Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution \begin{document}$u = u(t,x)$ \end{document} , an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities \begin{document}$u$ \end{document} and \begin{document}$v = 2 \arctan u_x$ \end{document} along each characteristic, it is obtained that the Cauchy problem with general initial data \begin{document}$u_0∈ H^1(\mathbb{R})$ \end{document} has a unique global conservative solution.


(Communicated by Adrian Constantin)
Abstract. It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution u = u(t, x), an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities u and v = 2 arctan ux along each characteristic, it is obtained that the Cauchy problem with general initial data u 0 ∈ H 1 (R) has a unique global conservative solution.

Introduction.
In this paper, we consider the generalized Camassa-Holm equation, where g : R → R is a given W 1,∞ loc (R)-function with g(0) = 0, and γ is a constant. Peakons in a particular case of equation (1) are studied in [26] and their stability is discussed in [24]. Yin established the well-posedness, global solutions for (1) in [33]. Later, Coclite, Holden and Karlsen [6] established existence of a strongly continuous semigroup of global weak solutions for equation (1), and also presented a "weak equals strong" uniqueness result. Moreover, wave breaking is the only way singularities can develop in finite time for (1) (cf. [34]), beyond the occurrence of wave breaking which exists global conservative solutions [37].
The early motivation to investigate the equation (1) is that it can be regarded as a generalization of the well-known Camassa-Holm equation (equation (1) with γ = 1, g(u) = 3 2 u 2 + ku) The equation (2) was first implicitly included in a bi-Hamiltonian generalization of KdV equation by Fokas and Fuchssteiner [20], and later derived as a model for unidirectional propagation of shallow water over a flat bottom by Camassa and Holm [4]. Analogous to the famous KdV equation, the Camassa-Holm equation (2) also has a bi-Hamiltonian structure [4,20], which is completely integrable not only in the sense of the existence of a Lax pair [4], but also (by means of inverse scattering and inverse spectral theory) as an infinite-dimensional Hamiltonian flow that can be linearised in suitable action-angle variables, cf. the considerations in the papers [7,8,12,13,18]. The orbital stability of solitary waves and the stability of the peakons (k = 0) for the Camassa-Holm equation are investigated by Constantin and Strauss [14,15]. The advantage of the CH equation in comparison with the famous KdV equation lies in the fact that the Camassa-Holm models the peculiar wave breaking phenomena, that is, the solution remains bounded but its slope becomes unbounded in finite time (cf. [5,11]). In addition to wave breaking, one of the most interesting aspects of the equation is the existence of peakon solutions, namely, the travelling wave solutions of greatest height of the governing equations for water waves have a peak at their crest (cf. [9,10,36]). Over the last three decades, there is a wide variety of patterns associated Camassa-Holm equation have been studied extensively. Such as, the hyperelasticrod wave equation (equation (1) with g(u) = 3 2 u 2 and γ ∈ R) The model (3) describes far-field, finite length, finite amplitude radial deformation waves in cylindrical compressible hyperelastic rods. The other special cases when g(u) = 3 2 u 2 + ku, γ ∈ R, can be found in [16,35]. The travelling waves solution, peaked solitary wave solutions for equation (1) with g(u) = au m + ku and some α, m, γ were showed in [28,27] and references therein. More related work in this fields can be found in [23,21,19,22,29,30,31,32,25].
In view of the possible development of singularities in finite time, it is natural to curious about the behavior of a solution beyond the occurrence of wave breaking. Bressan and Constantin showed that after wave breaking the solution of the solution of the Camassa-Holm equation can be continued uniquely as either global conservative or global dissipative solutions(cf. [1,2]). Recently, the uniqueness of conservation solution to the Camassa-Holm equation was direct proved in [3]. They develop a direct method to the uniqueness of conservative solutions for the Camassa-Holm equation by using characteristics, which extended the work of generalized characteristics [17], or so-called energy variable to the Camassa-Holm equation to separate different characteristics after singularity occurs. We known that the conservative solutions for equation (1) can be global existence (cf. [37]). The aim of this paper is to further prove that, the uniqueness of the global conservative weak solutions of equation (1) by suitable modifying recent result in [3] for the Camassa-Holm equation.
This paper is organized as follows. In Section 2, we recall basic notation and present our main theorem. In Section 3, the uniqueness of characteristic curve through each initial point is established. In Section 4, we investigate the gradient u x of a conservative solution varies a characteristic and obtain the proof of the main result.

Basic definitions and results. Let
and G * m = u. The system (1) can be rewritten as the following form with initial data u 0 (x) ∈ H 1 (R). For convenience, we set If u ∈ H 1 (R), Young's inequality ensure that Indeed, Due to the Sobolev's equality u L ∞ (R) ≤ u H 1 (R) and g ∈ W 1 loc (R) (g(0) = 0), we deduce that Thus, P x ∈ L 2 (R). Analogously, we can obtain P x ∈ L 2 (R), that is, P x ∈ L 2 (R).
Definition 2.1. We called u(t, x) is a solution of the Cauchy problem (4) on (0, T ], if a Hölder continuous function u(t, x) defined on (0, T ] × R with the following properties: (i) At each fixed t we have u(·, t) ∈ H 1 (R).
(ii) the map t → u(·, t) is Lipschitiz continuous from (0, T ] to L 2 (R), satisfying the initial condition u 0 (x) ∈ H 1 (R) together with for a.e. t. Here (5) is understood as an equality between functions in L 2 (R).

LI YANG, ZENG RONG, SHOUMING ZHOU AND CHUNLAI MU
For smooth solutions, we claim that the total energy is constant in time. In fact, by using ∂ 2 x G * f = G * f − f and differentiating the equation (4) with respect to x, we have Multiplying the first equation (4) by u, and the equation (7) by u x , we can get the following equations Therefore, the conservation law given by Next, we present the definition of the conservative solutions for the equation (4).
x provides a distributional solution to the following balance law namely, for every test function ϕ ∈ C 1 c (R).
In [37], we showed the existence of global conservation for (1) as follows. (ii) There exists a null set N ⊂ R with measN = 0 such that for t / ∈ N the measure µ (t) is absolutely continuous and has density u 2 x (t, ·) w.r.t. Lebesgue measure. (iii) The family {µ (t) ; t ∈ R} provides a measure-valued solution w to the linear transport equation with source The measure µ (t) has a nontrivial singular part for for a time t ∈ N . Owing to the conservative solution u is not smooth, generally, we only know that the energy E in (6) coincides a.e. with a constant, that is, The main purpose of this paper is to add the uniqueness in the above solution.
Theorem 2.4. Given any initial data u 0 ∈ H 1 (R), the Cauchy problem (4) has a unique conservative solution.
3. Uniqueness of characteristics. Set u = u(t, x) be a solution of Cauchy problem (4) with the additional balance law (13). By introducing the new coordinates (t, β), related to the original coordinates (t, x) from the following integral relation Here,we define x(t, β) to be the unique point x such that, for any time t and β ∈ R, Note that at every time where µ (t) is absolutely continuous with density u 2 x w.r.t Lebesgue measure. Next, combing the following Lemma 3.1 with Lemma 3.3, we establish the Lipschitz continuity of x and u as functions of the variables t, β.
Step 2. We claim that the map β → u(t, β) is Lipschitz continuity as β 1 < β 2 . Indeed, from (17), we get Step 3. We prove the Lipschitz continuity of the map t → x(t, β). Assure x(τ, β) = y. We note that the family of measure µ (t) satisfies the balance law (12), where for each t drift u and the source term 2u where the constant C s depending only on the H 1 (R) of norm and γ. Therefore, we have This yields x(t, β) ≥ y − (t) for all t > τ . A similar argument implies proving the uniform Lipschitz continuity of the map t → x(t, β).
The next result shows that characteristics can be uniquely determined by an integral equation combining the characteristic equation and balance law of µ (t) .
Lemma 3.2. Assume that u = u(t, x) be conservative solution of (4). Then, there exists a unique Lipschitz continuous map t → x(t) for anyỹ ∈ R, which the map satisfies both and d dt Additionally, for any 0 ≤ τ ≤ t one has Proof.
Step 1. Taking advantage of the adapted coordinates (t, β), we write the characteristic beginning withỹ in the form t → x(t) = x(t, β(t)), where β(·) is a map to be determined. Combing the two equation (21) and (22) integrating and w.r.t. time we obtain for t N , Defining the function and the constantβ Then we can rewrite the equation (24) in the following form: Step 2. For each fix t ≥ 0, since the maps x → u(t, x) and x → P (t, x) are both in H 1 (R), the function β → G(t, β) defined in (25) is uniformly bounded and absolutely continuous. Furthermore, we have for some constant C depending only on the H 1 norm of u. Hence the function G in (27) is uniformly Lipschitz continuous w.r.t. β.
Step 3. By the Lipschitz continuity of the function G, the existence of a unique solution to the integral (27) can be proved by a standard fixed point argument.
More details can be found in [3]. Step 4. Thanks to the previous construction, the map t → x(t) . = x(t, β(t)) provides the unique solution to (24). Since the Lipschitz continuity of β(t) and x(t) = x(t, β(t)), β(t) and x(t) are differentiable almost everywhere, so we only have to consider the time where x(t) is differentiable. It suffices to show that (21) holds at almost every time. Suppose, on the contrary, thatẋ(τ ) . = γu(τ, x(τ )). Without loss of generality, letẋ for some ε 0 > 0. The case ε 0 < 0 is entirely similar. To obtain a contradiction we find that, for all t ∈ (τ, τ + δ], with δ > 0 small enough one has We also observe that if ϕ is Lipschitz continuous with compact support then (13) is still true.

LI YANG, ZENG RONG, SHOUMING ZHOU AND CHUNLAI MU
For any > 0, we can still use the blow test functions used in [3].
Using ϕ as test function in (13) we obtain If t is sufficiently close to τ , one has In fact, for s ∈ [τ + , t − ], we have since γu(s, x) < γ(τ, x(τ )) + ε 0 and ϕ x ≤ 0. Due to the family of measures µ (t) depends continuously on t in the topology of weak convergence, taking the limit of (33) as → 0, we obtain The above inequality implies Note that the last term is higher order infinitesimal, satisfying o1(t−τ ) On the other hand, using (25) and (27) a linear approximation we obtain as t → τ . For t sufficiently close to τ , we have Relying on (36) and (37), observe that Subtracting common terms, dividing both sides by t − τ and letting t → τ , we get a contradiction, namely, (21) must hold.
Step 5. Now, we prove (22). From (5), one has for every test function φ ∈ C ∞ c (R). Let φ = ϕ x which ϕ ∈ C ∞ c is given. Owing to the map x → u(t, x) is absolutely continuous, integrating by part w.r.t. x and get By an approximation argument, the identity (40) still holds for any test function ϕ which is Lipschitz continuous with compact support. We now consider the functions for any > 0 is sufficiently small. Define with χ (s) defined in (31). We then use the test function ϕ = ψ in (40) and let → 0. Noticing that the function P x is continuous, we obtain that To complete the proof it suffices to show that the last term on the right hand side of (42) vanishes. The Cauchy's inequality implies since u x ∈ L 2 . Consider the function For each > 0. Observe that all functions ς are uniformly bounded. Furthermore, we have ς (t) → 0 pointwise at a.e. time t as → 0. Therefore, in view of the dominated convergence theorem, we have Combing (45) and (46), this prove that integral in (43) approaches zero as → 0. We now estimate the integral near the corners of the domain: as → 0. The above analysis shows that Therefore, we achieve (20) by (42).
Step 6. The uniqueness of the solution t → x(t) now becomes clear. Relying on (23) we can show Lipschitz continuity of u w.r.t. t, in the auxiliary coordinate system.
x) be a conservative solution of equation (4). Then the map (t, β) → u(t, β) . = u(t, x(t, β)) is Lipschitz continuous, with a constant depending only on the norm u 0 H 1 .
The next result shows that the solutions β(·) of (27) depend Lipschitz continuously on the initial data.
the G is defined in (25). Then there exists a constant C such that, for any two initial dataβ 1 ,β 2 and any t, τ ≥ 0 the corresponding solutions satisfy The proof is easy because of Lipschitz continuity of G with respect to β. We leave it to reader.
Lemma 3.5. Suppose that u ∈ H 1 (R). Then P x is absolutely continuous and satisfies Proof. The function φ(x) = 1 2 e −|x| satisfies the distributional identity D 2 x φ = φ − δ 0 , where δ 0 denotes a unit Dirac mass at the origin. Then for every function f ∈ L 1 (R), the convolution satisfies 4. The uniqueness of conservative solutions for equation (4). In this section, we mainly prove the uniqueness of conservative solutions for equation (4) (the result presents in Theorem 2.4). The proof will be established in several steps.
= P x (t, x(t, β)) are also Lipschitz continuous. In view of the Rademacher's theorem, the partial derivatives x t , x β , u t , u β and P x,β exist almost everywhere. In addition, for theses derivatives, a.e. point (t, β) is a Lebesgue point. Calling t → β(t,β) the unique solution to the integral equation (27), from Lemma 3.4 for a.eβ the following holds.
If the above condition holds, we say that t → β(t,β) is a good characteristic.
In particular, the quantities within square brackets on the left hand sides of (56) are absolutely continuous. From (56), using Lemma 3.5 along a good characteristic we obtain (57) Step 3. We now revert to the original (t, x) coordinates and deduce an evolution equation for the partial derivative u x along a "good" characteristic curve. Fix a point (τ,x) with τ / ∈ N . Suppose thatx is a Lebesgue point for the map x → u x (τ, x). Letβ be such thatx = x(τ,β) and assure that t → β(t; τ,β) is a good characteristic, so that (GC) is valid. We observe that If only x β > 0, along the characteristic though (τ,x) the partial derivative u x can be calculated as From the two ODEs (54)-(55) representing through the evolution of u β and x β , we obtain that the map t → u x (t, x(t, β(t; τβ))) is absolutely continuous (as long as x β = 0) and satisfies d dt u x (t, x(t, β(t; τβ))) = d dt In turn, as long as x β > 0 this means Step 4. Now we consider the function v .
This implies Then, v will be regarded as map taking values in the unit circle Ω . = [−π, π] with endpoints identified. We claim that, along each good characteristic, the map t → v(t) .
= v(t, x(t, β(t; τβ))) is absolutely continuous and satisfies In fact, denote by x β (t), u β (t) and u x (t) = u β (t) x β (t) the values of x β , u β (t) and u x along this particular characteristic. By (GC) we have x β (t) > 0 for a.e. t > 0. Assure that τ is any time where x β (τ ) > 0, we can find a neighborhood I = [τ − δ, τ + δ] such that x β (τ ) > 0 on I. From (60) and (62), v = 2 arctan( u β x β ) is absolutely continuous restricted to I and satisfies (63). To prove our claim, there remains to verify that t → v(t) is continuous on the null set N of times where x β (t) = 0. Let x β (t 0 ) = 0, by the following identity valid as long as x β > 0, it is clear that u 2 x → ∞ as t → t 0 and x β (t) → 0. This indicates v(t) = 2 arctan u x (t) → ±π. Since we identify the points ±π in Ω , this establishes the continuity of v for all t ≥ 0, proving our claim.
We recall that P and G, the function P is a representation in term of the variable β, namely (67)