Existence and Uniqueness of Global Weak solutions of the Camassa-Holm Equation with a Forcing

In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.


Introduction
In this paper, we consider the C-H equation with a forcing in the following form: where u := u(t, x) : R × R → R. The differential operators are defined by u t := ∂u ∂t , u x := ∂u ∂x , u xx := ∂ 2 u ∂x 2 , u txx := ∂ 3 u ∂t∂x 2 and u xxx := ∂ 3 u ∂x 3 . ku is the forcing term, where k ∈ R is a constant. In particular, when k = 0, Eq.(1.1) is a well-known integrable equation describing the velocity dynamics of shallow water waves, named Camassa-Holm equation, which models the propagation of unidirectional shallow water waves over a flat bottom by approximating the Green-Naghdi equations (see [14]), with formal Hamiltonian derivations provided in [10,20].
In the last two decades, the C-H equation has attracted many attentions, and a lot of interesting properties have been found, including integrability( [10,12,15]), existence of peaked solitons and multi-peakons( [1,3]), breaking waves( [11]). It should point out that tsunami modelling has been connected to the dynamics of the Camassa-Holm equation(see the discussion in [13,23]). Among these important properties, the following three conservation laws are crucial in studying the C-H equation.
This implies that the C-H equation can be written in bi-Hamiltonian form which means a second, compatible structure: where ′ denotes the Fréchet derivative. Then, the orbital stability of the peakon (a special global solution of the C-H equation in the form u(t, x) = ce −|x−ct| , where c ∈ R is wave speed) is well understood in [16,17,18]. With regards to the global existence of weak solutions for the Cauchy problem, in [5,24,25], the global existence and uniqueness of dispersive solutions is obtained as weak limits of viscous regularizations of the C-H equation in H 1 (R). The global existence of conservative solutions is studied in [4,21] by the new characteristic method. Recently, the uniqueness of conservative solutions is proved in [6] by using the technique of a generalized characteristic method.
However, for Eq.(1.1), due to the existence of the forcing term ku, the structure of Eq.(1.1) changes dramatically. For example, the peakon u(t, x) = ce −|x−ct| is no longer a solution of Eq.(1.1), and in fact, from [22], Eq. Motivated by those questions, we try to apply the new characteristic method established in [4,6,7], to study the global well-posedness for the forcing C-H equation (1.1). We remark that in [4], the authors provided a new characteristic method to study the existence of global conservative solutions for the C-H equation without forcing, relying heavily on the conservation of energy. But this is not essential. The essential structure is the study of a balance law for the C-H equation. Now, we briefly introduce our results and the key ideas of the proof. We first study the structure of Eq.(1.1). Let p(x) = 1 2 e −|x| , where x ∈ R. We collect some important properties of p(x) in the following (ii) F[p − p xx ] = 1 implies p − p xx = δ(x), where δ is the Dirac function.
where F denotes the Fourier transformation in R. Then, Eq.(1.1) can be expressed in the following form.
u t + uu x + P x − kQ = 0, (1.2) where the singular intergral operators P and Q are defined by By multiplying 2u x , fortunately, we find that Eq.(1.2) satisfies the following balance law in the following form.
Indeed, (1.5) is called the balance law due to the third term in (1.5) only includes first-order derivative of u, although some nonlocal operators are involved. In the present paper, we extend the arguments for the C-H equation without a forcing in [4,6] to the forcing C-H equation (1.2) by using only the balance law (1.5). Actually, there are two main difficulties to study the existence and uniqueness of global weak solutions for the forcing C-H equation (1.2): one is the loss of conservation of energy, that is, the other is the appearance of a new nonlocal singular operator Q.
We supplement Eq.(1.2) with the initial data x ∈ R} Furthermore, the weak solution satisfies the following properties.
(i) u(t, x) is 1 2 -Hölder continuous with respect to t and x, for t in any bounded interval.
(iii) The balance law (1.5) is satisfied in the following sense: there exists a family of Radon measures {µ (t) , t ∈ R}, depending continuously on time and w.r.t the topology of weak convergence of measures, and for every t ∈ R, the absolutely continuous part of µ (t) w.r.t. Lebesgue measure has density u 2 x (t, ·), which provides a measure-valued solution to the balance law (iv) Some continuous dependence result holds. Consider a sequence of initial data u 0,n such that u 0,n − u 0 H 1 → 0, as n → +∞. Then the corresponding solutions u n (t, x) converge to u(t, x) uniformly for (t, x) in any bounded sets.
Finally, we prove the uniqueness of the global weak solutions of Eq.(1.2) satisfying the balance law (1.9) in the last section of this paper, as follows.
We point out that in this paper we verified that the methods for both existence and uniqueness of weak solutions used in [4,6] can be extended to Eq.(1.2) by exploring only a balance law (1.5) instead of a conservation of energy. However, the jump for using a balance law instead of a conservation law is very nontrivial. This idea has also been used for a generalized wave equation with a higher-order nonlinearity in our forthcoming paper [9].

The Transferred System
In this paper, we firstly study the global weak solutions of (1.2)-(1.7) in H 1 (R), in terms of Bressan and Constantin's arguments in [4](see also [8]). First, we shall use the characteristic method to transfer the quasi-linear Eq.(1.2) to a semi-linear system in terms of smoothing solutions. Then, we prove the existence of global weak solutions for the transferred system by the ODE's argument. Finally, by the inverse transformation, we return to the original problem. More precisely, we introduce a new coordinate (T, Y ) defined The equation of the characteristic is Here, Y = Y (t, x) is a characteristic coordinate and satisfies Y t +uY x = 0 for any (t, x) ∈ R × R. After some computations, we see that for any smoothing function f := f (T, Y (t, x)) have the following properties, under the new coordinates (T, Y ): Take the transformation: v := 2 arctan u x and ξ := Under this transformation, we see that Then, we will consider Eq.(1.2) under the new characteristic coordinate In conclusion, from the new transformation (2. (2.6) We supplement (2.6) with the initial data under the new coordinate (T, Y ) Next, we show the expression of P , P x , Q and Q x under the new coordinate (T, Y ). It follows from the last formula in (2.5) that We take x = x(T, Y ′ ), then dx = ξ 1+u 2 x dY ′ , the validity of which will be checked below. Thus, P , P x , Q and Q x has the following form under (T, Y ).
(2.11) We remark that the equivalent semi-linear system (2.6) is invariant under translation by 2π in v. It would be more precise to use e iv as variable. For simplicity, we use v ∈ [−π, π] with endpoints identified.

Local Existence of the Transferred System
In this section, we will prove the existence of weak solutions of the transferred system (2.6)-(2.7) by the contracting mapping theory. Here, the work space X is defined by As usual, we shall prove the existence of a fixed point of the integral transformation: where P , P x , Q and Q x are defined by (2.8)-(2.11).
As the standard ODE's theory in the Banach space, we shall prove that all functions on the right hand side of (2.6) are locally Lipschitz continuous with respect to (u, v, ξ) in X. Then, we can obtain the local existence of weak solutions of (2.6)-(2.7), as follows. Proof. Let K ⊂ X be a bounded domain and defined by for a.e. Y ∈ R and constants A, B, C − , C + > 0. Then, if the mapping Φ(u, v, ξ) is Lipschitz continuous on K, then Φ(u, v, ξ) has a fixed point by contraction argument and the existence of local solutions will be followed. In order to do so, first, it follows from the Sobolev embedding H 1 (R) ֒→ L ∞ (R) with any q > 1 that u L ∞ ≤ C u H 1 . Due to the uniform boundness of v and ξ, the following maps: are all Lipschitz continuous as maps from K into L 2 (R), as well as from K into L ∞ (R). In order to estimate the singular integrals in P , P x , Q and But this is not enough to control the above singular integrals. Here, we claim the following estimate: For any z 1 < z 2 , we deduce that . The above estimate guarantees proper control on the singular integrals in P , P x , Q and Q x , which decreases quickly as |Y − Y ′ | → +∞. Therefore, (3.1) follows from the Young inequality. In addition, This completes the proof of Claim (3.1). Thirdly, by (3.1), we give a priori bounds on P , P x , Q and Q x , which implies P , P x , Q and Q x ∈ H 1 (R). More precisely, we deduce that P , ∂ Y P , . Indeed, we give estimates for Q, and the estimates for P can be obtain by similar argument. By applying (3.1), we deduce that Then, we have By the same arguments, we can obtain the similar estimates for P in the following.
Using the Sobolev inequality, we have u L ∞ ≤ C u H 1 and this implies P , P x , Q and Q x ∈ H 1 (R). Then, according to the above discussion, we show that the mapping Φ(u, v, ξ) is from K to K. Fourthly, we shall prove the mapping Φ(u, v, ξ) is Lipschitz continuity with respect to (u, v, ξ). More precisely, we need to prove that the following partial derivatives ∂P ∂u , ∂P ∂v , ∂P ∂ξ , ∂Px ∂u , ∂Px ∂v , ∂Px ∂ξ , ∂Q ∂u , ∂Q ∂v , ∂Q ∂ξ , ∂Qx ∂u , ∂Qx ∂v , ∂Qx ∂ξ are uniformly bounded as (u, v, ξ) range inside the domain K. We observe these derivatives are bounded linear operators from the appropriate spaces into H 1 (R). For sake of illustration, we shall give the detail estimates for Q.
And the estimates for P will be similar obtained. In other words, we shall prove ∂Q ∂u and ∂Qx ∂u are linear operators from H 1 (R) into H 1 (R); ∂Q ∂v and ∂Qx ∂v are linear operators from L 2 (R) L ∞ (R) into L 2 (R); ∂Q ∂ξ and ∂Qx ∂ξ are linear operators from L ∞ (R) into L 2 (R).
Here, we shall take ∂Q ∂u and ∂(∂ Y Q) ∂u as examples to illustrate the main idea. Indeed, Finally, we prove the mapping Φ(u, v, ξ) is uniformly Lipschitz continuity on a neighborhood K of the initial data in the space X. Then, we can apply the standard theory of ODE's local existence in Banach spaces, there exist a solutions to the Cauchy problem (2.6)-(2.7) on small time interval [−T, T ]. This completes the proof of Proposition 3.1.

Global Existence of the Transferred System
In this section, we shall prove that the local solution for the Cauchy problem (2.6)-(2.7) can be extended to the global one. To ensure this, we need to show that there exists a bound C(E 0 ) > 0 such that for all T ∈ R. Then, we obtain the following proposition.
We claim that for all time T , Indeed, from the first identity in (2.6), we have From the last two identity in (2.6), we see that for all time T 3) where in the last two estimates, we use the facts: for any u ∈ H 1 (R), lim Then, from (4.3), we can deduce that there exists a constant D 0 := D 0 (E 0 , T ) > 0 such that the new energy E(T ) has a priori on bounded intervals of time by the Gronwall inequality. That is, for any T on bounded intervals, For all the solutions of the Cauchy problem (2.6)-(2.7), it follows from (4.2) and (4.4) that which implies that u(T ) L ∞ has a priori bound D 0 . More precisely, we have the following estimate: By the definition of P , P x , Q and Q x in (2.8)-(2.11), using (4.4) and (4.5), we deduce that Inject (4.6) and (4.7) into the last identity in (2.6), we see that And from the initial condition ξ(0, Y ) = 1, we get where (4.11) In order to obtain the boundness of u(T ) H 1 , we need the following lemma to estimate P x L 1 , Q L 1 , (P x ) Y L 1 and Q Y L 1 .
Proof. We remark that ξ sin 2 v(T ) 2 L 1 ≤ D 2 0 does not imply that v(T ) L 2 is bounded, which is the main difficulty. As proved in above, we see that v(T ) L ∞ is bounded, and we may assume |v| ≤ 3π 2 because v(T ) is invariant under multiplier of 2π. Then, when π (4.13) For any z 1 < z 2 , we have Then, for every q ≥ 1, (4.12) follows from the Young inequality. In addition, by some computation, we have Now, we return to the proof of Proposition 4.1. By using Lemma 4.2, we have the following estimates: (4.14) Injecting (4.14) into (4.10) and (4.11), we can deduce that u(T ) 2 L 2 and (u(T )) Y 2 L 2 are bounded on any bounded interval of time T . That is, there exists a constant D 3 := D 3 (E 0 , T ) > 0 such that Finally, we estimate v(T ) L 2 with respect to T . Multiplying v(T ) to the second identity in (2.6) and integrating, we deduce that (4.16) Inject (4.4), (4.5), (4.8), (4.9) and (4.15) into the estimates in P L 2 and Q x L 2 . We have   Proof. First, we prove that the function u = u(t, x) is well-defined. From (4.5), we see that |u(T, Y )| ≤ D 0 . By (5.1), we get x 0 (Y ) = ±∞, this yields the image of the map (T, Y ) → (T, x(T, Y )) is the entire plane R 2 . We claim for all t and a.e. Y ∈ R. Indeed, from (2.6), we deduce that And by differentiating (5.1) with T and Y , we deduce that Moreover, from the fact that x → 2 arctan(u 0 ) x (x) is measurable, we see that the claim (5.3) is true for almost every Y ∈ R at T = 0. Then, (5.3) remains true for all times T ∈ R and a.e. Y ∈ R. Then, for any Y 1 = Y 2 (without loss of generality, we assume . Inject this into (4.2).
This proves that the map (t, x) → u(t(T ), x(T, Y )) at (5.2) is well defined for all (t, x) ∈ R 2 .
Secondly, we prove the regularity of u(t, x) and energy equation. From (4.4), we see that for t in any bounded interval, Now, applying the Sobolev inequality: u C 0,γ ≤ C u H 1 , where γ = 1 − 1 2 . From (5.4), we get u C 0, 1 2 ≤ C u H 1 ≤ CD 0 , which implies u of x is Hölder continuous with index 1 2 . On the other hand, it follows from the first identity in (2.6), (4.6) and (4.7) that u t L ∞ ≤ C( P x L ∞ + Q L ∞ ) ≤ C(D 0 ). Then, the map t → u(t, x(t)) is Lipschitz continuous along every characteristic curve t → x(t). Therefore, u = u(t, x) is Hölder continuous on the any bounded interval of times.
Thirdly, we prove that the L 2 (R)-norm of u(t) is Lipchitz continuous with respect to t on any bounded interval. Denote [τ, τ + h] to be any small interval and τ in any bounded interval. For a given point (τ,x), we choose the characteristic t → x(t, Y ) : {T → x(T, Y )} passes through the point (τ,x), i.e. x(τ ) =x. Since the characteristic speed u satisfies u L ∞ ≤ C u H 1 ≤ CD 0 , we have the following estimate.
Then, This implies that the map t → u(t) is Lipchitz continuous, in terms of the x-variable in L 2 (R).
Fourthly, we prove that the function u provides a weak solution of (1.2). Denote Γ := {(t, x)|t ∈ R, x ∈ R}. For any test function φ(t, x) ∈ C 1 c (Γ), the first equation of (2.6) has the following weak form.
which implies that the weak form (1.8) is true. Now, we introduce the Radon measures {µ (t) , t ∈ R}: for any Lebesgue measurable set {x ∈ A} in R. Supposing the corresponding pre-image set of the transformation is {Y ∈ G(A)}, we have For every t ∈ R \ N , the absolutely continuous part of µ (t) w.r.t. Lebesgue measure has density u 2 x (t, ·) by (5.3). It follows from (2.6) that for any test function φ(t, x) ∈ C 1 c (Γ), Then, (1.9) is true due to meas (N ) = 0.
Finally, let u 0,n be a sequence of initial data converging to u 0 in H 1 (R). From (2.1) and (2.7), at time t = 0 this implies uniformly for T, Y in bounded sets. Returning to the original coordinates, this yields the convergence uniformly on bounded sets, because all functions u, u n are Hölder continuous. This completes the proof.

Uniqueness of the Global Weak Solution
We shall give the proof of Theorem 1.2 in the case t ≥ 0, and the case t < 0 can be handled by the similar argument.
Step 1. We define x(t, β) to be the unique point x such that Recalling that at every time, µ (t) is absolutely continuous with density u 2 x w.r.t. Lebesgue measure, the above definition gives We study the Lipschitz continuity of x and u as functions of t, β.
Proof. (i) From the definition of β in (6.2) , we remark that for any time t ≥ 0, x → β(t, x) is right continuous and strictly increasing. Then, its inverse β → x(t, β) is well-defined, and is also continuous and nondecreasing. For any β 1 < β 2 , we deduce that We remark that the family of measures µ (t) satisfies the balance law (1.9), where for each t, the source term 2(u 2 −P +kQ x )u x in (1.9) has the following estimate by the Hölder inequality.
Step 2. The next lemma shows that characteristics can be uniquely determined by an integral equation combining the characteristic equation and balance law of u 2 x , which is crucial to study the uniqueness of the conservative solution of Eq.(1.2). Lemma 6.2. Let u = u(t, x) be the weak solution of Eq.(1.2) satisfying (1.9). Then, for any x 0 ∈ R there exists a unique Lipschitz continuous map t → x(t) which satisfies d dt x(t) = u(t, x(t)), and d dt for a.e. t ≥ 0. Furthermore, for any 0 ≤ τ ≤ t, we have Proof. Firstly, by the adapted coordinates (t, β), we write the characteristic starting at x 0 in the form t → x(t) = x(t, β(t)), where β(·) is a map to be determined. Sum up (6.4) and (6.5) and integrate w.r.t. time. We get For each fixed t ≥ 0, since the maps x → u(t, x), x → P (t, x) and x → Q x (t, x) are both in H 1 (R), the function β → G(t, β) defined at (6.8) is uniformly bounded and absolutely continuous. Moreover, for some constant C depending only on the H 1 -norm of u. Hence the function G in (6.8) is Lipschitz continuous w.r.t. β. We can apply the ODE's theory in the Banach space of all continuous functions β : By the Lipchitz continuity of G, Then, [Φβ] is a strict contraction. (6.8) has a unique solution t → β(t), and the corresponding function t → x(t, β(t)) satisfies (6.4) and (6.5). Secondly, from the integral equation (6.7), we can determine a Lipchitz continuous characteristic x(t) of (6.4). By the previous construction, the map t → x(t) := x(t, β(t)) provides the unique solution to (6.7). Being the composition of two Lipschitz functions, the map t → x(t, β(t)) is also Lipschitz continuous. To prove that it satisfies the ODE for the characteristics of (6.4), it suffices to show that (6.4) holds at each time τ > 0 such that (ii) the measure µ (τ ) is absolutely continuous.
Thus, lim ǫ→0 t τ ( x + (s)+ǫ x + (s)−ǫ u 2 x (ϕ ǫ t + uϕ ǫ x )(s, x) dx)ds ≥ 0 as t is sufficiently close to τ . Since the family of measure µ (t) depends continuously on t in the topology of weak convergence, by taking ǫ → 0 in (6.11), we have τ, t / ∈ N For every t > τ with t / ∈ N , when t is sufficiently close to τ , by injecting (6.9) and (6.12) into β(t), we deduce that On the other hand, from (6.7) and (6.8), a linear approximation yields By combining (6.14) and (6.13), we see that Subtracting common terms and dividing both sides by t − τ and letting t → τ , we get ε 0 ≤ 0, which is a contradiction. Then, (6.4) must hold.
Thirdly, we prove (6.6). In (1.8), let φ = ϕ x and ϕ ∈ C ∞ c . Since the map x → u(t, x) is absolutely continuous, we can integrate by parts w.r.t. x.
Then, we can obtain (6.19) by the Gronwall inequality.
Step 4. We give the proof of Theorem 1.2.
By differentiating w.r.t. β 0 the identity (6.6), we obtain (6.22) Combining (6.20)-(6.22), we thus obtain the system of ODEs (6.23) In particular, the quantities within square brackets on the left hand sides of (6.23) are absolutely continuous. Recall the fact P xx = P − ( 1 2 u 2 x + u 2 ) and x β . From (6.23), along a good characteristic we obtain where the first equation is obtained by the first two equations in (6.23) and the second equation is obtained by the first and third equations in (6.23) Secondly, we go back to the original (t, x) coordinates and derive an evolution equation for the partial derivative u x along a "good" characteristic curve. Fix a point (τ, x 0 ) with τ ∈ N . Assume that x 0 is a Lebesgue point for the map x → u x (τ, x). Let β 0 be such that x 0 = x(τ, β 0 ) and assume that t → Z(t) is a good characteristic, so that (GC) holds. We observe that u 2 x (τ, x) = 1 x β (τ,β 0 ) − 1 ≥ 0, which implies x β (τ, β 0 ) > 0. As long as x β > 0, along the characteristic through (τ, x 0 ) the partial derivative u x can be computed as u x (t, x(t, Z(t))) = u β (t,Z(t)) x β (t,Z(t)) . Using (6.21)-(6.22) describing the evolution of u β and x β , we conclude that the map t → u x (t, x(t, Z(t))) is absolutely continuous (as long as x β = 0) and satisfies d dt u x (t, x(t, Z(t))) = d dt ( u β (t, Z(t)) x β (t, Z(t)) ) = u 2 − P + kQ x − 1 2 We remark u 2 β = x 2 β u 2 x = x β (1 − x β ), and as long as x β > 0 this implies d dt arctan u x (t, x(t, Z(t))) = (u 2 − P + kQ x )x β − 1 2 (1 − x β ). Then, we see that In the following, v will be regarded as a map taking values in the unit circle S := [−π, π] with endpoints identified. We claim that, along each good characteristic, the map t → v(t) := v(t, x(t, β(t; τ, β 0 ))) is absolutely continuous and satisfies d dt v(t) = 2(u 2 − P + kQ x ) cos 2 v 2 − sin 2 v 2 .
(6.27) Indeed, 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 , and u x along this particular characteristic. By (GC) we have x β (t) > 0 for a.e. t > 0. If τ is any time where x β (τ ) > 0, we can find a neighborhood I = [τ − δ, τ + δ] such that x β (t) > 0 on I. By (6.25) and (6.26), v = 2 arctan(u β /x β ) is absolutely continuous restricted to I and satisfies (6.27). To prove our claim, it thus remains to show that t → v(t) is continuous on the null set N of times where x β (t) = 0. Suppose x β (0) = 0. From the fact that the identity u 2 x (t) = 1−x β (t) x β (t) valids as long as x β > 0, it is clear that u 2 x → +∞ as t → 0 and x β (t) → 0. This implies v(t) = 2 arctan u x (t) → ±π. Since we identify the points ±π, we get the continuity of v for all t ≥ 0, proving our claim.