Global Dissipative Solutions of the Camassa-Holm Equation

Transcription

1 Global Dissipative Solutions of the Camassa-Holm Equation Alberto Bressan Department of Mathematics, Pennsylvania State University, University Park 168, U.S.A. an Arian Constantin Department of Mathematics, Lun University, 1 Lun, Sween Trinity College, Department of Mathematics, Dublin, Irelan arian.constantin@math.lu.se Abstract. This paper is concerne with the global existence of issipative solutions to the Camassa-Holm equation after wave breaking. By introucing a new set of inepenent an epenent variables, the evolution problem is rewritten as an O.D.E. in an L space, containing a non-local source term which is iscontinuous but has boune irectional variation along a suitable cone of irections. For a given initial conition, the Cauchy problem has a unique solution obtaine as fixe point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global issipative solutions, efine for every initial ata ū H 1, an continuously epening on the initial ata. The new variables resolve all singularities ue to possible wave breaking an ensure that energy loss occurs only through wave breaking. - Introuction The Camassa-Holm equation u t u txx + 3uu x = u x u xx + uu xxx t >, x,.1 moels the propagation of water waves in the shallow water regime, when the wavelength is consierably larger than the average water epth. Here ut, x represents the water s free surface over a flat be [CH]. The equation.1 was first consiere by Fokas an Fuchssteiner [FF] as an abstract bi-hamiltonian P.D.E. with infinitely many conservation laws. For a etaile iscussion of the conservation laws we refer to [I] an [L]. This equation attracte a lot of attention after Camassa an Holm [CH] erive it as a moel for shallow water waves an iscovere that it is formally integrable, in the sense that there is an associate Lax pair, an that its solitary waves are solitons, i.e. they retain their shape an spee after the interaction with waves of the same type. An alternative erivation of the equation as a moel for shallow water waves was subsequently given by Johnson [J]. The Camassa-Holm equation has a very rich structure. For a large class of initial ata the equation is an integrable infinite imensional Hamiltonian system: to each solution with initial ata in this class, one can associate some scattering ata that evolve in time linearly at constant 1

2 spee, an from which the solution can be reconstructe explicitly, see [BSS, CM1, C]. In contrast to the Korteweg-e Vries equation, which is a classical integrable moel for shallow water waves, the Camassa-Holm equation possesses not only solutions that are global in time but moels also wave breaking. When this happens, the solution remains Höler continuous an uniformly boune, but evelops an unboune slope in finite time [C1]. By looking at a functional of the initial ata, one can preict whether solutions will remain globally smooth [CE1] or prouce wave breaking in finite time [CE]. An aspect of consierable interest is the behavior of the solutions after wave breaking. In [XZ1, XZ] an more recently in [CHK], global issipative solutions to the Camassa-Holm equations, in a weak istributional sense, were obtaine as weak limits of viscous regularizations. only In [BC] a new approach in the analysis of the Camassa-Holm equation was evelope. By introucing a new set of inepenent an epenent variables, the equation was transforme into a semilinear hyperbolic system. This yiels an O.D.E. in a suitable Banach space, whose solutions can be obtaine as fixe point of a contractive transformation. Returning to the original variables, this provie a group of conservative solutions, globally efine forwar an backwar in time. Here conservative means that the total energy of the solution equals a constant, for almost every time t. An avantage of this approach, besies being more irect, is that it provies solutions in a stronger sense. Namely, these solutions are Lipschitz continuous when regare as mappings u : [, T ] L, an satisfy the equation at a.e. time t. A similar approach, base on the introuction of suitable Lagrangian variables, was pursue in [BZZ] an in [BZ], in connection with first an secon orer nonlinear wave equations respectively. The aim of this paper is to prove that, by a suitable moification in the efinition of the semilinear hyperbolic system, one can construct a continuous semigroup of issipative solutions forwar in time. We remark that the issipative case is more elicate, because the corresponing O.D.E. now contains a iscontinuous non-local source term. The existence an uniqueness of solutions can still be be establishe, observing that all iscontinuities are crosse transversally. More precisely, we will write hyperbolic system as an O.D.E. in an L space, having locally boune variation in the irection of a suitable cone. The abstract result in [BS] thus provies the well-poseness of the Cauchy problem. Reverting to the original coorinates, we obtain a local, issipative solution of the Camassa- Holm equation. Thanks to a uniform boun on the H 1 norm, this solutions can be extene forwar in time for all t. Our construction yiels a continuous semigroup on the space H 1. We observe that energy loss for our issipative solutions can occur only uring wave breaking. This last feature is physically relevant. Inee, it is known that while water waves break a certain amount of heat is prouce. This is inicative of a loss of energy uring the process. The uniqueness of issipative solutions to the Camassa-Holm equation is a elicate issue. What we prove here is that: Our constructive proceure via coorinate transformations yiels a unique semigroup of solutions, efine on the entire space H 1. All of our solutions satisfy the Oleinik type inequality u x t, x C 1 + t 1 t >,. with a constant C epening only on the norm of the initial ata ū H 1. This leaves open the possibility that other constructive proceures yiel ifferent issipative solutions. We believe that this is not the case, an that the inequality. uniquely characterizes

3 our semigroup issipative solutions. A uniqueness proof might be achieve by constructing a suitable istance functional relate to optimal transportation, as in [BF] an [BC1]. However, this is outsie the scope of the present paper. 1 - The basic equations We consier the following form of the Camassa-Holm equation where the source term P is efine as a convolution: P =. 1 e x u + u x. The initial ata is specifie as u t + u / x + P x =, 1.1 u, x = ūx, x, 1. with ū H 1, the space of absolutely continuous functions u L with erivative u x L. This space is naturally enowe with the norm. [ u H 1 = u x + u xx ] 1/. x For u H 1, Young s inequality implies P = 1 x 1 u + 1 u x H 1. Notice that the assumption ut, H 1 implies that the convolution in 1. is well efine. Definition 1. By a solution of the Cauchy problem on a time interval [, T ] we mean a Höler continuous function u = ut, x efine on [, T ] such that ut, H 1 at every t [, T ], an the map t ut, is Lipschitz continuous from [, T ] into L an satisfies the initial conition 1. together with the following equality between functions in L : t u = uu x P x for a.e. t [, T ]. 1.3 For smooth solutions, applying the operator 1 x to both sies of 1.3, we recover the original form.1 of the Camassa-Holm equation. Differentiating 1.1 w.r.t. x one obtains u xt + uu xx + u x u + u x + P =. 1.4 Multiplying 1.1 by u an 1.4 by u x, for smooth solutions we obtain the two conservation laws with source term u u u P = u x P, 1.5 t 3 x

4 u x uu + x t u3 = u x P x In particular, for smooth solutions the energy satisfies the conservation law u + u x uu + x + u P =. Hence the total energy Et =. t x u t, x + u xt, x x 1.7 is constant in time. As shown in [CE1, CE], there exist smooth initial ata ū with compact support such that the corresponing solution of.1-. evelops singularities at a finite time T >. For such solutions the energy 1.7 remains constant on [, T [, while } lim inf u xt, x =. x t T In [BC] we showe that the solution can be continue after the breaking time by requiring that the energy remains constant for a.e. t. The present paper is concerne with global global solutions, where wave breaking might inuce a partial or even total loss of energy. Definition. A solution of the Cauchy problem is sai to be issipative if it satisfies the inequality. for some constant C, an moreover its energy Et in 1.7 is a non-increasing function of time. - An associate semilinear system Given an initial ata ū H 1, we begin by introucing an energy variable ξ. This will play the role of a Lagrangian variable, remaining constant along characteristics. Define the increasing map ξ ȳξ implicitly by the relation ȳξ 1 + ū x x = ξ..1 We now write a semilinear system of equations in terms of the inepenent variables t, ξ. This was first use in [BC] to construct global conservative solutions to the Camassa-Holm equation. An appropriate moification of this system will here yiel global issipative solutions. Assuming for the time being that the solution u remains Lipschitz continuous, enote by t yt, ξ the characteristic curve starting at ȳξ, so that Throughout the following, we use the notation t yt, ξ = u t, yt, ξ, y, ξ = ȳξ.. ut, ξ. = u t, yt, ξ, P t, ξ. = P t, yt, ξ, an efine the variables v = vt, ξ an q = qt, ξ as v. = arctan u x, q. = 1 + u x y ξ,.3 4

5 with u x = u x t, yt, ξ. Since v is efine up to multiples of π, all subsequent equations involving v will be invariant uner aition of multiples of π. Notice that.1 yiels Moreover one has the useful ientities q, ξ u x = cos v, u x 1 + u x = 1 sin v, u x 1 + u x = sin v,.5 an From.6 it follows Furthermore, we have y ξ = yt, ξ yt, ξ = P t, ξ = P t, yt, ξ = 1 q 1 + u x 1 P x t, ξ = P x t, yt, ξ = yt,ξ ξ ξ = cos v q..6 vt, s cos qt, s s..7 e yt,ξ x u t, x + 1 u xt, x x, yt,ξ e yt,ξ x u t, x + 1 u xt, x x. In the above formulas, performing the change of variables x = yt, ξ, we can write the convolution as an integral over the variable ξ. Using the ientities.5.7, we obtain P ξ = 1 exp ξ ξ cos vs an P x ξ = 1 ξ exp ξ ξ ξ } qs s cos vs [ u ξ cos vξ + 1 ] sin vξ qξ ξ,.8 } [ qs s u ξ cos vξ + 1 ] sin vξ qξ ξ..9 By 1.1 an., the evolution equation for u in the new variables t, ξ takes the form t ut, ξ = u t + uu x = P x t, ξ.1 with P x given at.9. On the other han, from. an 1.6 we obtain ξ qt, ξ ξ = t ξ 1 t = yt,ξ yt,ξ 1 yt,ξ yt,ξ u xt, x x 1 + u x t + [ u1 + u x ] x 5 } x = yt,ξ yt,ξ 1 u + 1 P u x x.

6 Differentiation w.r.t. ξ yiels t qt, ξ = u x u + 1 P 1 + u x q = u + 1 P sin v q..11 Moreover, using.-.3 an 1.4, we obtain [ ] vt, ξ = t 1 + u u x x + u P = u P cos v v sin..1 In.11 an in.1, the function P = P t, ξ is compute by.8. To obtain global conservative solutions, one procees as follows. Given the initial ata ū H 1, consier the corresponing Cauchy problems.1-.1 for the variables u, v, q in the form of the semilinear system u t = P x v t = u P 1 + cos v sin v q t = u + 1 P sin v q.13 with u, ξ = ū ȳξ ȳξ v, ξ = arctan ū x q, ξ = 1. The system.13 can be regare as an O.D.E. in the Banach space.14 enowe with the norm X. = H 1 [ L L ] L,.15 u, v, q X. = u H 1 + v L + v L + q L, Observing that the right han sie is locally Lipschitz, one obtains the local existence an uniqueness of solutions by general O.D.E. theory. The local solution can be extene to a global solution since blowup is prevente by the conservation of the energy Et, expresse in the t, ξ-variables as Et = u vt, ξ t, ξ cos + sin vt, ξ qt, ξ ξ = E, t. Going back to the original variables ut, x, in [BC] the authors constructe a semigroup of global solutions to the Camassa-Holm equation. The solutions constructe via this approach are conservative, in the sense that their total energy, measure by the H 1 norm of ut,, is equal to a constant for a.e. time t. In orer to obtain global issipative solutions, a moification of the system.13 is neee. In essence, we require the following. Assume that, along a given characteristic t yt, ξ, the wave breaks at a first time t = τξ. Recalling our rescale variable v = arctan u x, this of course means u x t, ξ an v = arctan u x, as t τξ. For all t τξ we then set vt, ξ π 6

7 an remove the values of ut, ξ, vt, ξ, qt, ξ from the computation of P, P x. More precisely, in the efinition of P, P x we now replace.8-.9 by P ξ = 1 } exp vξ > π } cos vs s [ξ,ξ ], vs> π [ u ξ cos vξ + 1 sin vξ } qs s ] qξ ξ,.16 P x ξ = 1 } } exp ξ >ξ, vξ > π ξ <ξ, vξ > π [ u ξ cos vξ + 1 ] sin vξ The system of O.D.E s.13 is now replace by u t = P x, s [ξ,ξ ], vs> π qξ ξ. } } cos vs qs s.17 v u t = P 1 + cos v sin v if v > π, if v π,.18 q u t = + 1 P sin v q if v > π, if v π. Notice that, in the issipative case, the Camassa-Holm equation is reuce to an O.D.E. in a Banach space, where the right han sie is now iscontinuous. The iscontinuity occurs precisely when v = π. We observe that, by the secon equation in.18, v approaches the value π transversally, i.e. with strictly negative erivative v t 1. It is precisely this transversality conition that guarantees the well-poseness of the system Local solutions of the semilinear system The Cauchy problem for.18 can be written in more compact form as t Ut, ξ = F Ut, ξ + G ξ, Ut, ξ, 3.1 U, ξ = Uξ. 3. Here U = u, v, q 3, while F U =, u 1 + cos v sin v, u + 1 sin v q if v > π,, 1, if v π, 3.3 7

8 G ξ, U = P x, 1 + cos v P, sin v q P if v > π, P x,, if v π. The nonlocal operators P, P x are efine at Notice the slight iscrepancy between the systems an.18. To achieve exactly the same evolution, one shoul efine F =,, whenever v π. The moification in 3.3 has the avantage of renering the fiel F continuous across the value v = π. Moreover, as soon as a solution u, v, p of is obtaine, the mapping t, ξ ut, ξ, max vt, ξ, π }, qt, ξ provies a solution of.18. We regar 3.1 as an O.D.E. on the space L ; 3. Notice that the vector fiel F : 3 3 in 3.3 is uniformly boune an Lipschitz continuous as long as u remains in a boune set. However, the nonlocal operator G is iscontinuous. Inee, the integral terms P, P x efine at.16,.17 are iscontinuous, because the set ξ ; τξ > t } = ξ ; vt, ξ > π } 3.6 may suenly shrink, at a time τ such that meas ξ ; vτ, ξ = π } >. Aim of this section is to prove: Theorem 1. Let any ū H 1 be given. In connection with the corresponing initial ata U =. ū, arctan ū x, 1 efine at.14, the Cauchy problem has a unique local solution, efine on some time interval [, T ]. Proof. The local an even global existence an uniqueness of the solution can be obtaine by applying the general theorem on irectionally continuous O.D.E s in functional spaces recently prove in [BS]. We shall give here a self-containe proof, in several steps, still following [BS] in the main lines. 1. We begin with some a priori estimates on F an G in 3.1. Assume that U = u, v, q L ; 3 satisfies the inequalities u L C, 1 C meas ξ ; vξ > π, qξ C for all ξ, 3.7 vξ π } C. 3.8 for some constant C. Then there is a constant κ epening only on C such that P L + P x L κ, P L 1 + P x L 1 κ 1 + u L 1 + v L 1, 3.9 F U L κ, GU L κ. 3.1 Moreover, there exists a Lipschitz constant κ such that, if Ũ = ũ, ṽ, q satisfies the same bouns , then F U F Ũ L κ U Ũ L,

9 [ ξ GU GŨ L κ U Ũ } L + meas ; vξ > π, ṽξ π + meas ξ ; ṽξ > π, vξ π } ] Let ū H 1 be given. Since ū, ū x L, for every ε > the set x ; ūx x } ε has finite measure. We can thus fin a constant C > such that ū L C ξ, meas ; vξ π } C Let the inequalities hol. By Step 1, we can choose δ > small enough so that vξ ] π, δ π] implies Consier the sets t vt, ξ = u P 1 + cos v sin v Ω δ. = ξ ; vξ ] π, δ π] }, Ω. = \ Ω δ. By possibly reucing the size of δ >, we can assume that meas Ω δ 1 8κ The solution t Ut will be obtaine as the unique fixe point of a contractive transformation P : D D, on a suitable omain D C [, T ], L. More precisely, for a given T >, we efine the omain D as the set of all continuous mappings t Ut = ut, vt, qt from [, T ] into L, 3 with the following properties. vt, ξ vs, ξ t s The Picar operator P is efine as U = U, 3.16 Ut Us L κ t s, 3.17 PU t, ξ. = U + t ξ Ω δ, s < t T [ F Uτ, ξ + G ξ, Uτ, ] τ Choosing T > sufficiently small, it is clear that P maps the omain D into itself. We now show that P is a strict contraction. Inee, assume U, Ũ D an efine η =. max Ut Ũt t [,T ] L

10 Define the crossing time τξ. = sup t [, T ] ; vt, ξ > π }. Define τξ in the same way, replacing v by ṽ. Observe that, for each ξ Ω δ, the property 3.14 implies τξ τξ η. 3. For t [, T ] we then have PUt P Ũt L κ t t Uτ Ũτ L τ + κ t F Uτ F Ũτ L τ + t ξ } + κ meas ; ṽτ, ξ > π, vτ, ξ π τ κt η + κ τξ τξ ξ Ω δ κt η + κ measω δ η η, t GUτ G Ũτ L τ meas ξ ; vτ, ξ > π, ṽτ, ξ π } τ provie that T is small enough. This proves that P is a strict contraction, hence it has a unique fixe point, which yiels the esire local solution of the Cauchy problem Global solutions In this section we show that the local solutions of the semilinear system.18 can be globally extene for all times t. The basic ingreient is a global boun on the total energy: Et = u vt, ξ vt, ξ t, ξ cos + sin qt, ξ ξ 4.1 vt,ξ> π} We begin by showing that, as long as the local solution of.18 is efine, u ξ = q sin v. 4. Inee, the first equation in.18 an the efinition of P x at.17 imply u ξt = ξ P q u x = cos v + 1 sin v P cos v if vt, ξ > π, if vt, ξ > π. On the other han, from the last two equations in.18, when v > π we obtain q sin v = q t t sin v + q v t cos v = q u + 1 P sin v + u P cos v + u P cos v cos v sin v = q u cos v + 1 v sin P v cos,

11 while, if v = π, the same equations in.18 yiel q sin v Next, at the initial time t =, by.14 an.5-.6 we have t =. 4.5 u ξ = ū x ȳ ξ = ū x ū x = sin v = q sin v, 4.6 because q 1. By 4.6, the ientity 4. hols at time t =. By , for every ξ we have u ξ q sin v = t Hence the ientity 4. hols for all times t, as long as the solution is efine. Next, we claim that the extene energy Ẽt =. u cos v + v sin q ξ 4.7 remains constant in time. We remark that the integral in 4.7 in general is strictly larger than the energy Et in 4.1, because here the integration ranges over the entire real line. From.18 we euce that t u cos v + v sin = = vξ> π} q ξ u cos v + v sin u + 1 P sin v cos v u P x cos v + sin v cos v [ 1 u u P cos v v ] } sin q ξ P sin v cos v u P x cos v + 3 u sin v cos v } q ξ. Notice that on the right han sie of 4.8 we are again integrating over the entire real line. Of course, this oes not make a ifference because cos v = whenever v = π. On the other han, from we infer 4.8 P ξ = q P x cos v. 4.9 Notice that the case v = π a separate computation yiels P ξ =. This is still consistent with 4.9, because cos π/ =. Together, 4. an 4.9 yiel u P ξ = u ξ P + u P ξ = P sin v cos v + u P x cos v } q. In aition, we observe that 3 q u sin v cos v = 3 u u ξ = u 3 ξ. 11

12 Using the above ientities, we obtain u cos v t + v } sin q ξ = u 3 u P ξ =. 4.1 ξ The last equality is justifie because lim ξ uξ =, while P is uniformly boune. This proves our claim, namely Ẽt = u vt, ξ t, ξ cos + sin vt, ξ qt, ξ ξ 4.11 = Ẽ. = E along any solution of.18. As long as the solution is efine, using 4. an 4.11 we obtain the boun sup u t, ξ uu ξ ξ u ξ sin v cos v q ξ E. 4.1 This provies a uniform a priori boun on ut L. Recalling the efinitions , from the estimate 4.11 we recover the uniform bouns P t L C, Px t L C, 4.13 where the constant C epens only on the total energy E. Looking at the thir equation in.18, by we euce that, as long as the solution is efine, q t E C q. Since q, ξ 1, the previous ifferential inequality yiels exp E + 1 } + C t By the secon equation in.18 it is clear that qt exp E + 1 } + C t π vt, ξ < π Finally, the lower boun on q at 4.14 together with the energy estimate 4.11 imply that, for every η >, there exists a constant C η epening only on C, E an T such that meas ξ ; vt, ξ } η C η t [, T ] Accoring to 4.1, 4.13 an 4.16, the a priori bouns which we neee to construct a local solution are satisfie with a constant C uniformly vali over any given time interval [, T ]. Therefore, the local solution can be globally extene for all times t. 5 - Continuous epenence 1

13 The local existence theorem prove in Section 3 was obtaine by representing the solution of as the fixe point of a contraction in a suitable L space. This yiele uniqueness an continuous epenence w.r.t. to convergence on the initial ata in L. Our eventual goal is to show the continuous epenence of solutions to the Camassa-Holm equation 1.1, as the initial ata converge in the H 1 norm. This requires further estimates. Inee, if ū n ū in H 1, setting v = arctan u x one has u n ū L, v n v L. 5.1 However, v n v L oes not converge to zero, in general. In this section we prove a result on continuous epenence of solutions of.18, base on the weaker assumptions 5.1. Theorem. Consier a sequence of initial ata ū n converging to ū in H 1. Let u n, u be the corresponing solutions of.18 with initial ata.14. Then, for any T >, the convergence hols uniformly for t, ξ [, T ]. u n t, ξ ut, ξ 5. Proof. 1. Let u, v, q an ũ, ṽ, q be any two solutions of.18, corresponing to initial ata of the form.14. Call E an upper boun for the energies of the two solutions. Assuming that at time t = u ũ L δ, v, ξ ṽ, ξ L δ, 5.3 we shall establish an a-priori boun on epening only on δ, T an E.. Define the set ut ũt L for t [, T ], 5.4 Λ. = ξ ; vt, ξ = π } ξ ; ṽt, ξ = π }. Notice that α. = measλ 5.5 is a number uniformly boune by a constant epening only on T an E. 3. For each ξ Λ, let τξ be the first time at which one of the two solutions reaches the value π, namely τξ =. inf t [, T ] ; min vt, ξ, ṽt, ξ } } = π. 5.6 Since the map ξ τξ is measurable, we can construct a measure-preserving, measurable map α ξα from the interval [, α ] onto the set Λ, having the aitional property α α if an only if τ ξα τ ξα. 5.7 The inverse mapping: Λ [, α ], which is of course still measure-preserving, will be enote as ξ αξ. 13

14 4. In connection with the map [, α ] Λ consiere at 5.7, we now efine the istance functional J u, v, q, ũ, ṽ, q. = u ũ L + v ṽ L + q q L For notational convenience, set α + K. = J + K J #. e Kα v ξα ṽ ξα α 5.8 Jt. = J ut, vt, qt, ũt, ṽt, qt = J t + K J # t. 5.9 We claim that, for some constants K, K, M large enough epening only on T an E, one has Jt M Jt. 5.1 t This will of course imply Jt e Mt J t [, T ] 5.11 thus proviing an a-priori estimate on the istance at To complete the proof, we nee to show that 5.1 hols, for suitable constants K, K, M epening only on T an E. For each fixe t [, T ], efine the sets Γt Γ + t Γ t. =. =. = } ξ Λ : vt, ξ > π, ṽt, ξ = π } ξ Λ : vt, ξ = ṽt, ξ = π, } ξ Λ : vt, ξ > π, ṽt, ξ > π = Notice that the above three sets are isjoint, an } ξ Λ : vt, ξ = π, ṽt, ξ > π, ξ Λ : } τξ > t. 5.1 Γt Γ + t Γ t = Λ for each t [, T ] Moreover, setting the property 5.7 implies Γ t = mt. = meas Γ t, ξα ; α [, mt ]} As in the proof of Theorem 1, from the equations.18 we erive the estimate ut ũt t L + vt ṽt L + qt qt L κ ut ũt L + vt ṽt L + qt qt L + meas Γt

15 7. Next, using 5.14 we compute t α = e Kα v t, ξα ṽ t, ξα α = Γt e Kαξ vt, ξ ṽ t, ξ t ξ + Γt Γ t Γ + t mt e Kα t e Kαξ vt, ξ ṽ t, ξ ξ t v t, ξα ṽ t, ξα α Inee, the integral over Γ + t is zero. As in 3.14, we can now choose δ > small enough epening only on T, E such that, for ξ Γt, vt, ξ ṽt, ξ δ implies vt, ξ ṽt, ξ 1 t On the other han, choosing a constant κ sufficiently large we easily obtain vt, ξ ṽt, ξ δ implies vt, ξ ṽt, ξ 1 t + κ vt, ξ ṽt, ξ Finally, for ξ Γ t, we have the estimate vt, ξ ṽt, ξ κ ut ũt L + vt ṽt L + qt qt L t so that mt + meas Γt + vt, ξ ṽt, ξ e Kα v t, ξα ṽ t, ξα α κ J t + meas Γt mt t κ J t + meas Γt mt 8. We rewrite 5.15 in the form Moreover, observing that from we euce t J # t 1 Γt + κ t J t κ + κ mt e Kα α e Kα vt, ξα ṽt, ξα α e Kα α + κ e Kαξ vt, ξ ṽt, ξ ξ Γ t J t + meas Γt. 5. ξ Γt implies αξ mt, e Kαξ ξ + κ Γt Γ t mt J t + meas Γt 1 ekmt meas Γt + κ J # t + κj t e Kαξ vt, ξ ṽt, ξ ξ e Kα α α + κ meas Γt mt e Kmt e Kα mt α 1 4 ekmt meas Γt + κ J # t + κ K ekα J t, 15 e Kα α 5.1

16 provie that we choose the constant K. = 4κ, so that κ mt e K α mt α κ K = In the efinition of the istance functional at we now choose K. = 4κ. Combining 5. an 5.1 we obtain J t + 4κ J # t κ J t + meas Γt + 4κ 1 t 4 meas Γt + κ J # t + κ K ekα J t κ J t + 4κ J # t + κe 4κα J t. This proves the ifferential inequality 5.1, with M = κ + κ e 4κα. 6 - Global issipative solutions to the Camassa-Holm equation Reverting to the original variables t, x, we now show that the global solution of the system.18 yiels a global issipative solution to the Camassa-Holm equation 1.1. Given any ū H 1, let u, v, q be the corresponing solution of.18 with initial ata.14. Define yt, ξ. = ȳξ + t uτ, ξ τ. 6.1 For each fixe ξ, the function t yt, ξ thus provies a solution to the Cauchy problem yt, ξ = ut, ξ, y, ξ = ȳξ. 6. t We claim that a solution of 1.1 can be obtaine by setting ut, x. = ut, ξ if yt, ξ = x. 6.3 Theorem 3. Let u, v, q provie a global solution to the Cauchy problem.18,.14. Then the function u = ut, x efine by 6.1, 6.3 provies a solution to the initial value problem for the Camassa-Holm equation. Proof. The argument given here relies on similar techniques as in [BC]. 1. By 4.1 we have the uniform boun ut, ξ E 1/. Hence 6.1 implies ȳξ E 1/ t yt, ξ ȳξ + E 1/ t. For each t, this yiels lim ȳt, ξ = ±. ξ ± Therefore, the image of the continuous map t, ξ t, yt, ξ covers the entire omain [, [. 16

17 . For reaer s convenience, we collect here the basic relations between the t, x an the t, ξ variables. u qt, ξ y vt, ξ t, ξ = sin vt, ξ, t, ξ = qt, ξ cos, 6.4 ξ ξ u x t, x = sin vt, ξ 1 + cos vt, ξ if x = yt, ξ, cos vt, ξ Notice that 6.4 is consistent with the fact that u ξ = y ξ = when v = π. By the secon equality in 6.4, for each fixe t the map ξ yt, ξ is non-ecreasing. Moreover, if ξ < ξ but yt, ξ = yt, ξ, then ξ ξ y ξ t, s s = ξ ξ vt, s qt, s cos s =. Hence cosv/ throughout the interval of integration. The first equality in 6.4 yiels ut, ξ ut, ξ = ξ ξ qt, s sin vt, s s =. Therefore the the map t, x ut, x at 6.3 is well efine, for all t, x [, [. 3. We now observe that, for every fixe t, the image of the singular set where v = π has measure zero in the x-variable. Inee yt, } meas ξ ; vt, ξ = π = y ξ t, ξ ξ = vt,ξ= π} vt,ξ= π} vt, ξ qt, ξ cos ξ =. 6.6 Next, using 6.5 to change the variable of integration, we compute u t, x + u xt, x x = u vt, ξ vt, ξ t, ξ cos + sin qt, ξ ξ E, vt,x > π} 6.7 because of By a Sobolev inequality [EG], this implies the uniform Höler continuity with exponent 1/ of u as a function of x. By the first equation in.18 an the uniform boun on P x L, it follows that the map t u t, yt, ξ is uniformly Lipschitz continuous along every characteristic curve t yt, ξ. Therefore, u = ut, x is globally Höler continuous on the entire t-x plane. 4. We now prove that the map t ut is Lipschitz continuous with values in L. Inee, consier any interval [τ, τ + h]. For a given point x, choose ξ such that the characteristic t yt, ξ passes through the point τ, x. By 3.1 an the boun 3.7 it follows uτ + h, x uτ, x uτ + h, x u τ + h, yτ + h, ξ + u τ + h, yτ + h, ξ uτ, x τ+h sup ut + h, y uτ + h, x x + y x E 1/ h τ 17 Px t, ξ t.

18 Integrating over the whole real line we obtain uτ + h, x uτ, x x = 4 E 1/ h x+e 1/ h x E 1/ h E 1/ h u x τ + h, y y x + x+e 1/ h x E 1/ h y+e 1/ h y E 1/ h 8 E h u x τ + h L + h qτ L C h τ+h ux τ + h, y y x + τ P x t, ξ vτ, ξ t qτ, ξ cos h ux τ + h, y x y + h qτ L τ+h τ P x t t L τ+h τ ξ Px t, ξ t qτ L ξ τ+h τ Px t, ξ t ξ for some constant C uniformly vali as t ranges on boune set, in view of 3.14, 3.9 an 4.8. This clearly implies the Lipschitz continuity of the map t ut, in terms of the x-variable. 5. Since L is a reflexive space, in view of the infinite-imensional version of Raemacher s theorem [A] the map t ut is ifferentiable for a.e. t. Since the right han sie of 1.3 clearly lies in L, to establish the equality it suffices to prove the following. For every smooth function with compact support φ Cc, at almost every time t one has t ut, x φx x = ut, xu x t, x P x t, x φx x Towar this goal, for each ξ efine Observe that, for almost every time t one has = u t, x φ x P x t, x φx + ut, xu x t, x φx x. 6.8 τξ. = inf t > ; vt = π }. 6.9 meas ξ ; τξ = t } =. 6.1 Choose a time t such that 6.1 hols. Integrating w.r.t. the variable ξ an recalling 6.4 an.5 we obtain ut, ξ φ yt, ξ [ vt, ξ ] qt, ξ cos ξ t = u t φq cos v + uφ y t q cos v + uφq t cos v } uφqv t sin v ξ = P x φq cos v + u φ q cos v u + uφ + 1 P sin vq cos v vt,ξ> π} uφq u P cos v v } sin v sin ξ = P x φq cos v + u φ q cos v + 1 } uφ sin v q ξ = vt,ξ> π} vt,ξ> π} P x φ + u φ q + uu x φ q cos v ξ. 18

19 This establishes 6.8, thus proving that u is a solution of the Camassa-Holm equation in the sense of Definition A semigroup of issipative solutions Given an initial ata ū H 1, we enote by ut = S t ū the corresponing global solution of the Camassa-Holm equation constructe in Theorem 3. Aim of this final section is to prove: Theorem 4. The map S : H 1 [, [ H 1 is a semigroup. In aition, the following properties hol. i The total energy is a non-increasing function of time, namely ut H1 ut H1 whenever t t. 7.1 ii For a constant C, epening only on the total energy ū H 1, one has u x t, x C 1 + t 1 t >, x. 7. iii Given a sequence of initial ata ū n such that ū n ū H1, the corresponing solutions u n = u n t, x converge to ut, x uniformly for t, x in boune sets. Proof. 1. To prove 7.1, for each ξ efine τξ as in 6.9. Recalling 6.7 we compute ut H 1 = vt,ξ> π} = E = E E τξ t} = ut H 1. u vt, ξ vt, ξ t, ξ cos + sin qt, ξ ξ u vt, ξ vt, ξ t, ξ cos + sin qt, ξ ξ sin v τξ, ξ q τξ, ξ ξ sin v τξ, ξ q τξ, ξ ξ τξ t } t <τξ t} τξ t } 7.3. We now prove the Oleinik type estimate 7.. As in the proof of Theorem 1, we can choose δ > so that v [π δ, π[ implies t vt, ξ 1. Therefore vt, ξ < min By 6.5, this yiels 7., for a suitable constant C. π δ, π t }. 19

20 3. Next, let ū n be a sequence of initial ata converging to ū in H 1. By Theorem this this implies the convergence u n t, ξ ut, ξ uniformly for t, ξ in compact sets. Returning to the original t-x coorinates, we obtain the convergence y n t, ξ yt, ξ, ut, x u n t, x, uniformly on boune sets, because all functions u, u n are uniformly Höler continuous. 4. To complete the proof, it remains to establish the semigroup property. Fix ū H 1 an τ >. For all t > we nee to show that S t Sτ ū = S τ+t ū. 7.4 Let t, ξ u, v, q t, ξ be the corresponing solution of.18 with initial ata.14. Call û =. S τ ū. To construct the trajectory t S t u, we consier a new energy variable σ an efine the map ξ σξ as a solution to the O.D.E. with initial ata ξ σξ = qτ, ξ if vτ, ξ > π, if vτ, ξ = π, Here the value ξ is chosen so that yτ, ξ =. We then efine û t, σ = uτ + t, ξσ, ˆv t, σ = vτ + t, ξσ, ˆq t, σ = 7.5 σξ =. 7.6 qτ + t, ξσ q τ, ξσ, where σ ξσ provies an a.e. inverse to the map in , say ξσ. = sup s ; σs σ }. 5. For every σ, we claim that the previous efinitions imply yτ, ξ 1 + u xτ, x x = σξ. 7.8 Inee, by 7.6 this is true when ξ = ξ, σ =. Moreover, recalling 6.4 an.5 one obtains ξ yτ, ξ 1 + u x τ, yτ, σξ = qτ, ξ = σξ. 7.9 ξ By an integration, in view of 7.5 this establishes To establish the semigroup property, it now suffices to show that the functions 7.7 provie a solution to the system.18. Towar this goal, we write the ientities qτ + t, ξ ξ = ˆqt, σξ qτ, σξ σξ σ = ˆqt, σξ σ. ξ These imply that the corresponing integral source terms in.18 satisfy 7.7 P t, σ = P τ + t, ξσ, Px t, σ = P x τ + t, ξσ. 7.1 The conclusion now follows from 7.7 an 7.1, because the thir equation in.18 is linear w.r.t. the variable q.

21 We remark that the semigroup S generate by the Camassa-Holm equation is NOT continuous as a map with values in H 1. In fact, even a single trajectory t ut, may fail to be continuous as a map from [, [ into H 1. This happens, for example, when a peakon an an antipeakon of the same strength collie an completely annihilate each other. Example Peakon-antipeakon interaction. The Camassa-Holm equation amits for every spee c the solitary wave solution [CH] u c x, t = c e x ct, calle peakon. Notice that the amplitue of a peakon equals to its spee. The peakon-antipeakon interaction is expresse by the following Ansatz [CH]: ut, x = p 1 t e x q 1t + p t e x q t, x, t, 7.11 with p 1, q 1, p, q Lipschitz continuous such that p 1 >, q 1 <, an Letting p 1 t = p t, q 1 t = q t. 7.1 pt = p 1 t, qt = q 1 t, 7.13 we see that 7.11 is a solution in the sense of Definition 1 if the variables qt an pt satisfy the following Hamiltonian system of orinary ifferential equations with iscontinuous right-han sie p = p eq, 7.14 q = p 1 e q, with Hamiltonian H = p 1 e q interprete in the sense of Carathéoory see [BC]. For some maximal time T > the breaking time we will have pt > an qt < on [, T, while More precisely, with we have see [BC] T = 1 H ln p + H p H, lim pt = an lim qt =. t T t T H = p 1 e q, 7.16 [p + H ] + [p H ] e H t pt = H [p + H ] [p H ] e H, t [, T, t qt = q ln [p + H ] e Ht/ + [p H ] e H t/, t [, T. p 7.17 Therefore see [BC] lim ut, x = for every x, 7.18 t T 1

22 while the total amount of energy concentrate in the interval between the two extremes crest an trough of the wave 7.11 equals q t q 1 t u t, x + u xt, x x = p t 1 e qt, t [, T, with But lim t T ut, H 1 q t q 1 t u t, x + u xt, x x = H. = p t 1 e qt = H, t [, T, so that, as t approaches the breaking time T, an increasingly large portion of the energy is concentrate within the interval [q 1 t, q t]. In the limit t T all the energy becomes concentrate at the single point x =. In [BC] the solution was continue past the breaking time as a conservative solution by setting, ut + ε, x = ut ε, x, x, for ε >. That is, in the conservative scenario the peakon an the antipeakon pass trough each other at breaking time an no energy is lost in the process. We now show that within the issipative framework total annihilation occurs at breaking time, namely ut, x = for t > T, x Inee, using in combination with., for t [, T an ξ, we have t yt, ξ = ln e ξ + ps sinh qs s if ξ > q, yt, ξ = ln eξ e ξ 1 e t e ξ + 1 e ξ 1 e t pse qs s pse qs s if < ξ < q, 7. while yt, = an yt, q = qt Using.3, we infer that for t [, T, for t [, T [. while vt, ξ = arctan Since 7.17 ensures lim t T t vt, ξ = arctan ps sinh qs pt e qt s = 1 e pt sinh qt e ξ + t qs ps sinh s if ξ q, e ξ e ξ 1 e 4 t e ξ + 1 e ξ 1 e 4 t qs ps e s qs ps e s if < ξ < q. q, lim pt e qt =, lim pt sinh qt t T t T =,

23 from the previous relations an their analogues for ξ < we euce that while ut, ξ, u x t, ξ, vt, ξ as t T if ξ ut, = on [, T an lim t T vt, = π. Therefore the solution U = u, v, q of.18 at t = T equals,, 1 an by Theorem 1 we infer that Ut =,, 1 for all t T. That is, for the peakon-antipeakon interaction in the issipative case we have ut, for t T. Acknowlegements. The work of the first author was supporte by the N.S.F., Grant DMS Applie Mathematics: Hyperbolic systems of conservation laws. The secon author gratefully acknowleges the support of the G. Gustafsson Founation for Research in Natural Sciences an Meicine. References [A] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Stuia Math , [BSS1] R. Beals, D. Sattinger an J. Szmigielski, Multi-peakons an a theorem of Stieltjes, Inverse Problems , L1-L4. [BSS] R. Beals, D. Sattinger an J. Szmigielski, Multipeakons an the classical moment problem, Av. Math. 154, [BSS3] R. Beals, D. Sattinger an J. Szmigielski, Peakon-antipeakon interaction, J. Nonlinear Math. Phys. 8 1, 3-7. [B] A. Bressan, Unique solutions for a class of iscontinuous ifferential equations, Proc. Amer. Math. Soc , [BC1] A. Bressan an A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 5, [BC] A. Bressan an A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., to appear. [BF] A. Bressan an M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation, Methos Appl. Anal., to appear. [BS] A. Bressan an W. Shen, Unique solutions of irectionally continuous O.D.E s in Banach spaces, Analysis an Applications, to appear. [BZZ] A. Bressan, P. Zhang an Y. Zheng, On asymptotic variational wave equations, Arch. Rat. Mech. Anal. 6, to appear. [BZ] A. Bressan, an Y. Zheng, Conservative solutions to a nonlinear variational wave equation Comm. Math. Phys. 6, to appear. 3

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