Wave Breaking Phenomena and Stability of Peakons for the Degasperis-Procesi

Transcription

1 Wave Breaking Phenomena and Stability of Peakons for the Degasperis-Procesi Yue Liu Technical Report

2 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-Procesi equation, which can be derived as a member of a one-parameter family of asymptotic shallow-water wave approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. We will focus on some important results including wave breaking phenomena, blow-up structure, global weak solutions and the orbital 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-up structure; Stability of peakons 1 Introduction Considered herein is the Degasperis-Procesi (DP) equation (1.1) y t + y x u + 3yu x = 0, x R, t > 0, with y = u u xx. Degasperis and Procesi [27] studied a family of third order dispersive nonlinear equations (1.2) u t α 2 u xxt + γu xxx + c 0 u x = (c 1 u 2 + c 2 u 2 x + c 3 uu xx ) x. with six real constants c 0, c 1, c 2, c 3, γ, α R. They [27] found that there are only three equations from this family were asymptotically integrable up to third order, that is, the Korteweg-de Vries (KdV) equation (α = c 2 = Department of Mathematics, University of Texas, Arlington, TX 76019, yliu@uta.edu 1

3 c 3 = 0), the Camassa-Holm (CH) equation (c 1 = 3c 3, c 2α 2 2 = c 3 2 ), and one new equation (c 1 = 2c 3, c α 2 2 = c 3 ), which is called the Degasperis-Procesi equation. By rescaling, shifting the dependent variable, and finally applying a Galilean transformation, those three completely integrable 1 equations can be transformed into the following forms, the Korteweg-de Vries (KdV) equation u t + u xxx + uu x = 0, the Camassa-Holm (CH) shallow water equation [5, 28, 40], (1.3) y t + y x u + 2yu x = 0, y = u u xx, and the Depasperis-Procesi equation of the form (1.1). These three cases are all the completely integrable candidates for (1.2) [5, 25, 27]. Applying a reciprocal transformation to the Degasperis-Procesi equation, Degasperis, Holm and Hone [25] used the Painlevé analysis to show the formal integrability of the DP equation as Hamiltonian systems by constructing a Lax pair and 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 shallow water under the Kodama transformation [29, 38, 39] and its asymptotic accuracy is the same as that of the Camassa-Holm (CH) shallow water equation, where u(t, x) is considered as the fluid velocity at time t in the spatial x-direction with momentum density y. More interestingly, the DP equation is recently observed as a model supporting shock waves [49]. More recently, Constantin and Lannes [18] give a rigorous proof of both the CH equation and the DP equation are valid approximation to the governing equations for water waves (see also [40] for the formal asymptotic procedures) and also show the relevance of these two equations as models for the propagation of shallow water waves. To see this rigorous justification of the derivation, one can consider the water wave equations for one-dimensional surfaces in nondimensionalized form µ xψ 2 + z 2 Ψ = 0, in Ω t, z Ψ = 0, at z = 1, t ξ 1 µ ( µ xξ x Φ + z Ψ) = 0, at z = ɛξ, t Ψ + ɛ 2 ( xψ) 2 + ɛ 2µ ( zψ) 2 = 0, at z = ɛξ, where x ɛξ(t, x) parameterizes the elevation of the free surface at time t, Ω t = {(x, z); 1 < z < ɛξ(t, x)} is the fluid domain delimited by the 1 Integrability is meant in the sense of the infinite-dimensional extension of a classical completely integrable Hamiltonian system: there is a transformation which converts the equation into an infinite sequence of linear ordinary differential equations which can be trivially integrated [52]. 2

4 free surface and the flat bottom {z = 1}, Ψ(t, ) is the velocity potential associated to the flow, and ɛ and µ are two dimensionless parameters defined by ɛ = a h, µ = h2 λ 2, where h is the mean depth, a is the typical amplitude, and λ is the typical wavelength of the waves. Define the vertically averaged horizontal component of the velocity by u(t, x) = 1 ɛξ x Ψ(t, x, z)dz. 1 + ɛξ 1 In the shallow-water scaling (µ 1), one can derive the Green-Naghdi equations [1, 35] for one-dimensional surfaces and flat bottoms without any assumption on ɛ(ɛ = O(1)). These equations couple the free surface elevation ξ to the vertically averaged horizontal component of the velocity u and can be written as { ξt + ((1 + ɛξ)u) x = 0 u t + ξ x + ɛuu x = µ ( ɛξ (1 + ɛξ) 3 (u xt + ɛuu xx ɛu 2 x) ) x, where O(µ 2 ) terms have been neglected. In the so-called long-wave regime µ 1, ɛ = O(µ), the right-going wave should satisfy the KdV equation u t + u x + ɛ 3 2 uu x + µ 1 6 u xxx = 0 with ξ = u + O(ɛ, µ), or a wider class of equations, referred as the BBM equations [2](sometimes also called the regularized long-wave equations), which provide an approximation of the exact water wave equations of the same accuracy as the KdV equation. u t + u x ɛuu x + µ(αu xxx + βu xxt ) = 0, with α β = 1 6. Consider now the so-called Camassa-Holm scaling, that is µ 1, ɛ = O( µ). With this scaling, one still has ɛ 1, the dimensionless parameter is, however, larger here than in the long wave scaling, and the nonlinear effects are therefore stronger and it is possible that a stronger nonlinearity could allow the appearance of breaking waves, which is a fundamental phenomenon in the theory of water waves that is not captured by the BBM equations. 3

5 Define the horizontal velocity u θ (θ [0, 1]) at the level line θ of the fluid domain by v u θ (x) = x Ψ z=(1+ɛξ)θ 1. Let p R and λ = 1 2 (θ2 1 3 ), with θ [0, 1]. Assume α = p + λ, β = p λ, γ = 2 3 p λ, δ = 9 2 p λ. Under the Camassa-Holm scaling, one should have the following class of equations for v u θ (θ [0, 1]), namely ( ) v t + v x ɛvv x + µ(αv xxx + βv xxt ) = ɛµ(γvv xxx + δv x v xx ), where O(ɛ 4, η 2 ) terms have been discarded. The vertically averaged horizontal velocity u and the free surface ξ satisfy u = u θ + µλu θ xx + 2µɛλu θ u θ xx, ξ = u + ɛ 4 u2 + µ 1 ( 1 6 u xt ɛµ 6 uu xx + 5 ) 48 u2 x. By rescaling, shifting the dependent variable, and applying a Galilean transformation, the Camassa-Holm equation U t + κu x + 3UU x U txx = 2U x U xx + UU xxx can be obtained from ( ) if the following conditions hold β < 0, α β, β = 2γ, δ = 2γ, where p = 1 3, θ2 = 1 2. The solution uθ of ( ) is transformed to the solution U of the CH equation by U(t, x) = 1 ( x a uθ b + ν c t, t ), c with a = 2 ɛκ (1 ν), b2 = 1 β µ, ν = α β, and c = b κ (1 ν). On the other hand, the DP equation U t + κu x + 4UU x U txx = 3U x U xx + UU xxx can also be derived if the following conditions hold β < 0, α β, β = 8 γ, δ = 3γ, 3 where p = , θ2 = The solution uθ of ( ) is also transformed to the solution U of the DP equation by U(t, x) = 1 ( x a uθ b + ν c t, t ), c 4

6 with a = 8 3ɛκ (1 ν), b2 = 1 β µ, ν = α β, and c = b κ (1 ν). A detailed derivation of the CH and DP equations can be found in [18, 40]. It is well known that the KdV equation is an integrable Hamiltonian equation that possesses smooth solitons as traveling waves. In the KdV equation, the leading order asymptotic balance that confines the traveling wave solitons occurs between nonlinear steepening linear dispersion. However, the nonlinear dispersion and nonlocal balance in the CH equation and the DP equation, even in the absence of linear dispersion, can still produce a 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 collisions under the CH dynamics, or the DP dynamics, respectively. The peakons of the CH equation and the DP equation are orbitally stable [22, 44]. The result of the stability of the DP equation will be discussed in the last section. The DP equation can be rewritten as the form (1.4) u t u txx + 4uu x = 3u x u xx + uu xxx, 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 3 (1.5) u t + x 2 u2 + p 2 u2 = 0, t > 0, x R, where p(x) = 1 2 e x, stands for convolution with respect to the spatial variable x R, and p f = (1 2 x) 1 f. Since p(x) = ϕ(x), in view of the structure of Eq.(1.5), it is quite clear why the peakons can be understood as solutions. Wave breaking is one of the most intriguing long-standing problems of water wave theory [61]. For models describing water waves we say that wave breaking holds if the wave profile remains bounded, but its slope becomes unbounded in finite time [61]. Breaking waves are commonly observed in the ocean and important for a variety of reasons, but surprisingly little is known about them. Indeed, breaking waves place large hydrodynamic loads on 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 gases and particulate matter [7, 57]. To further understand why waves break and what happens during and after breaking themselves, we must first investigate the dynamics of wave breaking. Research work on breaking waves can be divided into three categories: those concerning waves (1) before, (2) during, and (3) after breaking. Although we are now understanding much 5

7 about the processes leading up to breaking, there are still some aspects of these questions unanswered, in particular, question (3), what happens after breaking of those waves. In this review we shall concentrate on some of the latest results for the DP equation in the first two categories. The KdV equation is well-known a model for water-motion on shallow water 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 L 2 [42, 59]). On the other hand, wave-breaking phenomena have been observed for certain solutions to the Whitham equation [61], u t + uu x + K(x ξ)u x (t, ξ)dξ = 0 with the singular kernel R K(x) = 1 2π R ( ) tanh ξ 1/2 e iξx dξ, (see [16] for a rigorous proof). However, the numerical calculations carried out for the Whitham equation do not support any strong claim that soliton interaction can be expected [26]. As mentioned by Whitham [61], it is intriguing to know which mathematical models for shallow water waves exhibit both phenomena of soliton interaction and wave breaking. It is found that both of the CH equation and DP equation could be first such equations and have the potential to become the new master equations for shallow water wave theory [34], modeling the soliton interaction of peaked traveling waves, wave breaking, admitting as solutions permanent waves, and being integrable Hammiltonian systems. For the CH equation, a procedure to understand the continuation of solutions past wave breaking has been recently presented by Bressan and Constantin [3]. As far as we know, the case of the Camassa-Holm equation (first derived by Fokas and Fuchssteiner [33] using the method of recursion operators as an abstract bi-hamiltonian equation) is well understood by now [12, 15, 16, 19] and the citations therein, while the Degasperis-Procesi equation case is the subject of this article. The main mathematical questions concerning with the DP equation are the well-posedness of the initial-value problem, wavebreaking phenomena, existence of global weak solutions, and stability of peakons 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 scattering approach for computing n-peakon solutions to Eq.(1.5). Holm and Staley [38] studied stability of solitons and peakons numerically to Eq.(1.5). More recently, Liu and Yin [47] proved that the first blow-up for Eq.(1.4) must ξ 6

8 occur as wave breaking and shock waves possibly appear afterwards. It is shown in [47] that the lifespan of solutions of the DP equation is not affected by the smoothness and size of initial profiles, but affected by the shape of initial profiles. This can be viewed as a significant difference between the DP 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. For 1 p, the norm in the Lebesgue space L p (R) will be written L p, while H s, s 0 will stand for the norm in the classical Sobolev spaces H s (R). 2 Comparisons between the Camassa-Holm equation and the Degasperis-Procesi equation The DP equation is presently of great interest due to its structure (integrability, special solutions presenting interesting features). While Eq.(1.1) has an apparent similarity to Eq.(1.3), which both are important model equations for shallow water waves with the breaking phenomena, there are major structural differences and it is not much to know about its qualitative aspects. One of the novel features of the DP equation is that it has not only peakon solitons [25], u(t, x) = ce x ct, c > 0 but also shock peakons [11, 49] of the form u(t, x) = 1 t + k sgn(x)e x, k > 0. It is easy to see from [49] that the above shock-peakon solutions can be observed by substituting (x, t) (ɛx, ɛt) to Eq.(1.4) and letting ɛ 0 so that it yields the derivative Burgers equation (u t + uu x ) xx = 0, from which shock waves form. In the periodic case of the spatial variable, both the CH equation and DP equation have periodic peakons [64] of the form u c (t, x) = c cosh(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 the periodic DP equation possesses the periodic shock waves given by ( cosh( 1 ) 2 ) 1 u c (t, x) = sinh( 1 2 ) t + c sinh(x [x] 1 2 ) sinh( 1 2 ), x R \ Z, c > 0, 0, x Z. On the other hand, the isospectral problem in the Lax pair for the DP equation is of third-order instead of second [25], and consequently is not 7

9 self-adjoint, ψ x ψ xxx λyψ = 0, and ψ t + 1 ( λ ψ xx + uψ x u x + 2 ) ψ = 0, 3λ while the isospectral problem for the CH equation is of second order [5], ψ xx 1 4 ψ λyψ = 0 and ( ) 1 ψ t 2λ u ψ x 1 2 u xψ = 0 (in both cases y = u u xx ). The spectral analysis and the inverse spectral theory for the CH equation are presented by Constantin and McKean [14, 19] and Johnson [41]. Lundmark and Szmigielski [50] presented an inverse scattering transform (IST) method for computing n peakon solutions of the DP equation. The approach is similar to that used by Beals, Sattinger and Szmigielski [4] to obtain n peakon solutions of the CH equation, but the present case does involve substantially new features as mentioned above. It is also noted that the CH equation is a re-expression of geodesic flow on the diffeomorphism group [17] or on the Bott-Virasoro group [55], while no such geometric derivation of the DP equation is available. Another indication of the fact that there is no simple transformation of the DP equation into the CH equation is the entirely different form of conservation laws for these two equations [5, 25]. The following are three useful conservation laws of the DP equation. E 1 (u) = y dx, E 2 (u) = yv dx, E 3 (u) = u 3 dx, R where y = (1 x)u 2 and v = (4 x) 2 1 u, while the corresponding three useful conservation laws of the CH equation are the following. F 1 (u) = y dx, F 2 (u) = (u 2 + u 2 x) dx, F 3 (u) = (u 3 + uu 2 x) dx. R R It is observed that the corresponding conservation laws of the DP equation are much weaker than those of the CH equation. Therefore, the issue of if and how particular initial data generate a blow-up in finite time is more subtle. It is worth noticing the following result obtained by Henry [37] and Mustafa [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 u 0 is a smooth function with compact support. If the solution u with initial data u 0 of (1.4) exists on some time interval [0, ɛ) with ɛ > 0 and, at any time instant t [0, ɛ), the solution u(t, ) has compact support, then u is identically zero. R R R 8

10 3 The Cauchy problem Recall p(x) = 1 2 e x and (1 2 x) 1 f = p f. Then Eq.(1.1) can be rewritten as the following form (3.1) { ut + uu x + x p ( 3 2 u2 ) = 0, t > 0, x R, u(0, x) = u 0 (x), x R. The local well-posedness of the Cauchy problem (3.1) with initial data u 0 H s (R), s > 3 2 can be obtained by applying the Kato s semigroup theory for quasilinear evolution equations. Lemma 3.1. [63] Given u 0 H s (R), s > 3 2, there exist a maximal T = T (u 0 ) > 0 and a unique solution u to initial-value problem (3.1), such that u = u(, u 0 ) C([0, T ); H s (R)) C 1 ([0, T ); H s 1 (R)). Moreover, the solution depends continuously on the initial data, i.e. the mapping u 0 u(, u 0 ) : H s (R) C([0, T ); H s (R)) C 1 ([0, T ); H s 1 (R)) is continuous and the maximal time of existence T > 0 can be chosen to be independent of s. In view of Lemma 3.1, using the energy method, one can establish the following precise blow-up scenario of strong solutions to (3.1) [63], which is similar to the case of the CH equation. Proposition 3.2. [63] Given u 0 H s (R), s > 3 2, blow up of the solution u = u(, u 0 ) in finite time T < + occurs if and only if lim inf { inf [u x(t, x)]} =. t T x R Proposition 3.3. [47] Assume u 0 H s (R), s > 3 2. Let T be the maximal existence time of the solution u to (3.1) guaranteed by Lemma 3.1. Then we have u(t, x) L 3 u 0 (x) 2 L t + u 2 0 (x) L, t [0, T ]. Remark. It is observed that if u is the solution of (1.3) with initial data u 0 H 1 (R), then we have for all t > 0, u(t, ) L (R) 2 u(t, ) H 1 (R) 2 u 0 ( ) H 1 (R). The advantage of the CH equation in comparison with the KdV equation lies in the fact that the CH equation has peaked solitons and models wave breaking (the wave profile remains bounded, but its slope becomes unbounded in finite time wave) [6]. However, note that the conservation laws of the DP 9

11 equation are much weaker than the those of the CH equation. In particular, one can see that the conservation law E 2 (u) for the DP equation is equivalent to u 2 L 2. Indeed, by the Fourier transform, we have (3.2) E 2 (u) = R yvdx = R 1 + ξ ξ 2 û(ξ) 2 dξ û 2 L 2 = u 2 L 2. It seems that the estimate in Proposition 3.3 is the best way to control the L norm of the solution for the DP equation. It then follows from Proposition 3.2 and 3.3 that the DP equation also models wave breaking in any finite time and one can expect that all the points at which the wave breaking 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 u 0 H s (R), s 3, and let T > 0 be the maximal existence time of the corresponding solution u to (3.1). Then Eq.(3.3) has a unique solution q C 1 ([0, T ) R, R). Moreover, the map q(t, ) is an increasing diffeomorphism of R with ( t ) q x (t, x) = exp u x (s, q(s, x))ds > 0, (t, x) [0, T ) R. 0 The following result, analogous to the case of the CH equation, plays a crucial 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 is well-defined. We then infer by means of the geometric interpretation of q a very important invariant for the solutions to (3.1). Lemma 3.5. [65] Let u 0 H s (R), s 3, and let T > 0 be the maximal existence time of the corresponding solution u to (3.1). Setting y = u u xx, we have y(t, q(t, x))q 3 x(t, x) = y 0 (x), (t, x) [0, T ) R. Proof. The result can be obtained by differentiating the left-hand side with respect 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 of the 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 (Theorem 3.6 and Theorem 4.1). 10

12 Theorem 3.6. [47] Assume u 0 H s (R), s > 3 2 and there exists x 0 [, ] such that { y0 (x) 0 if x x 0, y 0 (x) 0 if x x 0. Then initial-value problem (3.1) has a unique global strong solution u = u(., u 0 ) C([0, ); H s (R)) C 1 ([0, ); H s 1 (R)). Moreover, E 2 (u) = R yv dx = E 2(u 0 ), where y = (1 2 x)u and v = (4 2 x) 1 u, and for all t R + we have (i) u x (t, ) u(t, ) on R, (ii) u u 0 4 L 2 t u 0 2 L 2 u 0 L t + u Proof. Note that the solution u of the DP equation satisfies (3.4) u(t, x) = e x 2 x e η y(t, η)dη + ex 2 x e η y(t, η)dη and (3.5) u x (t, x) = e x 2 x e η y(η)dη + ex 2 x e η y(η)dη. From the above two equations, we deduce that (3.6) { ux (t, x) u(t, x) if x q(t, x 0 ), u x (t, x) u(t, x) if x q(t, x 0 ). This implies that u x (t, ) u(t, ) on R for all t [0, T ). Therefore, in view of Proposition 3.2, the global existence result can be obtained by using the estimate of u L in Proposition Wave breaking phenomena Taking into account invariant of y in Lemma 3.5, one can establish some wave breaking results with different shapes of initial profiles. Theorem 4.1. [47] Let u 0 H s (R), s > 3 2. Assume there exists x 0 R such that { y0 (x) = u 0 (x) u 0,xx (x) 0 if x x 0, y 0 (x) = u 0 (x) u 0,xx (x) 0 if x x 0, and y 0 changes sign. Then, the corresponding solution to (3.1) blows up in finite time T < + and satisfies lim inf { inf [u x(t, x)]} =. t T x R 11

13 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 time evolution of u x (t, q(t, x 0 )) which can be used to prove that T <. We give only an outline and refer to [47] for the full details. Proof. Firstly, by differentiating Eq.(1.5) with respect to x, we have u tx + uu xx = u 2 x + 3 ( ) 3 2 u2 p 2 u2 = u 2 x + 3 ( ) 1 2 u2 p 2 u2 x + u p ( u 2 x u 2). taking into account p ( 1 2 u2 x + u 2) (t, x) 1 2 u2 (t, x), it then follows that (4.1) u tx + uu xx u 2 x + u p ( u 2 x u 2). 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, x 0 ). Next we define x (4.3) M(t, x) := e x e η y(t, η)dη, t [0, T ), and (4.4) N(t, x) := e x e η y(t, η)dη, t [0, T ). x Then it is easy to see that (4.5) M(t, q(t, x 0 ))N(t, q(t, x 0 )) = u 2 (t, q(t, x 0 )) u 2 x(t, q(t, x 0 )), and (4.6) M(t, q(t, x 0 ))N(t, q(t, x 0 )) = q(t,x0 ) e η y(t, η)dη q(t,x 0 ) e η y(t, η)dη < 0, t [0, T ). 12

14 On the other hand, we have (4.7) dm(t, q(t, x 0 )) dt = u(t, x 0 )M(t, q(t, x 0 )) 1 2 u2 (t, q(t, x 0 )) u(t, q(t, x 0 ))u x (t, q(t, x 0 )) + u 2 x(t, q(t, x 0 )) + e q(t,x 0) = u 2 x(t, q(t, x 0 )) e q(t,x 0) 3 2 u2 (t, q(t, x 0 )) + e q(t,x 0) q(t,x0 ) q(t,x0 ) M(t, q(t, x 0 ))N(t, q(t, x 0 )) e q(t,x 0) q(t,x0 ) e η ( u 2 (t, η) u 2 η(t, η) ) dη ( ) 1 e η 2 u2 η(t, η) + u 2 (t, η) dη q(t,x0 ) The following estimate is crucial to ensure increase of M at t. (4.8) q(t,x0 e q(t,x ) 0) 3 2 eη u 2 (t, η)dη e η ( u 2 (t, η) u 2 η(t, η) ) dη. e η ( u 2 (t, η) u 2 η(t, η) ) dη M(t, q(t, x 0 ))N(t, q(t, x 0 )). Combining (4.17) with (4.18), we obtain (4.9) dm(t, q(t, x 0 )) dt 1 2 M(t, q(t, x 0))N(t, q(t, x 0 )) > 0. In an analogous way, we have the estimate of N. (4.10) and e q(t,x 0) (4.11) Therefore, q(t,x 0 ) e η ( u 2 (t, η) u 2 η(t, η) ) dη M(t, q(t, x 0 ))N(t, q(t, x 0 )) dn(t, q(t, x 0 )) dt 1 2 M(t, q(t, x 0))N(t, q(t, x 0 )) < 0. (4.12) M(t, q(t, x 0 ))N(t, q(t, x 0 )) < M(0, x 0 )N(0, x 0 ) < 0, t [0, T ). On the other hand, it follows from (4.8) and (4.10) that (4.13) p (u 2 u 2 x)(t, q(t, x 0 )) = 1 2 e q(t,x 0) η ( u 2 (t, η) u 2 η(t, η) ) dη M(t, q(t, x 0 ))N(t, q(t, x 0 )). In view of (4.12) and (4.13), it is then inferred from (4.1) that (4.14) df(t) dt 1 2 ( u 2 (t, q(t, x 0 )) u 2 x(t, q(t, x 0 )) ) = 1 2 M(t, q(t, x 0))N(t, q(t, x 0 )) < 1 2 M(0, x 0)N(0, x 0 ) < 0. 13

15 where the function f(t) is defined by f(t) = u x (t, q(t, x 0 )). 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 that there exists t 1 > 0 such that (4.15) f 2 (t) 2u 2 (t, q(t, x 0 )), t t 1. Applying Gronwall s inequality to (4.9) and (4.11) yields that M(t, q(t, x 0 )) M(0, x 0 )e 1 2 N(0,x 0)t > 0, N(t, q(t, x 0 )) N(0, x 0 )e 1 2 M(0,x 0)t > 0. The above two estimates then imply that (4.16) u 2 x(t, q(t, x 0 )) u 2 (t, q(t, x 0 )) M(0, x 0 )N(0, x 0 )e 1 2 (M(0,x 0) N(0,x 0 ))t. The proof of estimate (4.15) then follows from Proposition 3.3. Combining (4.14) with (4.15), we obtain (4.17) Note that d dt f(t) 1 2 u2 (t, q(t, x 0 )) 1 2 f 2 (t) 1 4 f 2 (t), f(0) = u x (0, x 0 ) = 1 2 e x 0 x0 e η y 0 (η)dη ex 0 t [t 1, ). x 0 e η y 0 (η)dη < 0. The proof of blow-up T < + then results from the classical inequality (4.17). Remark. Assume the initial profile y 0 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, y 0 0 on R and y 0 0 on R +, implies u 0 (0) < 0. The following blow-up result improves Theorem 4.1 and may also include the case of u 0 (0) 0. Theorem 4.2. [48] Assume u 0 H s (R), s > 3 2 and y 0(x) = u 0 (x) u 0,xx (x) is odd. If there is only one point x 0 (0, ) such that y 0 (x 0 ) = 0, then the corresponding solution u to initial-value problem (3.1) blows up in finite time. Proof. The key point is how to use the assumptions to derive a differential inequality for the time evolution equation of u x (t, q(t, x 0 )). By the symmetry, we only need to consider the case that there is a x 0 > 0 such that { y0 (x) > 0 x (, x 0 ), y 0 (x) < 0 x ( x 0, 0), 14

16 and y 0 (x 0 ) = 0. Since y 0 is odd, it follows that u 0 = p y 0 is odd. Thus, the corresponding 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 symmetry of the solution, we then deduce that (4.18) { y(t, x) > 0 x (, q(t, x0 )), y(t, x) < 0 x ( q(t, x 0 ), 0), and y(t, q(t, x 0 )) = 0 for all t [0, T ). In view of (4.18), we have for all t [0, T ) (4.19) (u u x )(t, q(t, x 0 )) = e q(t, x 0) q(t, x0 ) e η y(t, η)dη > 0 and (4.20) (u x + u)(t, q(t, x 0 )) = e q(t, x 0) q(t, x 0 ) ( 0 = e q(t, x 0) [e η e η ]y(t, η)dη + q(t, x 0 ) e q(t, x 0) q(t,x 0 ) e η y(t, η)dη < 0. e η y(t, η)dη q(t,x 0 ) e η y(t, η)dη ) From the above two relations (4.18) and (4.19), we may also obtain (4.21) u x (t, q(t, x 0 )) < 0. On the other hand, for η (, q(t, x 0 )), t 0 we have (4.22) u 2 (t, η) u 2 x(t, η) = η ( q(t, x0 ) e ξ y(t, ξ)dξ q(t, x0 ) e ξ y(t, ξ)dξ q(t, x0 ) e ξ y(t, ξ)dξ e ξ y(t, ξ)dξ q(t, x 0 ) = u 2 (t, q(t, x 0 )) u 2 x(t, q(t, x 0 )). η ( q(t, x0 ) η e ξ y(t, ξ)dξ + q(t, x 0 ) ) e ξ y(t, ξ)dξ e ξ y(t, ξ)dξ q(t, x 0 ) e ξ y(t, ξ)dξ ) 15

17 Hence we infer from the above inequality that (4.23) d dt (u u x)(t, q(t, x 0 )) = q t (t, x 0 )(u u x )(t, q(t, x 0 )) + e q(t,x 0) = u 2 x(t, q(t, x 0 )) e q(t, x 0) 3 2 u2 (t, q(t, x 0 )) e q(t, x 0) q(t, x0 ) (u 2 x u 2 )(t, q(t, x 0 )) e q(t, x 0) q(t, x0 ) 1 2 [(u u x)(u + u x )](t, q(t, x 0 )) > 0. It then follows from (4.19) and (4.23) that (4.24) d ( ) e q(t, x0) (u u x )(t, q(t, x 0 )) dt q(t, x0 ) e η y t (t, η)dη e η ( u 2 (t, η) u 2 η(t, η) ) dη q(t, x0 ) e η ( u 2 η(t, η) + 2u 2 (t, η) ) dη e η ( u 2 (t, η) u 2 η(t, η) ) dη = e q(t, x 0) q t (u u x )(t, q(t, x 0 )) + e q(t, x 0) d dt (u u x)(t, q(t, x 0 )) e q(t, x 0) (u 2 uu x )(t, q(t, x 0 )) eq(t, x 0) (u 2 x u 2 )(t, q(t, x 0 )) 1 2 eq(t, x 0) (u u x )(t, q(t, x 0 ))[(u u x )(0, x 0 )] > 0. This in turn implies that ( ) e q(t, x0) (u u x )(t, q(t, x 0 )) e x 0 [(u u x )(0, x 0 )]e 1 2 [(u u x)(0, x 0 )]t. Consequently, we deduce that u x (t, q(t, x 0 )) (4.25) [(u u x )(0, x 0 )]e ( 1 2 [(u ux)(0, x 0)]t+q(t,x 0 ) x 0) u(t, q(t, x0 )) [(u u x )(0, x 0 )]e ( 1 2 [(u ux)(0, x 0)]t x 0) ( 3 u0 2 L 2 t + u 0 L ). The rest of the proof can be obtained by following the outline of proof in Theorem 4.1. Remark. It was shown by McKean [53, 54] that the solutions of the CH equation breaks down if and only if some portion of the positive part of y = u 2 xu initially lies to the left of some portion of its negative part. The 16

18 problem whether or not the DP equation has these wave breaking phenomena still remains open. Because of the structural difference between these two equations, it is difficult to use the machinery of McKean [Mc2] in study of the associated spectral problem with the corresponding eigenvalues. The issue of if and how these particular initial data generate a global solution or blow-up in finite time is more subtle. In contrast with the conditions of the blow-up solution of the DP equation defined on the line R, one can see that the criteria of blow-up for periodic solutions of the DP equation are quite different. Let us consider periodic solutions of (3.1), i.e, u : S [0, T ) R where S is the unit circle and T > 0 is the maximal existence time of the solution. The interest in periodic solutions is motivated by the observation that the majority of the waves propagating on a channel are approximately periodic. Define G(x) by G(x) = cosh(x [x] 1 2 ), where [x] stands for the integer part 2 sinh( 1 2 ) of x R, then (1 x) 2 1 f = G f for all f L 2 (S). Using this identity, we can rewrite (3.1) as a quasi-linear evolution equation of hyperbolic type, namely, u t + uu x + x G ( 3 2 u2 ) = 0, t > 0, x R, (4.26) u(0, x) = u 0 (x), x R, u(t, x) = u(t, x + 1), t 0, x R, Theorem 4.3. [31] Assume that u 0 H s (S), s > 3 2, u 0 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 from that of the line case (Teeorem ). An outline of the proof is given in the following. By assumption, for each t [0, T ) there is a ξ t [0, 1] such that u(t, ξ t ) = 0. Then for x S we have ( x ) 2 x [ (4.27) u 2 (t, x) = u x dx (x ξ t ) u 2 x dx, x ξ t, ξ t + 1 ]. ξ t 2 ξ t Hence the above relation and an integration by parts yield ξt+ 1 2 ξt+ 1 ( u 2 u 2 2 x ) ( ) x dx (x ξ t )u 2 x u 2 x dx 1 ξt u 2 x dx. ξ t ξ t ξ t 4 ξ t Combining this estimate with a similar estimate on [ξ t + 1 2, ξ t +1], we obtain (4.28) u 2 u 2 x dx 1 ( 2 u 2 x dx). S 4 S 17

19 By (4.19), we also have (4.29) sup x S u 2 (t, x) 1 2 S u 2 x dx. Let us assume that the solution u(t, x) exists globally in time. Note that 1 G(x) for all x S. It then follows from (4.20) and (4.21) that 2 sinh( 1 ) 2 (4.30) ( d u 3 x dx = 3 u 2 x u 2 x uu xx + 3 ( 3 dt S S 2 u2 G = 3 u 4 x dx 3 u 2 xuu xx dx + 9 u 2 S S 2 xu 2 dx 9 S 2 S 2 u 4 x dx + 9 ( S 8 2 u 2 x dx) S 9 2 sinh( 1 2 ) Applying the Cauchy-Schwartz inequality, we have (4.31) ( ) 9 2 u 2 x dx 9 u 4 x dx. 8 S 8 S S u 2 xdx On the other hand, in view of (4.20), it is easy to see that 2 u2 S )) dx u 2 xg ( u 2) dx u 2 dx. ( ) 2 (4.32) u 2 xdx u 2 dx 2 u 2 dx 1 S S S 8 u 0 4 L. 2 It is thus deduced from (4.21)-(4.23) that d u 3 x dx 7 u 4 9 x dx dt S 8 S 16 sinh( 1 2 ) u 0 4 L, t 0. 2 Hence an application of Hölder s inequality yields (4.33) d u 3 x dx 7 ( ) 4 u x dx dt S 8 S 16 sinh( 1 2 ) u 0 4 L, t 0. 2 If we define V (t) := S u3 x(t, x) dx for all t 0, then V (t) V (0) 9 16 sinh( 1 2 ) u 0 4 L 2 t, t 0. Since u 0 0, the above inequality implies that there exists some t 0 0 such that V (t) < 0 for all t t 0. It is then inferred from (4.24) that d dt V (t) 7 8 (V (t)) 4 3, t > t0. 18

20 Thus we have ( 7(t t 0 ) (V (t 0 )) 1 3 ) 3 1 V (t) < 0, t t 0. Since V (t 0 ) < 0, the above inequality will lead to a contradiction as t t 0 is big enough, which implies T <. As immediate consequences of Theorem 4.3, we have Corollary 4.4. If u 0 H 3 (S), u 0 0 and S u 0 dx = 0 or S y 0 dx = 0, then the corresponding solution u to (4.26) blows up in finite time. Proof. Note u(t, x) dx = S S y(t, x) dx = S y 0 (x) dx = S u 0 (x) dx = 0. The above relation shows that u(t, x) has a zero for all t S. It follows from Theorem 4.3 that the solution u to (4.26) blows up in finite time. Corollary 4.5. If u 0 H s (S), s > 3 2, u 0 0 and S u3 0 dx = 0, then the corresponding solution u(t, x) of (4.26) blows up in finite time. Proof. The result can be obtained immediately from the conservation law E 3 (u) = u 3 dx. S Corollary 4.6. If u 0 (x) or y 0 is odd, then the corresponding solution u to (4.26) blows up in finite time. Proof. If u 0 (x) or y 0 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 3.3 in [47]) where the corresponding solution u with u 0 H s (R), s > 3/2, and u 0 (x) being odd, may exist globally in time, we find from Theorem 4.3 that there 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 unbounded in finite time. In this section we give a quite detailed description of its phenomena. The following theorem shows that there is only one point where the slope of the solution becomes infinity exactly at breaking time. 19

21 Theorem 5.1. [47] Assume u 0 H s (R), s > 3 2 and there exists x 0 R such that { y0 (x) = u 0 (x) u 0,xx (x) 0 if x x 0, y 0 (x) = u 0 (x) u 0,xx (x) 0 if x x 0, and y 0 changes sign. Let T < be the finite blow-up time of the corresponding solution u to (3.1). Then we have Proof. For any x q(t, x 0 ), we have lim u x(t, q(t, x 0 )) =. t T q(t,x0 u x (t, x) = u(t, x) + e x ) e η y(t, η)dη + e x On the other hand, x u(t, x) + u x (t, q(t, x 0 )) + u(t, q(t, x 0 )). q(t,x 0 ) u x (t, x) u(t, x) + u x (t, q(t, x 0 )) u(t, q(t, x 0 )). e η y(t, η)dη From the above two inequalities we deduce that for (t, x) [0, T ) R, (5.1) u x (t, x) u x (t, q(t, x 0 )) 2 u(t, ) L u x (t, q(t, x 0 )) 2 ( 3 u 0 (x) 2 L 2 t + u 0 (x) L ). In view of T <, it follows from Proposition 3.2 that lim inf ( inf u x(t, x)) =. t T x R Consequently, lim t T u x (t, q(t, x 0 )) =. Theorem 5.2. [30] Let T < be the blow-up time of the solution u of (3.1) with initial data u 0 H s (R), s > 3 2 such that the associated potential y 0 = u 0 u 0,xx satisfies y 0 (x) 0 on (, x 0 ] and y 0 (x) 0 on [x 0, ) for some points x 0 R and y 0 does not have a constant sign. Then ( ) ( ) lim inf {u x(t, x)}(t t) = 1 and lim sup{u x (t, x)}(t t) = 0. t T x R t T x R The following result gives some information about the blow-up set of a breaking wave to (3.1) with a large class of initial data. Theorem 5.3. [30] Assume that u 0 H s (R), s > 3 2 and u 0 0 is odd such that the associated potential y 0 = u 0 u 0,xx is nonnegative on R. Then the solution u to (3.1) with initial data u 0 blows up in finite time only at zero point. 20

22 For the periodic solutions of (4.26) we have Theorem 5.4. [31] Assume that u 0 H s (S), s > 3 2 and that the corresponding associated potential y 0 := u 0 u 0,xx 0 is odd. (a) Suppose that (x 1 2 )y 0(x) 0 on S. Then the solution to (4.26)) with initial data u 0 blows up in finite time only at the point x = 1 2 and lim u x(t, 1 t T 2 ) =. (b) Suppose that (x 1 2 )y 0(x) 0 on S. Then the solution to (4.26) with initial data u 0 blows up in finite time only at two points x = 0 and x = 1, and lim u x(t, 0) = lim u x (t, 1) =. t T t T 6 Global weak solutions As mentioned in Introduction, the DP equation possesses the peaked solitons of the form (6.1) u(t, x) = cϕ(x ct), where ϕ(x) = e x. It is easy to see that (1 2 x)ϕ = 2δ (here δ is the Dirac distribution). Note these peakons are not the strong solutions in H s, s 3 2, but the global weak solutions in H 1 [25, 65]. Using the conservation law of the DP equation, a partial integration result in the Bochner spaces, and the Helly theorem, one can obtain the following global weak solutions in H 1. Theorem 6.1. [30] Let u 0 H 1 (R). Assume y 0 = (u 0 u 0,xx ) M(R) and there is a x 0 R such that supp y 0 (, x 0) and supp y + 0 (x 0, ). Then initial-value problem (3.1) has a unique global weak solution with u W 1, loc (R + R) L loc (R +; H 1 (R)) y(t, ) = u(t, ) u xx (t, )) L loc (R +, M(R). Moreover, E 1 (u) and E 2 (u) are two conservation laws. Example 1.(Peaked solitons) Let u 0 (x) = ce x, x R, c > 0. Then y 0 = u 0 u 0,xx = 2cδ(x) and u(t, x) = ce x ct is the unique global weak solution with the initial data u 0. Remark. More interestingly, we find in [32] that there are no travelingwave solutions u C([0, ); H 3 ) C 1 ([0, ); H 2 ) to (3.1). Arguing by 21

23 contradiction, we assume that w H 3 and u(t, x) = w(x ct), c 0 is a strong solution of (3.1). Then we have We find that cw cw 4ww + 3w w + ww = 0 in L 2. ( cw w 2 2w 2 + (w ) 2 + ww ) = 0 in L2 and therefore or, what is same, cw w 2 2w 2 + (w ) 2 + ww = 0 in H 1 (c w)(w w ) ( w 2 (w ) 2) = 0 in H 1 since w H 3 C 2 0 (R). Multiplying this identity with 2w yields that (6.2) (c w) ( w 2 (w ) 2) 2w ( w 2 (w ) 2) = 0. Since w H 3 C0 2(R), we have w c, a.e. and w2 (w ) 2, a.e. Let w 0 = w(ξ) = max x R w(x) > 0. Then taking integration for (6.2) in [ξ, x] yields This implies that x ξ d ( w 2 (w ) 2) x w 2 (w ) 2 = ξ 2dw c w, x R. (6.3) (w c) 2 w 2 (w ) 2 = w 2 0(w 0 c) 2 x R. If we take into account w, w 0 as x, it is then inferred from relation (6.3) that w 2 0(w 0 c) 2 = 0 which also implies from (6.3) that (w c) 2 w 2 (w ) 2 = 0, x R. This leads a contradiction since w H 3. Example 2. Let u 0 (x) = c 1 e x x 1 + c 2 e x x 2, x R, with c 1 < 0, c 2 > 0 and x 1 < x 2. It is easily found that y 0 = u 0 u 0,xx = 2c 1 δ(x x 1 ) + 2c 2 δ(x x 2 ). 22

24 By Theorem 6.1, there exists a unique global weak solution u to (3.1) with the initial data u 0. It has the explicit form [25] u(t, x) = p 1 (t)e x q 1(t) + p 2 (t)e x q 2(t), (t, x) R + R, for some p 1, p 2, q 1, q 2 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 to the left and the peakon moves off to the right so that no collision occurs. Remark. In addition, if the initial data u 0 (x) = c 1 e x x 1 + c 2 e x x 2, with c 1 > 0, c 2 < 0 and x 1 + x 2 = 0, x 2 > 0, then the collision occurs at x = 0 and the solution u(x, t) = p 1 (t)e x q 1(t) + p 2 (t)e x q 2(t), (x, t) R + R, only satisfies the DP equation for t < T. The unique continuation of u into an entropy weak solution is then given by the stationary decaying 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 DP equation shows that the DP equation would behave radically different from the Camassa-Holm equation, but similar to the inviscid Burgers equation, which implies that a well-posedness theory should depend on some functional spaces which contain discontinuous functions. Indeed, this observation was confirmed by Coclite and Karlsen [8, 9]. In [8, 9], they [10] proved the global existence and uniqueness of L 1 BV entropy weak solutions satisfying an infinite family of Kru zkov-type entropy inequalities. Recently, they proved existence of bounded weak solutions by an Oleĭnik-type estimate for L solutions to (3.1). Theorem 6.2. [10] Suppose u 0 L (R). Then there exists a unique entropy weak solution u L ((0, T ) R), for any T > 0 to initial-value problem (3.1) and for each T > 0 there exists a positive constant K T such that the estimate ( ) u(t, x) u)t, y) 1 K T x y t + 1 holds for any x, y R, x y, 0 < t < T. Indeed, it is recently shown by Coclite and Karlsen [10] that the infinite family of entropy inequality is equivalent to the one-side Lipshitz inequality. 23

25 Therefore the well-posedness in L 1 L of the Cauchy problem for the DP equation can be established [10]. The relevance of these solutions of the DP equation is supported by Lundmark [49], who found some explicit shock solutions of the DP equation which are entropy weak solutions. Numerical schemes for computing entropy weak solutions of the DP equation is developed and analyzed in [11]. On the other hand, these discontinuous solutions of the DP equation are also investigated by Feng and Liu [32] using an operator splitting method, which consists of a second-order total variation diminishing (TVD) scheme and a second-order linearized finite difference scheme. Here, we first show an example for symmetric peakon-antipeakon collision, in which initial condition u 0 is taken as (6.2) u 0 (x) = e x+4 e x 4. As shown in Figure 1, when t > 4, a shockpeakon is formed and continues to decay as a shockpeakon solution mentioned previously. Actually, we observed that shock formation is generic if u 0 (x) < 0 and u 0(x) = 0 for some x. We illustrate this by a numerical example with initial condition (6.3) u 0 (x) = e 0.5x2 sin(πx), for x [ 2, 2], where u 0 is extended periodically outside this interval. Figure 2 shows the numerical results up to t = 0.9. It is seen that two shocks are formed at t 0.2, then a collision between them produces a stationary shockpeakon at x = u(x,t) t=0 t=2 t=4 t= x Figure 1: The numerical solution of a symmetric peakon antipeakon collision 24

26 4 3 t=0.0 t=0.2 t=0.6 t= u(x,t) x Figure 2: Wave breaking from a smooth initial condition 7 Stability of peakons The stability of solitary waves is one of the fundamental qualitative properties of the solutions of nonlinear wave equations. Numerical simulations [25, 49] suggest that the sizes and velocities of the peakons do not change as a result of collision so these patterns are expected to be stable. Furthermore, it is observed that the shape of the peakons remains approximately the same as time evolves. stability of the peakons for the Camassa-Holm equation is well understood by now [21, 22], while the the case of the Degasperis-Procesi equation is recently studied in [44]. As mentioned before, the corresponding conservation laws of the DP equation are much weaker than those of the CH equation. In particular, one can see that the conservation law E 2 (u) for the DP equation is only equivalent to u 2 L 2. Therefore, the stability issue of the peaked solitons of the DP equation is more subtle. Probably, one can only expect to obtain the orbital stability of peakons in the sense of L 2 norm due to a weaker conservation law E 2. The solutions of the DP equation usually tend to be oscillations which spread out spatially in a quite complicated way. In general, a small perturbation of a solitary wave can yield another one with a different speed and phase shift. We define the orbit of traveling-wave solutions cϕ to be the set U(ϕ) = {cϕ( + x 0 ), x 0 R}, and a peaked soliton of the DP equation is called orbitally stable if a wave starting close to the peakon remains close to some translate of it at all later times. Let us denote E 2 (u) = u 2 X. Theorem 7.1. [44] Let cϕ be the peaked soliton defined in (6.1). Then cϕ 25

27 is orbitally stable in the following sense. If u 0 H s for some s > 3/2, y 0 = u 0 2 xu 0 is a nonnegative Radon measure of finite total mass, and u 0 cϕ X < cε, E 3 (u 0 ) E 3 (cϕ) < c 3 ε, 0 < ε < 1 2, then the corresponding solution u(t) of (3.1) with initial value u(0) = u 0 satisfies sup u(t, ) cϕ( ξ 1 (t)) X < 3c ε 1/4, t 0 where ξ 1 (t) R is the maximum point of the function v(t, ) = (4 2 x) 1 u(t, ). Moreover, let M 1 (t) = v(t, ξ 1 (t)) M 2 (t) M n (t) 0 and m 1 (t) m n 1 (t) 0 be all local maxima and minima of the nonnegative function v(t, ), respectively. Then (7.1) M 1 (t) c c 2ε 6 and (7.2) n ( M 2 i (t) m 2 i 1 (t) ) < 2c 2 ε. i=2 Remark 1. Under the assumption y 0 = u 0 2 xu 0 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 y 0 that it is a nonnegative measure is quite natural for a small perturbation of the peakons. Existence of global weak solution in H 1 of the DP equation is also proved in [30]. Note that peakons cϕ are not strong solutions, since ϕ H s, only for s < 3/2. Remark 2. The above theorem of orbital stability states that any solution starting close to peakons cϕ remains close to some translate of cϕ in the norm X, at any later time. More information about this stability is contained in (7.1) and (7.2). Notice that for peakons cϕ, the function v cϕ is single-humped with the height 1 6c. So (7.1) and (7.2) imply that the graph of 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 M 1 = max v u E 2 (u) /12. For peakons cϕ, we have max v cϕ = E 2 (cϕ) /12 = 1 6c. So among all waves of a fixed energy E 2, the peakon is tallest in terms of v u. Remark 4. In addition, it is found in the proof of [44] that the peakons are energy minimizers with a fixed invariant E 3, which explains their stability, i. e. if E 3 (u) = E 3 (ϕ), then E 2 (u) E 2 (ϕ). The same remark also applies 26

28 to the CH equation and shows that the CH-peakons are energy minima with fixed F 3. There are two standard methods to study stability issues of dispersive wave equations. One is the variational approach which constructs the solitary waves as energy minimizers under appropriate constraints, and the stability automatically follows. However, without uniqueness of the minimizer, one can only obtain the stability of the set of minima. The variational approach is used in [21] for the CH equation. It is shown in [21] that the each peakon cϕ is the unique minimum (ground state) of constrained energy, from which its orbital stability is proved for initial data u 0 H 3 with y 0 = (1 x)u Their proof strongly relies on the fact that the conserved energy F 2 of the CH equation is the H 1 norm of the solution. However, for the DP equation the energy E 2 is only the L 2 norm of the solution. Consequently, it is more difficult to use such a variational approach for the DP equation. Another approach to study stability is to linearize the equation around the solitary waves, and it is commonly believed that nonlinear stability is governed by the linearized equation. However, for the CH and DP equations, the nonlinearity plays the dominant role rather than being a higher-order correction to linear terms. Thus it is unclear how one can get nonlinear stability of peakons by studying the linearized problem. Moreover, the peaked solitons cϕ are not differentiable, which makes it difficult to analyze the spectrum of the linearized operator around cϕ. To establish the stability result for the DP equation, we extend the approach in [22] for the CH equation. The idea in [22] is to directly use the energy F 2 as the Liapunov functional. By expanding F 2 around the peakon cϕ, the error term is in the form of the difference of the maxima of cϕ and the perturbed solution u. To estimate this difference, they establish two integral relations g 2 = F 2 (u) 2 (max u) 2 and ug 2 = F 3 (u) 4 (max u)3 3 with a function g. Relating these two integrals, one can get F 3 (u) MF 2 (u) 2 3 M 3, M = max u(x) and the error estimate M max ϕ then follows from the structure of the above polynomial inequality. To extend the above approach to nonlinear stability of the DP peakons, one has to overcome several difficulties. By expanding the energy E 2 (u) around the peakon cϕ, the error term turns out to be max v cϕ max v u, with v u = (4 x) 2 1 u. We can derive the following two integral relations 27

29 for M 1 = max v u, E 2 (u) and E 3 (u) by g 2 = E 2 (u) 12M1 2 and hg 2 = E 3 (u) 144M 3 1 with some functions g and h related to v u. To get the required polynomial inequality from the above two identities, we need to show h 18 max v u. However, since h is of the form 2 xv u ± 6 x v u + 16v u, generally it can not be bounded by v u. This new difficulty is due to the more complicated nonlinear structure and weaker conservation laws of the DP equation. To overcome it, we introduce a new idea. By constructing g and h piecewise according to monotonicity of the function v u, we then establish two new integral identities related to E 2, E 3 and all local maxima and minima of v u. The crucial estimate h 18 max v u can now be shown by using this monotonicity structure and properties of the DP solutions. Then one can obtain not only the error estimate M 1 max v cϕ but more precise stability information from (7.2). It is observed that the same approach can also be used for the CH equation to gain more stability information. The detailed proof of stability of peakons for the DP equation can be found in [44]. References [1] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D waterwaves and asymptotics, Invent. Math., 171 (2008), [2] T.b. Benjamin, J. L. Bona and J. J. Mahoney, Lmodel equations for long waves in nonlinear dispersive system, Philos. trans. R. Soc. Lond. A, 227 (1972), [3] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), [4] R. Beals, D.H. Sattinger and J. Szmigielski, Multipeakons and the classical moment problem, Adv. Math., 154 (2000), [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), [6] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), [7] E. D. Cokelet, Breaking waves, Nature, 267 (1997), [8] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006),