ , and c = b

κ(1 − ν). A detailedderivation of the CH and DP equations can be found in [18, 40].

It is well known that the KdV equation is an integrable Hamiltonianequation that possesses smooth solitons as traveling waves. In the KdVequation, the leading order asymptotic balance that confines the travelingwave solitons occurs between nonlinear steepening linear dispersion. How-ever, the nonlinear dispersion and nonlocal balance in the CH equation andthe DP equation, even in the absence of linear dispersion, can still producea confined solitary traveling waves

u(t, x) = cϕ(x− ct) = ce−|x−ct|,

traveling at constant speed c > 0, which are called the peakons [5, 25].Peakons of both equations are true solitons that interact via elastic collisionsunder the CH dynamics, or the DP dynamics, respectively. The peakons ofthe CH equation and the DP equation are orbitally stable [22, 44]. The resultof the stability of the DP equation will be discussed in the last section.

The DP equation can be rewritten as the form

(1.4) ut − utxx + 4uux = 3uxuxx + uuxxx, t > 0, x ∈ R.

It is noted that the peaked solitons are not classical solutions of (1.4).They satisfy the Degasperis-Procesi equation in conservation law form

(1.5) ut + ∂x

(12u2 + p ∗

(32u2

))= 0, t > 0, x ∈ R,

where p(x) = 12e−|x|, ∗ stands for convolution with respect to the spatial

variable x ∈ R, and p ∗ f = (1− ∂2x)−1f.

Since p(x) = ϕ(x), in view of the structure of Eq.(1.5), it is quite clearwhy the peakons can be understood as solutions.

Wave breaking is one of the most intriguing long-standing problems ofwater wave theory [61]. For models describing water waves we say that wavebreaking holds if the wave profile remains bounded, but its slope becomesunbounded in finite time [61]. Breaking waves are commonly observed inthe ocean and important for a variety of reasons, but surprisingly little isknown about them. Indeed, breaking waves place large hydrodynamic loadson man-made structure, transfer horizontal momentum to surface currents,provide a source of turbulent energy to mix the upper layers of the ocean,move sediment in shallow water, and enhance the air-sea exchange of gasesand particulate matter [7, 57]. To further understand why waves break andwhat happens during and after breaking themselves, we must first inves-tigate the dynamics of wave breaking. Research work on breaking wavescan be divided into three categories: those concerning waves (1) before, (2)during, and (3) after breaking. Although we are now understanding much

5

about the processes leading up to breaking, there are still some aspects ofthese questions unanswered, in particular, question (3), what happens afterbreaking of those waves. In this review we shall concentrate on some of thelatest results for the DP equation in the first two categories.

The KdV equation is well-known a model for water-motion on shallowwater with a flat bottom and admits interaction for its solitary waves. It,however, does not describe breaking of wave as physical water waves do(the KdV equation is globally well-posed for initial data in L2 [42, 59]). Onthe other hand, wave-breaking phenomena have been observed for certainsolutions to the Whitham equation [61],

ut + uux +∫

RK(x− ξ)ux(t, ξ)dξ = 0

with the singular kernel

K(x) =12π

∫

R

(tanh ξ

ξ

)1/2

eiξxdξ,

(see [16] for a rigorous proof). However, the numerical calculations carriedout for the Whitham equation do not support any strong claim that soli-ton interaction can be expected [26]. As mentioned by Whitham [61], it isintriguing to know which mathematical models for shallow water waves ex-hibit both phenomena of soliton interaction and wave breaking. It is foundthat both of the CH equation and DP equation could be first such equationsand have the potential to become the new master equations for shallow wa-ter wave theory [34], modeling the soliton interaction of peaked travelingwaves, wave breaking, admitting as solutions permanent waves, and beingintegrable Hammiltonian systems. For the CH equation, a procedure to un-derstand the continuation of solutions past wave breaking has been recentlypresented by Bressan and Constantin [3].

As far as we know, the case of the Camassa-Holm equation (first derivedby Fokas and Fuchssteiner [33] using the method of recursion operators as anabstract bi-Hamiltonian equation) is well understood by now [12, 15, 16, 19]and the citations therein, while the Degasperis-Procesi equation case is thesubject of this article. The main mathematical questions concerning withthe DP equation are the well-posedness of the initial-value problem, wave-breaking phenomena, existence of global weak solutions, and stability ofpeakons and their role in the dynamics.

Since its discovery, there has been considerable interest in the Degasperis-Procesi equation, cf. [17, 37, 43, 49, 51, 64, 65] and the citations therein.For example, Lundmark and Szmigielski [50] presented an inverse scatteringapproach for computing n-peakon solutions to Eq.(1.5). Holm and Staley[38] studied stability of solitons and peakons numerically to Eq.(1.5). Morerecently, Liu and Yin [47] proved that the first blow-up for Eq.(1.4) must

6

occur as wave breaking and shock waves possibly appear afterwards. It isshown in [47] that the lifespan of solutions of the DP equation is not affectedby the smoothness and size of initial profiles, but affected by the shape ofinitial profiles. This can be viewed as a significant difference between theDP equation (or the CH equation ) and the KdV equation.

Our goal is to present a review of some significant work of them.

Notation. As above and henceforth, we denote by ∗ the convolution. For1 ≤ p ≤ ∞, the norm in the Lebesgue space Lp(R) will be written ‖ · ‖Lp ,while ‖ · ‖Hs , s ≥ 0 will stand for the norm in the classical Sobolev spacesHs(R).

2 Comparisons between the Camassa-Holm equa-tion and the Degasperis-Procesi equation

The DP equation is presently of great interest due to its structure (inte-grability, special solutions presenting interesting features). While Eq.(1.1)has an apparent similarity to Eq.(1.3), which both are important modelequations for shallow water waves with the breaking phenomena, there aremajor structural differences and it is not much to know about its qualitativeaspects. One of the novel features of the DP equation is that it has notonly peakon solitons [25], u(t, x) = ce−|x−ct|, c > 0 but also shock peakons[11, 49] of the form

u(t, x) = − 1t + k

sgn(x)e−|x|, k > 0.

It is easy to see from [49] that the above shock-peakon solutions can beobserved by substituting (x, t) 7−→ (εx, εt) to Eq.(1.4) and letting ε → 0so that it yields the “derivative Burgers equation” (ut + uux)xx = 0, fromwhich shock waves form.

In the periodic case of the spatial variable, both the CH equation andDP equation have periodic peakons [64] of the form

uc(t, x) = ccosh(x− ct− [x− ct]− 1

2)sinh(1

2), x ∈ R, t ≥ 0, c > 0.

However, it is recently shown by Escher, Liu and Yin [31] that the theperiodic DP equation possesses the periodic shock waves given by

uc(t, x) =

(cosh( 1

2)

sinh( 12)t + c

)−1 sinh(x−[x]− 12)

sinh( 12)

, x ∈ R \ Z, c > 0,

0, x ∈ Z.

On the other hand, the isospectral problem in the Lax pair for the DPequation is of third-order instead of second [25], and consequently is not

7

self-adjoint,ψx − ψxxx − λyψ = 0,

and

ψt +1λ

ψxx + uψx −(

ux +23λ

)ψ = 0,

while the isospectral problem for the CH equation is of second order [5],

ψxx − 14ψ − λyψ = 0

and

ψt −(

12λ

− u

)ψx − 1

2uxψ = 0

(in both cases y = u− uxx). The spectral analysis and the inverse spectraltheory for the CH equation are presented by Constantin and McKean [14,19] and Johnson [41]. Lundmark and Szmigielski [50] presented an inversescattering transform (IST) method for computing n−peakon solutions of theDP equation. The approach is similar to that used by Beals, Sattinger andSzmigielski [4] to obtain n−peakon solutions of the CH equation, but thepresent case does involve substantially new features as mentioned above.

It is also noted that the CH equation is a re-expression of geodesic flowon the diffeomorphism group [17] or on the Bott-Virasoro group [55], whileno such geometric derivation of the DP equation is available.

Another indication of the fact that there is no simple transformationof the DP equation into the CH equation is the entirely different form ofconservation laws for these two equations [5, 25]. The following are threeuseful conservation laws of the DP equation.

E1(u) =∫

Ry dx, E2(u) =

∫

Ryv dx, E3(u) =

∫

Ru3 dx,

where y = (1 − ∂2x)u and v = (4 − ∂2

x)−1u, while the corresponding threeuseful conservation laws of the CH equation are the following.

F1(u) =∫

Ry dx, F2(u) =

∫

R(u2 + u2

x) dx, F3(u) =∫

R(u3 + uu2

x) dx.

It is observed that the corresponding conservation laws of the DP equationare much weaker than those of the CH equation. Therefore, the issue ofif and how particular initial data generate a blow-up in finite time is moresubtle.

It is worth noticing the following result obtained by Henry [37] andMustafa [56]. which implies that, analogous to the case of the CH equation,smooth solutions of the DP equation have infinite propagation speed.

Proposition 2.1. Assume u0 is a smooth function with compact support.If the solution u with initial data u0 of (1.4) exists on some time interval[0, ε) with ε > 0 and, at any time instant t ∈ [0, ε), the solution u(t, ·) hascompact support, then u is identically zero.

8

3 The Cauchy problem

Recall p(x) = 12e−|x| and (1−∂2

x)−1f = p∗f. Then Eq.(1.1) can be rewrittenas the following form

(3.1)

ut + uux + ∂xp ∗ (32u2) = 0, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R.

The local well-posedness of the Cauchy problem (3.1) with initial data u0 ∈Hs(R), s > 3

2 can be obtained by applying the Kato’s semigroup theory forquasilinear evolution equations.

Lemma 3.1. [63] Given u0 ∈ Hs(R), s > 32 , there exist a maximal T =

T (u0) > 0 and a unique solution u to initial-value problem (3.1), such that

u = u(·, u0) ∈ C([0, T );Hs(R)) ∩ C1([0, T );Hs−1(R)).

Moreover, the solution depends continuously on the initial data, i.e. themapping u0 7→ u(·, u0) : Hs(R) → C([0, T );Hs(R)) ∩ C1([0, T );Hs−1(R)) iscontinuous and the maximal time of existence T > 0 can be chosen to beindependent of s.

In view of Lemma 3.1, using the energy method, one can establish thefollowing precise blow-up scenario of strong solutions to (3.1) [63], which issimilar to the case of the CH equation.

Proposition 3.2. [63] Given u0 ∈ Hs(R), s > 32 , blow up of the solution

u = u(·, u0) in finite time T < +∞ occurs if and only if

lim inft↑T

infx∈R

[ux(t, x)] = −∞.

Proposition 3.3. [47] Assume u0 ∈ Hs(R), s > 32 . Let T be the maximal

existence time of the solution u to (3.1) guaranteed by Lemma 3.1. Then wehave

‖u(t, x)‖L∞ ≤ 3‖u0(x)‖2L2t + ‖u0(x)‖L∞ , ∀t ∈ [0, T ].

Remark. It is observed that if u is the solution of (1.3) with initial datau0 ∈ H1(R), then we have for all t > 0,

‖u(t, ·)‖L∞(R) ≤√

2‖u(t, ·)‖H1(R) ≤√

2‖u0(·)‖H1(R).

The advantage of the CH equation in comparison with the KdV equation liesin the fact that the CH equation has peaked solitons and models wave break-ing (the wave profile remains bounded, but its slope becomes unbounded infinite time wave) [6]. However, note that the conservation laws of the DP

9

equation are much weaker than the those of the CH equation. In partic-ular, one can see that the conservation law E2(u) for the DP equation isequivalent to ‖u‖2

L2 . Indeed, by the Fourier transform, we have

(3.2) E2(u) =∫

Ryvdx =

∫

R

1 + ξ2

4 + ξ2|u(ξ)|2dξ ∼ ‖u‖2

L2 = ‖u‖2L2 .

It seems that the estimate in Proposition 3.3 is the best way to controlthe L∞− norm of the solution for the DP equation. It then follows fromProposition 3.2 and 3.3 that the DP equation also models wave breaking inany finite time and one can expect that all the points at which the wavebreaking occurs should be peaked points [47].

Consider the following differential equation

(3.3)

qt = u(t, q), t ∈ [0, T ),q(0, x) = x, x ∈ R.

Applying classical results in the theory of ordinary differential equations,one can obtain the following result on q.

Lemma 3.4. [65] Let u0 ∈ Hs(R), s ≥ 3, and let T > 0 be the maximalexistence time of the corresponding solution u to (3.1). Then Eq.(3.3) hasa unique solution q ∈ C1([0, T ) × R,R). Moreover, the map q(t, ·) is anincreasing diffeomorphism of R with

qx(t, x) = exp(∫ t

0ux(s, q(s, x))ds

)> 0, ∀(t, x) ∈ [0, T )× R.

The following result, analogous to the case of the CH equation, plays acrucial role in our considerations on global existence and blow-up solutions.It roughly says that y(t, ·) does not change on the time interval where it iswell-defined. We then infer by means of the geometric interpretation of q avery important invariant for the solutions to (3.1).

Lemma 3.5. [65] Let u0 ∈ Hs(R), s ≥ 3, and let T > 0 be the maximalexistence time of the corresponding solution u to (3.1). Setting y = u−uxx,we have

y(t, q(t, x))q3x(t, x) = y0(x), ∀(t, x) ∈ [0, T )× R.

Proof. The result can be obtained by differentiating the left-hand side withrespect to time and taking inton account (3.1), (3.3) and Lemma 3.4.

Remark. From Lemma 3.5, one can expect that the lifespan of solutions ofthe DP equation is not affected by the smoothness and size of initial profiles,but affected by the shape of initial profiles.

By Lemma 3.5, one can obtain the criteria for breaking-waves (Theorem3.6 and Theorem 4.1).

10

Theorem 3.6. [47] Assume u0 ∈ Hs(R), s > 32 and there exists x0 ∈

[−∞, ∞] such that

y0(x) ≤ 0 if x ≤ x0,y0(x) ≥ 0 if x ≥ x0.

Then initial-value problem (3.1) has a unique global strong solution

u = u(., u0) ∈ C([0,∞);Hs(R)) ∩ C1([0,∞);Hs−1(R)).

Moreover, E2(u) =∫R yv dx = E2(u0), where y = (1 − ∂2

x)u and v = (4 −∂2

x)−1u, and for all t ∈ R+ we have(i) ux(t, ·) ≥ −|u(t, ·)| on R,(ii) ‖u‖2

1 ≤ 6‖u0‖4L2t

2 + 4‖u0‖2L2‖u0‖L∞t + ‖u0‖2

1.

Proof. Note that the solution u of the DP equation satisfies

(3.4) u(t, x) =e−x

2

∫ x

−∞eηy(t, η)dη +

ex

2

∫ ∞

xe−ηy(t, η)dη

and

(3.5) ux(t, x) = −e−x

2

∫ x

−∞eηy(η)dη +

ex

2

∫ ∞

xe−ηy(η)dη.

From the above two equations, we deduce that

(3.6)

ux(t, x) ≥ u(t, x) if x ≤ q(t, x0),ux(t, x) ≥ −u(t, x) if x ≥ q(t, x0).

This implies that ux(t, ·) ≥ −|u(t, ·)| on R for all t ∈ [0, T ). Therefore, inview of Proposition 3.2, the global existence result can be obtained by usingthe estimate of ‖u‖L∞ in Proposition 3.3.

4 Wave breaking phenomena

Taking into account invariant of y in Lemma 3.5, one can establish somewave breaking results with different shapes of initial profiles.

Theorem 4.1. [47] Let u0 ∈ Hs(R), s > 32 . Assume there exists x0 ∈ R

such that

y0(x) = u0(x)− u0,xx(x) ≥ 0 if x ≤ x0,y0(x) = u0(x)− u0,xx(x) ≤ 0 if x ≥ x0,

and y0 changes sign. Then, the corresponding solution to (3.1) blows up infinite time T < +∞ and satisfies lim inf

t↑T inf

x∈R[ux(t, x)] = −∞.

11

The proof of the theorem is inspired by the argument of Constantin[12]. The idea of the proof is to obtain a differential inequality for the timeevolution of ux(t, q(t, x0)) which can be used to prove that T < ∞. We giveonly an outline and refer to [47] for the full details.

Proof. Firstly, by differentiating Eq.(1.5) with respect to x, we have

utx + uuxx = −u2x +

32u2 − p ∗

(32u2

)

= −u2x +

32u2 − p ∗

(12u2

x + u2

)+

12p ∗ (

u2x − u2

).

taking into account p ∗ (12u2

x + u2)(t, x) ≥ 1

2u2(t, x), it then follows that

(4.1) utx + uuxx ≤ −u2x + u2 +

12p ∗ (

u2x − u2

).

It is then deduced from the assumption of the theorem that for t ∈ [0, T )

(4.2)

y(t, x) ≥ 0 if x ≤ q(t, x0),y(t, x) ≤ 0 if x ≥ q(t, x0).

Next we define

(4.3) M(t, x) := e−x

∫ x

−∞eηy(t, η)dη, t ∈ [0, T ), and

(4.4) N(t, x) := ex

∫ ∞

xe−ηy(t, η)dη, t ∈ [0, T ).

Then it is easy to see that

(4.5) M(t, q(t, x0))N(t, q(t, x0)) = u2(t, q(t, x0))− u2x(t, q(t, x0)),

and

(4.6) M(t, q(t, x0))N(t, q(t, x0))

=∫ q(t,x0)

−∞eηy(t, η)dη

∫ ∞

q(t,x0)e−ηy(t, η)dη < 0, t ∈ [0, T ).

12

On the other hand, we have

dM(t, q(t, x0))dt

= −u(t, x0)M(t, q(t, x0))− 12u2(t, q(t, x0))

− u(t, q(t, x0))ux(t, q(t, x0)) + u2x(t, q(t, x0)) + e−q(t,x0)

∫ q(t,x0)

−∞

32eηu2(t, η)dη

= u2x(t, q(t, x0)) +

12e−q(t,x0)

∫ q(t,x0)

−∞eη

(u2(t, η)− u2

η(t, η))dη

− 32u2(t, q(t, x0)) + e−q(t,x0)

∫ q(t,x0)

−∞eη

(12u2

η(t, η) + u2(t, η))

dη

≥ −M(t, q(t, x0))N(t, q(t, x0)) +12e−q(t,x0)

∫ q(t,x0)

−∞eη

(u2(t, η)− u2

η(t, η))dη.

(4.7)

The following estimate is crucial to ensure increase of M at t.(4.8)

e−q(t,x0)

∫ q(t,x0)

−∞eη

(u2(t, η)− u2

η(t, η))dη ≥ M(t, q(t, x0))N(t, q(t, x0)).

Combining (4.17) with (4.18), we obtain

(4.9)dM(t, q(t, x0))

dt≥ −1

2M(t, q(t, x0))N(t, q(t, x0)) > 0.

In an analogous way, we have the estimate of N.(4.10)

eq(t,x0)

∫ ∞

q(t,x0)e−η

(u2(t, η)− u2

η(t, η))dη ≥ M(t, q(t, x0))N(t, q(t, x0))

and

(4.11)dN(t, q(t, x0))

dt≤ 1

2M(t, q(t, x0))N(t, q(t, x0)) < 0.

Therefore,

(4.12) M(t, q(t, x0))N(t, q(t, x0)) < M(0, x0)N(0, x0) < 0, t ∈ [0, T ).

On the other hand, it follows from (4.8) and (4.10) that

p ∗ (u2 − u2x)(t, q(t, x0)) =

12

∫ ∞

−∞e−|q(t,x0)−η| (u2(t, η)− u2

η(t, η))dη

≥ M(t, q(t, x0))N(t, q(t, x0)).(4.13)

In view of (4.12) and (4.13), it is then inferred from (4.1) that

df(t)dt

≤ 12

(u2(t, q(t, x0))− u2

x(t, q(t, x0)))

=12M(t, q(t, x0))N(t, q(t, x0)) <

12M(0, x0)N(0, x0) < 0.

(4.14)

13

where the function f(t) is defined by f(t) = ux(t, q(t, x0)).Suppose now the solution u(t) of (3.1) exists globally in time t ∈ [0,∞),

that is, T = ∞. We show this leads to a contradiction. We first claim thatthere exists t1 > 0 such that

(4.15) f2(t) ≥ 2u2(t, q(t, x0)), t ≥ t1.

Applying Gronwall’s inequality to (4.9) and (4.11) yields that

M(t, q(t, x0)) ≥ M(0, x0)e−12N(0,x0)t > 0,

−N(t, q(t, x0)) ≥ −N(0, x0)e12M(0,x0)t > 0.

The above two estimates then imply that(4.16)

u2x(t, q(t, x0))− u2(t, q(t, x0)) ≥ −M(0, x0)N(0, x0)e

12(M(0,x0)−N(0,x0))t.

The proof of estimate (4.15) then follows from Proposition 3.3.Combining (4.14) with (4.15), we obtain

(4.17)d

dtf(t) ≤ 1

2u2(t, q(t, x0))− 1

2f2(t) ≤ −1

4f2(t), t ∈ [t1,∞).

Note that

f(0) = ux(0, x0) = −12e−x0

∫ x0

−∞eηy0(η)dη +

12ex0

∫ ∞

x0

e−ηy0(η)dη < 0.

The proof of blow-up T < +∞ then results from the classical inequality(4.17).

Remark. Assume the initial profile y0 is odd. Then the solution of (3.1)blows up in finite time if u

′0(0) < 0 [63]. One can see that the assumption

for blow-up in Theorem 4.1, that is, y0 ≥ 0 on R− and y0 ≤ 0 on R+, impliesu′0(0) < 0.

The following blow-up result improves Theorem 4.1 and may also includethe case of u

′0(0) ≥ 0.

Theorem 4.2. [48] Assume u0 ∈ Hs(R), s > 32 and y0(x) = u0(x)−u0,xx(x)

is odd. If there is only one point x0 ∈ (0,∞) such that y0(x0) = 0, thenthe corresponding solution u to initial-value problem (3.1) blows up in finitetime.

Proof. The key point is how to use the assumptions to derive a differentialinequality for the time evolution equation of ux(t, q(t, x0)). By the symmetry,we only need to consider the case that there is a x0 > 0 such that

y0(x) > 0 x ∈ (−∞,−x0),y0(x) < 0 x ∈ (−x0, 0),

14

and y0(x0) = 0. Since y0 is odd, it follows that u0 = p ? y0 is odd. Thus, thecorresponding solution u(t, ·) and y(t, ·) are odd for any t ∈ [0, T ). Let q(t, ·)be defined in (3.3). Then q(t, ·) is also odd for any t ∈ [0, T ). By symmetryof the solution, we then deduce that

(4.18)

y(t, x) > 0 x ∈ (−∞,−q(t, x0)),y(t, x) < 0 x ∈ (−q(t, x0), 0),

and y(t, q(t,−x0)) = 0 for all t ∈ [0, T ). In view of (4.18), we have for allt ∈ [0, T )

(4.19) (u− ux)(t, q(t,−x0)) = e−q(t,−x0)

∫ q(t,−x0)

−∞eηy(t, η)dη > 0

and

(ux + u)(t, q(t,−x0)) = eq(t,−x0)

∫ ∞

q(t,−x0)e−ηy(t, η)dη

= eq(t,−x0)

(∫ 0

q(t,−x0)[e−η − eη]y(t, η)dη +

∫ ∞

q(t,x0)e−ηy(t, η)dη

)

≤ eq(t,−x0)

∫ ∞

q(t,x0)e−ηy(t, η)dη < 0.

(4.20)

From the above two relations (4.18) and (4.19), we may also obtain

(4.21) ux(t, q(t,−x0)) < 0.

On the other hand, for η ∈ (−∞, q(t,−x0)), t ≥ 0 we have

u2(t, η)− u2x(t, η) =

∫ η

−∞eξy(t, ξ)dξ

(∫ q(t,−x0)

ηe−ξy(t, ξ)dξ +

∫ ∞

q(t,−x0)e−ξy(t, ξ)dξ

)

≥(∫ q(t,−x0)

−∞eξy(t, ξ)dξ −

∫ q(t,−x0)

ηeξy(t, ξ)dξ

)∫ ∞

q(t,−x0)e−ξy(t, ξ)dξ

≥∫ q(t,−x0)

−∞eξy(t, ξ)dξ

∫ ∞

q(t,−x0)e−ξy(t, ξ)dξ

= u2(t, q(t,−x0))− u2x(t, q(t,−x0)).

(4.22)

15

Hence we infer from the above inequality that

d

dt(u− ux)(t, q(t,−x0))

= −qt(t,−x0)(u− ux)(t, q(t,−x0)) + e−q(t,x0)

∫ q(t,−x0)

−∞eηyt(t, η)dη

= u2x(t, q(t,−x0)) +

12e−q(t,−x0)

∫ q(t,−x0)

−∞eη

(u2(t, η)− u2

η(t, η))dη

− 32u2(t, q(t,−x0)) +

12e−q(t,−x0)

∫ q(t,−x0)

−∞eη

(u2

η(t, η) + 2u2(t, η))dη

≥ (u2x − u2)(t, q(t,−x0)) +

12e−q(t,−x0)

∫ q(t,−x0)

−∞eη

(u2(t, η)− u2

η(t, η))dη

≥ −12[(u− ux)(u + ux)](t, q(t,−x0)) > 0.

(4.23)

It then follows from (4.19) and (4.23) that

d

dt

(eq(t,−x0)(u− ux)(t, q(t,−x0))

)

= eq(t,−x0)qt(u− ux)(t, q(t,−x0)) + eq(t,−x0) d

dt(u− ux)(t, q(t,−x0))

≥ eq(t,−x0)(u2 − uux)(t, q(t,−x0)) +12eq(t,−x0)(u2

x − u2)(t, q(t,−x0))

≥ 12eq(t,−x0)(u− ux)(t, q(t,−x0))[(u− ux)(0,−x0)] > 0.

(4.24)

This in turn implies that(eq(t,−x0)(u− ux)(t, q(t,−x0))

)≥ e−x0 [(u− ux)(0,−x0)]e

12[(u−ux)(0,−x0)]t.

Consequently, we deduce that

− ux(t, q(t,−x0))

≥ [(u− ux)(0,−x0)]e(12[(u−ux)(0,−x0)]t+q(t,x0)−x0) − u(t, q(t,−x0))

≥ [(u− ux)(0,−x0)]e(12[(u−ux)(0,−x0)]t−x0) − (

3‖u0‖2L2t + ‖u0‖L∞

).

(4.25)

The rest of the proof can be obtained by following the outline of proof inTheorem 4.1.

Remark. It was shown by McKean [53, 54] that the solutions of the CHequation breaks down if and only if some portion of the positive part ofy = u−∂2

xu initially lies to the left of some portion of its negative part. The

16

problem whether or not the DP equation has these wave breaking phenom-ena still remains open. Because of the structural difference between thesetwo equations, it is difficult to use the machinery of McKean [Mc2] in studyof the associated spectral problem with the corresponding eigenvalues. Theissue of if and how these particular initial data generate a global solution orblow-up in finite time is more subtle.

In contrast with the conditions of the blow-up solution of the DP equa-tion defined on the line R, one can see that the criteria of blow-up forperiodic solutions of the DP equation are quite different. Let us considerperiodic solutions of (3.1), i.e, u : S× [0, T ) → R where S is the unit circleand T > 0 is the maximal existence time of the solution. The interest inperiodic solutions is motivated by the observation that the majority of thewaves propagating on a channel are approximately periodic.

Define G(x) by G(x) = cosh(x−[x]− 12)

2 sinh( 12)

, where [x] stands for the integer part

of x ∈ R, then (1 − ∂2x)−1f = G ∗ f for all f ∈ L2(S). Using this identity,

we can rewrite (3.1) as a quasi-linear evolution equation of hyperbolic type,namely,

(4.26)

ut + uux + ∂xG ∗ (32u2) = 0, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R,

u(t, x) = u(t, x + 1), t ≥ 0, x ∈ R,

Theorem 4.3. [31] Assume that u0 ∈ Hs(S), s > 32 , u0 6≡ 0, and the

corresponding solution u(t, x) of (4.26) has a zero for any time t ≥ 0. Then,the solution u(t, x) of(4.26) blows up in finite time.

Proof. Proof of blow-up solution for the periodic case is quite different fromthat of the line case (Teeorem4.1-4.2). An outline of the proof is given inthe following.

By assumption, for each t ∈ [0, T ) there is a ξt ∈ [0, 1] such that u(t, ξt) =0. Then for ∀x ∈ S we have

(4.27) u2(t, x) =(∫ x

ξt

ux dx

)2

≤ (x− ξt)∫ x

ξt

u2x dx, x ∈

[ξt, ξt +

12

].

Hence the above relation and an integration by parts yield

∫ ξt+12

ξt

u2u2x dx ≤

∫ ξt+12

ξt

(x− ξt)u2x

(∫ x

ξt

u2x

)dx ≤ 1

4

(∫ ξt+12

ξt

u2x dx

)2

.

Combining this estimate with a similar estimate on [ξt + 12 , ξt +1], we obtain

(4.28)∫

Su2u2

x dx ≤ 14

(∫

Su2

x dx

)2

.

17

By (4.19), we also have

(4.29) supx∈S

u2(t, x) ≤ 12

∫

Su2

x dx.

Let us assume that the solution u(t, x) exists globally in time. Note thatG(x) ≥ 1

2 sinh( 12)

for all x ∈ S. It then follows from (4.20) and (4.21) that

d

dt

∫

Su3

x dx = 3∫

Su2

x

(−u2

x − uuxx +32u2 −G ∗

(32u2

))dx

= −3∫

Su4

x dx− 3∫

Su2

xuuxx dx +92

∫

Su2

xu2 dx− 92

∫

Su2

xG ∗ (u2

)dx

≤ −2∫

Su4

x dx +98

(∫

Su2

x dx

)2

− 92 sinh(1

2)

∫

Su2

xdx

∫

Su2 dx.

(4.30)

Applying the Cauchy-Schwartz inequality, we have

(4.31)98

(∫

Su2

x dx

)2

≤ 98

∫

Su4

x dx.

On the other hand, in view of (4.20), it is easy to see that

(4.32) −∫

Su2

xdx

∫

Su2 dx ≤ −2

(∫

Su2dx

)2

≤ −18‖u0‖4

L2 .

It is thus deduced from (4.21)-(4.23) that

d

dt

∫

Su3

x dx ≤ −78

∫

Su4

x dx− 916 sinh(1

2)‖u0‖4

L2 , t ≥ 0.

Hence an application of Holder’s inequality yields

(4.33)d

dt

∫

Su3

x dx ≤ −78

(∫

Su3

x dx

) 43

− 916 sinh(1

2)‖u0‖4

L2 , t ≥ 0.

If we define V (t) :=∫S u3

x(t, x) dx for all t ≥ 0, then

V (t) ≤ V (0)− 916 sinh(1

2)‖u0‖4

L2 t, t ≥ 0.

Since u0 6≡ 0, the above inequality implies that there exists some t0 ≥ 0 suchthat V (t) < 0 for all t ≥ t0. It is then inferred from (4.24) that

d

dtV (t) ≤ −7

8(V (t))

43 , t > t0.

18

Thus we have(

7(t− t0)24

+1

(V (t0))13

)3

≤ 1V (t)

< 0, t ≥ t0.

Since V (t0) < 0, the above inequality will lead to a contradiction as t ≥ t0is big enough, which implies T < ∞.

As immediate consequences of Theorem 4.3, we have

Corollary 4.4. If u0 ∈ H3(S), u0 6≡ 0 and∫S u0 dx = 0 or

∫S y0 dx = 0,

then the corresponding solution u to (4.26) blows up in finite time.

Proof. Note∫

Su(t, x) dx =

∫

Sy(t, x) dx =

∫

Sy0(x) dx =

∫

Su0(x) dx = 0.

The above relation shows that u(t, x) has a zero for all t ∈ S. It follows fromTheorem 4.3 that the solution u to (4.26) blows up in finite time.

Corollary 4.5. If u0 ∈ Hs(S), s > 32 , u0 6≡ 0 and

∫S u3

0 dx = 0, then thecorresponding solution u(t, x) of (4.26) blows up in finite time.

Proof. The result can be obtained immediately from the conservation law

E3(u) =∫

Su3 dx.

Corollary 4.6. If u0(x) or y0 is odd, then the corresponding solution u to(4.26) blows up in finite time.

Proof. If u0(x) or y0 is odd, then the solution u(t, x) is odd for all t ≥ 0.This also shows that u(t, x) has a zero for all t ∈ S.

Remark. Compared with the line case (see Theorem 3.2 and Remark 3.3in [47]) where the corresponding solution u with u0 ∈ Hs(R), s > 3/2, andu0(x) being odd, may exist globally in time, we find from Theorem 4.3 thatthere is quite a difference in the blow-up phenomena of the Degasperis-Procesi equation between the periodic case and the line case.

5 Blow-up structure

As mentioned above, blow-up can occur only in the form of wave-breaking,i.e. the wave profile remains bounded but its slope becomes unboundedin finite time. In this section we give a quite detailed description of itsphenomena. The following theorem shows that there is only one point wherethe slope of the solution becomes infinity exactly at breaking time.

19

Theorem 5.1. [47] Assume u0 ∈ Hs(R), s > 32 and there exists x0 ∈ R

such that

y0(x) = u0(x)− u0,xx(x) ≥ 0 if x ≤ x0,y0(x) = u0(x)− u0,xx(x) ≤ 0 if x ≥ x0,

and y0 changes sign. Let T < ∞ be the finite blow-up time of the corre-sponding solution u to (3.1). Then we have

limt→T

ux(t, q(t, x0)) = −∞.

Proof. For any x ≤ q(t, x0), we have

ux(t, x) = −u(t, x) + ex

∫ q(t,x0)

xe−ηy(t, η)dη + ex

∫ ∞

q(t,x0)e−ηy(t, η)dη

≥ −u(t, x) + ux(t, q(t, x0)) + u(t, q(t, x0)).

On the other hand,

ux(t, x) ≥ u(t, x) + ux(t, q(t, x0))− u(t, q(t, x0)).

From the above two inequalities we deduce that for (t, x) ∈ [0, T )× R,

ux(t, x) ≥ ux(t, q(t, x0))− 2‖u(t, ·)‖L∞

≥ ux(t, q(t, x0))− 2(3‖u0(x)‖2

L2t + ‖u0(x)‖L∞).

(5.1)

In view of T < ∞, it follows from Proposition 3.2 that

lim inft→T

( infx∈R

ux(t, x)) = −∞.

Consequently, limt→T

ux(t, q(t, x0)) = −∞.

Theorem 5.2. [30] Let T < ∞ be the blow-up time of the solution u of(3.1) with initial data u0 ∈ Hs(R), s > 3

2 such that the associated potentialy0 = u0 − u0,xx satisfies y0(x) ≥ 0 on (−∞, x0] and y0(x) ≤ 0 on [x0,∞)for some points x0 ∈ R and y0 does not have a constant sign. Then

limt→T

(infx∈R

ux(t, x)(T − t))

= −1 and limt→T

(supx∈R

ux(t, x)(T − t))

= 0.

The following result gives some information about the blow-up set of abreaking wave to (3.1) with a large class of initial data.

Theorem 5.3. [30] Assume that u0 ∈ Hs(R), s > 32 and u0 6≡ 0 is odd such

that the associated potential y0 = u0−u0,xx is nonnegative on R−. Then thesolution u to (3.1) with initial data u0 blows up in finite time only at zeropoint.

20

For the periodic solutions of (4.26) we have

Theorem 5.4. [31] Assume that u0 ∈ Hs(S), s > 32 and that the corre-

sponding associated potential y0 := u0 − u0,xx 6≡ 0 is odd.(a) Suppose that (x − 1

2)y0(x) ≤ 0 on S. Then the solution to (4.26)) withinitial data u0 blows up in finite time only at the point x = 1

2 and

limt→T

ux(t,12) = −∞.

(b) Suppose that (x − 12)y0(x) ≥ 0 on S. Then the solution to (4.26) with

initial data u0 blows up in finite time only at two points x = 0 and x = 1,and

limt→T

ux(t, 0) = limt→T

ux(t, 1) = −∞.

6 Global weak solutions

As mentioned in Introduction, the DP equation possesses the peaked solitonsof the form

(6.1) u(t, x) = cϕ(x− ct),

where ϕ(x) = e−|x|. It is easy to see that (1−∂2x)ϕ = 2δ (here δ is the Dirac

distribution). Note these peakons are not the strong solutions in Hs, s ≥ 32 ,

but the global weak solutions in H1[25, 65]. Using the conservation law ofthe DP equation, a partial integration result in the Bochner spaces, and theHelly theorem, one can obtain the following global weak solutions in H1.

Theorem 6.1. [30] Let u0 ∈ H1(R). Assume y0 = (u0−u0,xx) ∈ M(R) andthere is a x0 ∈ R such that supp y−0 ⊂ (−∞, x0) and supp y+

0 ⊂ (x0,∞).Then initial-value problem (3.1) has a unique global weak solution

u ∈ W 1, ∞loc (R+ × R) ∩ L∞loc(R+;H1(R))

withy(t, ·) = u(t, ·)− uxx(t, ·)) ∈ L∞loc(R+,M(R).

Moreover, E1(u) and E2(u) are two conservation laws.

Example 1.(Peaked solitons) Let u0(x) = ce−|x|, x ∈ R, c > 0. Then y0 =u0−u0,xx = 2cδ(x) and u(t, x) = ce−|x−ct| is the unique global weak solutionwith the initial data u0.

Remark. More interestingly, we find in [32] that there are no traveling-wave solutions u ∈ C([0,∞);H3) ∩ C1([0,∞);H2) to (3.1). Arguing by

21

contradiction, we assume that w ∈ H3 and u(t, x) = w(x − ct), c 6= 0 is astrong solution of (3.1). Then we have

cw′ − cw′′′ − 4ww′ + 3w′w′′ + ww′′′ = 0 in L2.

We find that(cw − w2 − 2w2 + (w”)2 + ww′′

)′ = 0 in L2

and therefore

cw − w2 − 2w2 + (w”)2 + ww′′ = 0 in H1

or, what is same,

(c− w)(w − w′′)− (w2 − (w′)2

)= 0 in H1

since w ∈ H3 ⊂ C20 (R). Multiplying this identity with 2w′ yields that

(6.2) (c− w)(w2 − (w′)2

)′ − 2w′(w2 − (w′)2

)= 0.

Since w ∈ H3 ⊂ C20 (R), we have w 6= c, a.e. and w2 6= (w′)2, a.e.

Let w0 = w(ξ) = maxx∈Rw(x) > 0. Then taking integration for (6.2) in[ξ, x] yields ∫ x

ξ

d(w2 − (w′)2

)

w2 − (w′)2=

∫ x

ξ

2dw

c− w, x ∈ R.

This implies that

(6.3) (w − c)2∣∣w2 − (w′)2

∣∣ = w20(w0 − c)2 x ∈ R.

If we take into account w, w′ → 0 as x →∞, it is then inferred from relation(6.3) that

w20(w0 − c)2 = 0

which also implies from (6.3) that

(w − c)2|w2 − (w′)2| = 0, x ∈ R.

This leads a contradiction since w ∈ H3.

Example 2. Let

u0(x) = c1e−|x−x1| + c2e

−|x−x2|, x ∈ R,

with c1 < 0, c2 > 0 and x1 < x2. It is easily found that

y0 = u0 − u0,xx = 2c1δ(x− x1) + 2c2δ(x− x2).

22

By Theorem 6.1, there exists a unique global weak solution u to (3.1) withthe initial data u0. It has the explicit form [25]

u(t, x) = p1(t)e−|x−q1(t)| + p2(t)e−|x−q2(t)|, (t, x) ∈ R+ × R,

for some p1, p2, q1, q2 ∈ W 1,∞loc (R). Indeed, the solution u is the sum of a

peakon and an antipeakon. It is observed that the antipeakon moves off tothe left and the peakon moves off to the right so that no collision occurs.

Remark. In addition, if the initial data

u0(x) = c1e−|x−x1| + c2e

−|x−x2|,

with c1 > 0, c2 < 0 and x1 + x2 = 0, x2 > 0, then the collision occurs atx = 0 and the solution

u(x, t) = p1(t)e−|x−q1(t)| + p2(t)e−|x−q2(t)|,

(x, t) ∈ R+ × R, only satisfies the DP equation for t < T. The unique con-tinuation of u into an entropy weak solution is then given by the stationarydecaying shockpeakon [49]

u(x, t) =−sgn(x)e−|x|

k + (t− T )for t ≥ T.

Existence of these discontinuous (shock waves, [49]) solutions of the DPequation shows that the DP equation would behave radically different fromthe Camassa-Holm equation, but similar to the inviscid Burgers equation,which implies that a well-posedness theory should depend on some functionalspaces which contain discontinuous functions. Indeed, this observation wasconfirmed by Coclite and Karlsen [8, 9]. In [8, 9], they [10] proved the globalexistence and uniqueness of L1 ∩ BV entropy weak solutions satisfying aninfinite family of Kruzkov-type entropy inequalities. Recently, they provedexistence of bounded weak solutions by an Oleinik-type estimate for L∞

solutions to (3.1).

Theorem 6.2. [10] Suppose u0 ∈ L∞(R). Then there exists a unique en-tropy weak solution u ∈ L∞((0, T ) × R), for any T > 0 to initial-valueproblem (3.1) and for each T > 0 there exists a positive constant KT suchthat the estimate

u(t, x)− u)t, y)x− y

≤ KT

(1t

+ 1)

holds for any x, y ∈ R, x 6= y, 0 < t < T.

Indeed, it is recently shown by Coclite and Karlsen [10] that the infinitefamily of entropy inequality is equivalent to the one-side Lipshitz inequality.

23

Therefore the well-posedness in L1 ∩L∞ of the Cauchy problem for the DPequation can be established [10].

The relevance of these solutions of the DP equation is supported byLundmark [49], who found some explicit shock solutions of the DP equationwhich are entropy weak solutions. Numerical schemes for computing entropyweak solutions of the DP equation is developed and analyzed in [11]. Onthe other hand, these discontinuous solutions of the DP equation are alsoinvestigated by Feng and Liu [32] using an operator splitting method, whichconsists of a second-order total variation diminishing (TVD) scheme and asecond-order linearized finite difference scheme. Here, we first show an ex-ample for symmetric peakon-antipeakon collision, in which initial conditionu0 is taken as

(6.2) u0(x) = e−|x+4| − e−|x−4|.

As shown in Figure 1, when t > 4, a shockpeakon is formed and continuesto decay as a shockpeakon solution mentioned previously. Actually, weobserved that shock formation is generic if u′0(x) < 0 and u0(x) = 0 forsome x. We illustrate this by a numerical example with initial condition

(6.3) u0(x) = e0.5x2sin(πx),

for x ∈ [−2, 2], where u0 is extended periodically outside this interval. Figure2 shows the numerical results up to t = 0.9. It is seen that two shocks areformed at t ≈ 0.2, then a collision between them produces a stationaryshockpeakon at x = 0.

−10 −5 0 5 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x,

t)

t=0t=2t=4t=6

Figure 1: The numerical solution of a symmetric peakon antipeakon collision

24

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

x

u(x,

t)

t=0.0t=0.2t=0.6t=0.9

Figure 2: Wave breaking from a smooth initial condition

7 Stability of peakons

The stability of solitary waves is one of the fundamental qualitative prop-erties of the solutions of nonlinear wave equations. Numerical simulations[25, 49] suggest that the sizes and velocities of the peakons do not change asa result of collision so these patterns are expected to be stable. Furthermore,it is observed that the shape of the peakons remains approximately the sameas time evolves. stability of the peakons for the Camassa-Holm equation iswell understood by now [21, 22], while the the case of the Degasperis-Procesiequation is recently studied in [44].

As mentioned before, the corresponding conservation laws of the DPequation are much weaker than those of the CH equation. In particular,one can see that the conservation law E2(u) for the DP equation is onlyequivalent to ‖u‖2

L2 . Therefore, the stability issue of the peaked solitons ofthe DP equation is more subtle.

Probably, one can only expect to obtain the orbital stability of peakons inthe sense of L2−norm due to a weaker conservation law E2. The solutions ofthe DP equation usually tend to be oscillations which spread out spatially ina quite complicated way. In general, a small perturbation of a solitary wavecan yield another one with a different speed and phase shift. We define theorbit of traveling-wave solutions cϕ to be the set U(ϕ) = cϕ(·+ x0), x0 ∈R, and a peaked soliton of the DP equation is called orbitally stable if awave starting close to the peakon remains close to some translate of it at alllater times. Let us denote

E2(u) = ‖u‖2X .

Theorem 7.1. [44] Let cϕ be the peaked soliton defined in (6.1). Then cϕ

25

is orbitally stable in the following sense. If u0 ∈ Hs for some s > 3/2,y0 = u0 − ∂2

xu0 is a nonnegative Radon measure of finite total mass, and

‖u0 − cϕ‖X < cε, |E3(u0)− E3(cϕ)| < c3ε, 0 < ε <12,

then the corresponding solution u(t) of (3.1) with initial value u(0) = u0

satisfiessupt≥0

‖u(t, ·)− cϕ(· − ξ1(t))‖X < 3c ε1/4,

where ξ1(t) ∈ R is the maximum point of the function v(t, ·) = (4−∂2x)−1u(t, ·).

Moreover, let

M1 (t) = v(t, ξ1(t)) ≥ M2 (t) · · · ≥ Mn (t) ≥ 0 and m1 (t) ≥ · · · ≥ mn−1 (t) ≥ 0

be all local maxima and minima of the nonnegative function v(t, ·), respec-tively. Then

(7.1)∣∣∣M1 (t)− c

6

∣∣∣ ≤ c√

2ε

and

(7.2)n∑

i=2

(M2

i (t)−m2i−1 (t)

)< 2c2√ε.

Remark 1. Under the assumption y0 = u0 − ∂2xu0 ≥ 0 in Theorem 7.1, the

existence is global in time [47], that is T = +∞. For peakons cϕ with c > 0,we have

(1− ∂2

x

)(cϕ) = 2cδ (here δ is the Dirac distribution). Hence the

assumption on y0 that it is a nonnegative measure is quite natural for asmall perturbation of the peakons. Existence of global weak solution in H1

of the DP equation is also proved in [30]. Note that peakons cϕ are notstrong solutions, since ϕ ∈ Hs, only for s < 3/2.

Remark 2. The above theorem of orbital stability states that any solutionstarting close to peakons cϕ remains close to some translate of cϕ in thenorm ‖ ‖X , at any later time. More information about this stability iscontained in (7.1) and (7.2). Notice that for peakons cϕ, the function vcϕ issingle-humped with the height 1

6c. So (7.1) and (7.2) imply that the graphof v(t, ·) is close to that of the peakon cϕ with a fixed c > 0 for all times.

Remark 3. It is shown in [44] that M1 = max vu ≤√

E2 (u) /12. For peakonscϕ, we have max vcϕ =

√E2 (cϕ) /12 = 1

6c. So among all waves of a fixedenergy E2, the peakon is tallest in terms of vu.

Remark 4. In addition, it is found in the proof of [44] that the peakons areenergy minimizers with a fixed invariant E3, which explains their stability,i. e. if E3 (u) = E3 (ϕ), then E2(u) ≥ E2(ϕ). The same remark also applies

26

to the CH equation and shows that the CH-peakons are energy minima withfixed F3.

There are two standard methods to study stability issues of dispersivewave equations. One is the variational approach which constructs the soli-tary waves as energy minimizers under appropriate constraints, and thestability automatically follows. However, without uniqueness of the mini-mizer, one can only obtain the stability of the set of minima. The variationalapproach is used in [21] for the CH equation. It is shown in [21] that theeach peakon cϕ is the unique minimum (ground state) of constrained en-ergy, from which its orbital stability is proved for initial data u0 ∈ H3

with y0 = (1 − ∂2x)u0 ≥ 0. Their proof strongly relies on the fact that the

conserved energy F2 of the CH equation is the H1−norm of the solution.However, for the DP equation the energy E2 is only the L2 norm of the so-lution. Consequently, it is more difficult to use such a variational approachfor the DP equation.

Another approach to study stability is to linearize the equation aroundthe solitary waves, and it is commonly believed that nonlinear stability isgoverned by the linearized equation. However, for the CH and DP equations,the nonlinearity plays the dominant role rather than being a higher-ordercorrection to linear terms. Thus it is unclear how one can get nonlinear sta-bility of peakons by studying the linearized problem. Moreover, the peakedsolitons cϕ are not differentiable, which makes it difficult to analyze thespectrum of the linearized operator around cϕ.

To establish the stability result for the DP equation, we extend theapproach in [22] for the CH equation. The idea in [22] is to directly use theenergy F2 as the Liapunov functional. By expanding F2 around the peakoncϕ, the error term is in the form of the difference of the maxima of cϕ andthe perturbed solution u. To estimate this difference, they establish twointegral relations

∫g2 = F2 (u)− 2 (maxu)2 and

∫ug2 = F3 (u)− 4

3(maxu)3

with a function g. Relating these two integrals, one can get

F3(u) ≤ MF2(u)− 23M3, M = max u(x)

and the error estimate |M −maxϕ| then follows from the structure of theabove polynomial inequality.

To extend the above approach to nonlinear stability of the DP peakons,one has to overcome several difficulties. By expanding the energy E2 (u)around the peakon cϕ, the error term turns out to be max vcϕ − max vu,with vu = (4 − ∂2

x)−1u. We can derive the following two integral relations

27

for M1 = max vu, E2 (u) and E3(u) by∫

g2 = E2 (u)− 12M21 and

∫hg2 = E3 (u)− 144M3

1

with some functions g and h related to vu. To get the required polynomialinequality from the above two identities, we need to show h ≤ 18max vu.However, since h is of the form −∂2

xvu ± 6∂xvu + 16vu, generally it cannot be bounded by vu. This new difficulty is due to the more complicatednonlinear structure and weaker conservation laws of the DP equation. Toovercome it, we introduce a new idea. By constructing g and h piecewiseaccording to monotonicity of the function vu, we then establish two newintegral identities related to E2, E3 and all local maxima and minima ofvu. The crucial estimate h ≤ 18 max vu can now be shown by using thismonotonicity structure and properties of the DP solutions. Then one canobtain not only the error estimate |M1 −max vcϕ| but more precise stabilityinformation from (7.2). It is observed that the same approach can also beused for the CH equation to gain more stability information.

The detailed proof of stability of peakons for the DP equation can befound in [44].

References

[1] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485–541.

[2] T.b. Benjamin, J. L. Bona and J. J. Mahoney, Lmodel equations forlong waves in nonlinear dispersive system, Philos. trans. R. Soc. Lond. A,227 (1972), 47–78.

[3] A. Bressan and A. Constantin, Global conservative solutions of theCamassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215–239.

[4] R. Beals, D.H. Sattinger and J. Szmigielski, Multipeakons and theclassical moment problem, Adv. Math., 154 (2000), 229–257.

[5] R. Camassa and D. Holm, An integrable shallow water equation withpeaked solitons, Phys. Rev. Letters, 71 (1993), 1661–1664.

[6] R. Camassa, D. Holm and J. Hyman, A new integrable shallow waterequation, Adv. Appl. Mech., 31 (1994), 1–33.

[7] E. D. Cokelet, Breaking waves, Nature, 267 (1997), 769-774.

[8] G. M. Coclite and K. H. Karlsen, On the well-posedness of theDegasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60–91.

28

[9] G. M. Coclite and K. H. Karlsen, On the uniqueness of discontinuoussolutions to the Degasperis-Procesi equation, J. Differential Equations, 234(2007), 142–160.

[10] G. M. Coclite and K. H. Karlsen, Bounded solutions for the Degasperis-Procesi equation, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), toappear.

[11] G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numerical schemesfor computing discontinuous solutions of the Degasperis-Procesi equation,IMA J. Numer. Anal., to appear.

[12] A. Constantin, Global existence of solutions and breaking waves for a shal-low water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50(2000), 321–362.

[13] A. Constantin, Finite propagation speed for the Camassa-Holm equation,J. Math. Phys., 46 (2005), 023506, 4 pp.

[14] A. Constantin, On the scattering problem for the Camassa-Holm equation,Proc. R. Soc. lond. A 457 (2001), 953–970.

[15] A. Constantin and J. Escher, Global existence and blow-up for a shallowwater equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303–328.

[16] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shal-low water equations, Acta Mathematica, 181 (1998), 229–243.

[17] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism groupof the circle, Comment. Math. Helv., 78 (2003), 787–804.

[18] A. Constantin and D. Lannes, The hydrodynamical relevance of theCamassa-Holm and Degasperis-Procesi equations, Arch. Rantional Mech.Anal., (2008), to appear.

[19] A. Constantin and H. P. McKean, A shallow water equation on the circle,Comm. Pure Appl. Math., 52 (1999), 949–982.

[20] A. Constantin and L. Molinet, Global weak solutions for a shallow waterequation, Comm. Math. Phys., 211 (2000), 45–61.

[21] A. Constantin and L. Molinet, Obtital stability of solitary waves for ashallow water equation, Physica D, 157 (2001), 75–89.

[22] A. Constantin and W. A. Strauss, Stability of peakons, Comm. PureAppl. Math., 53 (2000), 603–610.

[23] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm soli-tons, J. Nonlinear Science, 12 (2002), 415–422.

[24] H. H. Dai, Model equations for nonlinear dispersive waves in a compressibleMooney-Rivlin rod, Acta Mechanica, 127(1998), 193–207.

29

[25] A. Degasperis, D. D. Holm, and A. N. W. Hone, A New IntegralEquation with Peakon Solutions, Theoretical and Mathematical Physics, 133(2002), 1463-1474.

[26] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitonsand nonlienar wave equations, Academic Press, New York, 1984.

[27] A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry andPerturbation Theory, edited by A. Degasperis and G. Gaeta, World Scientific(1999), 23–37.

[28] H. R. Dullin, G. A. Gottwald, and D. D. Holm, An integrable shallowwater equation with linear and nonlinear dispersion, Phys. Rev. Letters, 87(2001), 4501–4504.

[29] H. R. Dullin, G. A. Gottwald, and D. D. Holm, Camassa-Holm,Korteweg-de Vries-5 and other asymptotically equivalent equations for shallowwater waves, Fluid Dynam. Res., 33 (2003), 73–95.

[30] J. Escher, Y. Liu, and Z. Yin, Global weak solutions and blow-up structurefor the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.

[31] J. Escher, Y. Liu, and Z. Yin, Shock waves and blow-up phenomena forthe periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007),87–117.

[32] B. Feng and Y. Liu, The non-smooth solutions of the Degasperis-Procesiequation and its numerical aspects, preprint.

[33] A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlundtransformation and hereditary symmetries, Physica D, 4 (1981), 47–66.

[34] B. Fuchssteiner, Some tricks from the symmetrytoolbox for nonlinear equa-tions: generalizations of the Camassa-Holm equation, Physica D, 95 (1996),296-343.

[35] A.E. Green and P.M.Naghdi, A derivation of equations for wave propa-gation in water of variable depth, J. Fluid Mech., 78(1976), 237-246.

[36] G. Gui, Y. Liu, and L. Tian, Global existence and blow-up phenomena forthe peakon b-family of equations, Indiana Univ. Math. J., (2007), accepted.

[37] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation. J.Math. Anal. Appl., 311 (2005), 755–759.

[38] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in afamily of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. (electronic), 2 (2003),323–380.

[39] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stabilityin dynamics of solitons, peakons, ramps/cliffs and leftons in a 1− 1 nonlinearevolutionary PDE, Phys. Lett. A, 308 (2003), 437-444.

30

[40] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models forwater waves, J. Fluid Mech., 455 (2002), 63–82.

[41] R. S. Johnson, On solution of the Camassa-Holm equation, Proc. r. Soc.Lond., A 459 (2003), 1687–1708.

[42] C. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering resultsfor the generalized Korteweg-de Veris equation via the contraction principle,Comm. Pure Appl. Math., 46 (1993), 527–620.

[43] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J.Math. Anal. Appl., 306 (2005), 72–82.

[44] Z. Lin and Y. Liu, Stabiltiy of peakons for the Degasperis-Procesi equation,Comm. Pure Appl. Math., LXI (2008), 1-22.

[45] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrablenonlinearly dispersive model wave equation, J. Differential Equations, 162(2000), 27–63.

[46] Y. Liu, Global existence and blow-up solutions for a nonlinear shallow waterequation, Math. Ann., 335 (2006), 717-735.

[47] Y. Liu and Z. Yin, Global existence and blow-up phenomena for theDegasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.

[48] Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Projesiequation, Int. Math. Res. Not., 117 (2007), 22 pages.

[49] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.

[50] H. Lundmark and J. Szmigielski, Multi-peakon solutions of theDegasperis-Procesi equation, Inverse Problems, 19 (2003), 1241–1245.

[51] Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation andtheir peakon limit, Inverse Problems, 21 (2005), 1553–1570.

[52] H. P. Mckean, Integrable systems and algebraic curves, Global Analysis,Springer Lecture Notes in Mathematics, 755 (1979), 83–200.

[53] H. P. Mckean, Integrable systems and algebraicBreakdown of a shallowwater equation, Asian J. Math., 2 (1998), 867-874.

[54] H. P. Mckean, Breakdown of the Camassa-Holm equation, Comm. PureAppl. Math., LVII (2004), 0416-0418.

[55] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203–208.

[56] O. G. Mustafa, A note on the Degasperis-Procesi equation, J. NonlinearMath. Phys., 12 (2005), 10–14.

31

[57] D. H. Peregrine, Breaking waves on beaches, Ann. Rev. Fluid Mech., 15(1983), 149–178.

[58] V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutionsof the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059–1073.

[59] T. Tao, Low-regularity global solutions to nonlinear dispersive equations,Surveys in analysis and operator theory (Canberra,2001), 19-48, Proc. CentreMath. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002.

[60] L. Tian, G. Gui, and Y. Liu, On the Cauchy problem and scattering prob-lem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., 257 (2005),667-701.

[61] G. B. Whitham, Linear and Nonlinear Waves, J. Wiley & Sons, New York,1980.

[62] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,Comm. Pure Appl. Math., 53 (2000), 1411–1433.

[63] Z. Yin, On the Cauchy problem for an integrable equation with peakon solu-tions, Illinois J. Math., 47 (2003), 649-666.

[64] Z. Yin, Global weak solutions to a new periodic integrable equation withpeakon solutions, J. Funct. Anal., 212 (2004), 182-194.

[65] Z. Yin, Global solutions to a new integrable equation with peakons, IndianaUniv. Math. J., 53 (2004), 1189-1210.

32