Lipschitz metric for the Camassa-Holm equation on the line

We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$. The relationship between this metric and the usual norms in $H^1$ and $L^\infty$ is clarified. The method extends to the generalized hyperelastic-rod equation $u_t-u_{xxt}+f(u)_x-f(u)_{xxx}+(g(u)+\frac12 f"(u)(u_x)^2)_x=0$ (for $f$ without inflection points).


Introduction
The Cauchy problem for the Camassa-Holm (CH) equation [3,4], where κ ∈ R is a constant, has attracted much attention due to the fact that it serves as a model for shallow water waves [8] and its rich mathematical structure. For example, it has a bi-Hamiltonian structure, infinitely many conserved quantities and blow-up phenomena have been studied, see, e.g., [5], [6], and [7]. We here focus on the construction of the Lipschitz metric for the semigroup of conservative solutions on the real line. This problem has been recently considered by Grunert, Holden, and Raynaud [12] in the periodic case, and here we want to present how the approach used there has to be modified in the non-periodic case.
For simplicity, we will only discuss the case κ = 0, that is, and from now on we refer to (1.2) as the CH equation. However, the approach presented here can also handle the generalized hyperelastic-rod equation, see Remark 2.9. In particular, it includes the case with nonzero κ. The generalized hyperelastic-rod equation has been introduced in [15]. It is given by where f and g are smooth functions. 1 With f (u) = u 2 2 and g(u) = κu + u 2 , we recover (1.1) for any κ. With f (u) = γu 2 2 and g(u) = 3−γ 2 u 2 , we obtain the hyperelastic-rod wave equation: which has been introduced by Dai [9,10,11]. Equation (1.2) can be rewritten as the following system u t + uu x + P x = 0, (1.4) where we choose u to be an element of H 1 (R). The H 1 norm is preserved as for any smooth solution u. However, even for smooth initial data, the solution may break down in finite time. In this case, the solution experiences wave breaking ( [4,5]): The solution remains bounded while, at some point, the spatial derivative u x tends to −∞. This phenomenon can be nicely illustrated by the so called multipeakon solutions. These are solutions of the form Let us consider the case with n = 2 and one peakon p 1 (0) > 0 (moving to the right) and one antipeakon p 2 (0) < 0 (moving to the left). In the symmetric case (p 1 (0) = −p 2 (0) and q 1 (0) = −q 2 (0) < 0) the solution u will vanish pointwise at the collision time t * when q 1 (t * ) = q 2 (t * ), that is, u(t * , x) = 0 for all x ∈ R. At time t = t * , the whole energy is concentrated at the origin, and we have lim t→t * (u 2 + u 2 x ) dx = u 0 H 1 δ, with δ denoting the Dirac delta distribution at the origin. In general we have two possibilities to continue the solution beyond wave breaking, namely to set u identically equal to zero for t > t , which is called a dissipative solution, or to let the peakons pass through each other, which is called a conservative solution and which is depicted in Figure 1. We are interested in the latter case, for which solutions have been studied by Bressan and Constantin [1] and Holden and Raynaud [13,14]. Since the H 1 -norm is preserved, the space H 1 appears as a natural space for the semigroup of solutions. However, the previous multipeakon example reveals the opposite. Indeed, u(t * , x) = 0 for all x ∈ R. Thus, the trivial solutionū, that is,ū(t, x) = 0 for all t, x ∈ R, which is also a conservative solution, coincides with u at t = t * . To define a semigroup of conservative solutions, we therefore need more information about the solution than just its pointwise values, for instance, the amount and location of the energy which concentrates on sets of zero measure. This justifies the introduction of the set D of Eulerian coordinates, see Definition 4.1, for which a semigroup can be constructed [13].
Furthermore, the H 1 norm is not well suited to establish a stability result. Consider, e.g., the sequence of multipeakons u ε defined as u ε (t, x) = u(t − ε, x), see Figure 1. Then, assuming that u(0) H 1 (R) = 1, we have so that the flow is clearly discontinuous with respect to the H 1 norm. Figure 1. The dashed curve depicts the antisymmetric multipeakon solution u(t, x), which vanishes at t * , for t = 0 (left) and t = t * (middle) and t = 2t * (right). The solid curve depicts the multipeakon solution given by u ε (t, x) = u(t − ε, x) for some small ε > 0 (the CH equation is invariant with respect to time translation).
The aim of this article is to present a metric for which the semigroup of conservative solutions on the line is Lipschitz continuous. A more extensive discussion about Lipschitz continuity with examples from ordinary differential equations, can be found in [2]. A detailed presentation for the Camassa-Holm equation in the periodic case is presented in [12], thus we here focus on explaining the differences between the periodic case and the decaying case. However, we first present the general construction.
The construction of the metric is closely connected to the construction of the semigroup itself. Let us outline this construction. We rewrite the CH equation in Lagrangian coordinates and obtain a semilinear system of ordinary differential equations: Let u(t, x) denote the solution and y(t, ξ) the corresponding characteristics, thus y t (t, ξ) = u(t, y(t, ξ)). Then our new variables are y(t, ξ), as well as where U corresponds to the Lagrangian velocity while H can be interpreted as the Lagrangian cumulative energy distribution. The time evolution for any X = (y, U, H) is described by This system is well-posed as a locally Lipschitz system of ordinary differential equations in a Banach space, and we can define a semigroup of solution which we denote S t . From standard theory for ordinary differential equations we know that S t is locally Lipschitz continuous, that is, given T and M , where the constant C M,T depends only on M and T . The mapping from Lagrangian to Eulerian coordinates is surjective but not bijective. The discrepancy between the two sets of coordinates is due to the freedom of relabeling in Lagrangian coordinates. The relabeling functions form a group, which we denote G, and which basically consists of the diffeomorphisms of the line with some additional assumptions (see Definition 2.3). Given X = (y, U, H), the element X • f = (y • f, U • f, H • f ) is the relabeled version of X by the relabeling function f ∈ G. Using the fact that the semigroup S t is equivariant with respect to relabeling, that is, we can construct a semigroup of solutions on equivalence classes from S t . Finally, after establishing the existence of a bijection between the Eulerian coordinates and the equivalence classes in Lagrangian coordinates, we can transport the semigroup of solutions defined on equivalence classes and construct a semigroup, which we denote T t , of conservative solutions in D.
We want to find a metric which makes T t Lipschitz continuous. For that purpose, we introduce a pseudometric 2 in Lagrangian coordinates which does not distinguish between elements of the same equivalence class and which, at the same time, leaves the semigroup S t locally Lipschitz continuous. This strategy has been used in [2] for the Hunter-Saxton equation and in [12] for the Camassa-Holm equation in the periodic case. In [2], the pseudometric is defined by using ideas from Riemannian geometry. Here, we follow the approach of [12] and first introduce a pseudosemimetric 3 which also identifies elements of the same equivalence class and leaves S t Lipschitz continuous. A natural choice, which was applied in [12], is to consider the pseudometricJ defined as The pseudometricJ identifies elements of the same equivalence class, asJ(X, X • f ) = 0. Moreover, it is invariant with respect to relabeling, that is,J(X α • f, X β • g) =J(X α , X β ) for any f, g ∈ G. It remains to prove that the pseudosemimetric makes the semigroup S t locally Lipschitz, that is, given M and T , there exists a constant C depending on M and T such that for all t ∈ [0, T ] and X α , X β ∈ B M . The proof follows almost directly from the stability and equivariance of S t . We outline it here. For every ε > 0, there exist 2 By a pseudometric we mean a map d : for which the triangle inequality d(x, y) ≤ d(x, z) + d(z, y) holds, and satisfies d(x, x) = 0 for x, y, z ∈ X.
3 By a pseudosemimetric we mean a map d : X × X → [0, ∞) which is symmetric, d(x, y) = d(y, x) and satisfies d(x, x) = 0 for x, y ∈ X.
and (1.15) follows by letting ε tend to zero. However, the use of the Lipschitz stability of S t (1.12) relies on bounds on X α • f and X β • g that are unavailable. The problem is that the norm · of the Banach space is not invariant with respect to relabeling and therefore, since f and g are a priori arbitrary, we cannot obtain any bound depending on M for X α • f and X β • g . This motivates the introduction in this paper of the pseudosemimetric J defined as As expected, the pseudosemimetric J identifies equivalence classes (we have J(X, X• f ) = 0) but we lose the nice relabeling invariance property. At the same time, this definition of J implies some implicit restrictions on the diffeomorphisms f 1 and f 2 which allow us to bound the relabeled versions X α • f 1 and X β • f 2 so that the approach sketched in (1.16) can be carried out. It remains to explain why, in the periodic case [12], we could use the definition ofJ, which is a more natural definition and moreover simplifies the proofs. In the periodic case (we take the period equal to one), the stability of the semigroup S t is established in the space Note that, in order to keep these formal explanations as simple as possible, we just consider the second component of X = (y, U, H).
, for any f ∈ G, so that the norm defined in (1.18) is relabeling invariant. Now, if the norm of the Banach space is relabeling invariant, we have and the pseudosemimetrics J andJ are equivalent. However, the natural Banach but the corresponding embedding does not hold in the case of the real line. This also shows that the approach that we present here for the real line can also be used in the periodic case and that the novelty in this article is that we handle a norm which is not relabeling invariant. The final step consists of deriving a pseudometric from the pseudosemimetric J. This can be achieved by the following general construction: Let where the infimum is taken over all finite sequences {X n } N n=0 with X 0 = X α and X N = X β . The pseudometric d inherits the Lipschitz stability property (1.15) from J. Finally, identifying elements belonging to the same equivalence class, the pseudometric d turns into a metric on the set of equivalence classes. By bijection, it yields a metric in D which makes the semigroup of conservative solutions Lipschitz continuous.
In the last section, Section 5, we compare this new metric with the usual norms in H 1 and L ∞ .

Semigroup of solutions in Lagrangian coordinates
In this section, we recall from [13] the construction of the semigroup in Lagrangian coordinates. The Camassa-Holm equation reads and can be rewritten as the following system Next, we rewrite the equation in Lagrangian coordinates. Therefore we introduce the characteristics The Lagrangian velocity U reads We define the Lagrangian cumulative energy as As an immediate consequence of the definition of the characteristics we obtain The last term can be expressed uniquely in terms of y, U , and H. From (2.3) we obtain the following explicit expression for P , Setting Q(t, ξ) = P x (t, y(t, ξ)) and writing P (t, ξ) = P (t, y(t, ξ)), we obtain and (2.10) Moreover we introduce another variable ζ(t, ξ) = y(t, ξ) − ξ. Thus we have derived a new system of equations, which is up to that point only formally equivalent to the Camassa-Holm equation: Let V be the Banach space defined by Of course H 1 ⊂ V but the converse is not true as V contains functions that do not vanish at infinity. We will employ the Banach space E defined by Given a constant M > 0, we denote by B M the ball Theorem 2.2. For anyX = (ȳ,Ū ,H) ∈ G, the system (2.11) admits a unique global solution X(t) = (y(t), U (t), H(t)) in C 1 (R + , E) with initial dataX = (ȳ,Ū ,H).
We have X(t) ∈ G for all times. If we equip G with the topology induced by the E-norm, then the mapping S : is a continuous semigroup. More precisely, given M > 0 and T > 0, there exists a constant C M which depends only on M and T such that, for any two elements Proof. We have where F is locally Lipschitz (see proof [13,Theorem 2.3]). It implies, by Gronwall's lemma, that we have where Id denotes the identity function. Given κ > 0, we denote by G κ the subset G κ of G defined by The subsets G κ do not possess the group structure of G. The next lemma provides a useful characterization of G κ . We define the subsets F κ and F of G as follows As we shall see, the space F 0 will play a special role. These sets are relevant only because they are preserved by the governing equation (2.11) as the next lemma shows. In particular, while the mapping ξ → y(t, ξ) may not be a diffeomorphism for some time t, the mapping ξ → y(t, ξ) + H(t, ξ) remains a diffeomorphism for all times t.
Lemma 2.5. The space F is preserved by the governing equation (2.11). More precisely, given κ, T ≥ 0, there exists κ which only depends on T , κ and X such that S t (X) ∈ F κ for anyX ∈ F κ .
For the sake of simplicity, for any X = (y, U, H) ∈ F and any function f ∈ G, we denote (y • f, U • f, H • f ) by X • f . This operation corresponds to relabeling. Definition 2.6. We denote by Π(X) the projection of F into F 0 defined as The element Π(X) is the unique relabeled version of X that belongs to F 0 .
Lemma 2.7. The mapping S t is equivariant, that is, This follows from the governing equation and the equivariance of the mappings X → P (X) and X → Q(X), where P and Q are defined in (2.9) and (2.10), see [13] for more details. From this lemma we get that Definition 2.8. We define the semigroupS t on F 0 as The semigroup property ofS t follows from (2.19). From [13], we know thatS t is continuous with respect to the norm of E. It follows basically from the continuity of the mapping Π, but Π is not Lipschitz continuous and the goal of the next section is to find a metric that makesS t Lipschitz continuous. Remark 2.9. The details of the construction of the semigroup of solutions in Lagrangian coordinates for the generalized hyperelastic-rod equation (1.3) is given in [15]. The construction is based on a reformulation of the equation in Lagrangian coordinates which leads to a semilinear system of equations, similar to (2.11) for the Camassa-Holm equation. In the case of the generalized hyperelastic-rod equation, the equation can be rewritten as where G(v) is given by 3. Lipschitz metric for the semigroupS t Definition 3.1. Let X α , X β ∈ F, we define J(X α , X β ) as The mapping J is symmetric. Moreover, if X α and X β are equivalent, then J(X α , X β ) = 0. Our goal is to create a distance between equivalence classes, and that is the reason why we introduce the pseudosemimetricJ as follows in the periodic case ( [12]).
The pseudosemimetricJ is relabeling invariant, that is,J(X α • f, X β • g) = J(X α , X β ). With Definition 3.1, we lose this important property. However, Definition 3.1 allows us to obtain estimates that cannot be obtained by Definition 3.2, see the proof of Theorem 3.10. In addition, it turns out that we do not actually need the relabeling invariance property to hold strictly and the estimates contained in the following lemma are enough for our purpose. Lemma 3.3. Given X α , X β ∈ F and f ∈ G κ , we have for some constant C which depends only on κ.
Definition 3.4. Let X α , X β ∈ F 0 , we define d(X α , X β ) as where the infimum is taken over all sequences {X n } N n=0 ∈ F 0 which satisfy X 0 = X α and X N = X β . Lemma 3.5. For any X α , X β ∈ F 0 , we have Proof. First, we prove that, for any X α , X β ∈ F 0 , we have We have It follows from the definition of F 0 that 0 ≤ y ξ ≤ 1, 0 ≤ H ξ ≤ 1 and |U ξ | ≤ 1 so that X α,ξ L ∞ ≤ 3. We also have Hence, from (3.8), we get In the same way, we obtain X α − X β L ∞ ≤ 4 X α − X β • f L ∞ for any f ∈ G. After adding these two last inequalities and taking the infimum, we get (3.7). For any ε > 0, we consider a sequence {X n } N n=0 ∈ F 0 such that X 0 = X α and X N = X β and After letting ε tend to zero, we get (3.6).
Lemma 3.6. The mapping d : F 0 ×F 0 → R + is a distance on F 0 , which is bounded as follows Proof. The symmetry is embedded in the definition of J while the construction of d from J takes care of the triangle inequality. From Lemma 3.5, we get that d(X α , X β ) = 0 implies X α = X β . The first inequality in (3.9) follows from Lemma 3.5 while the second one follows from the definition of J and d. Indeed, we have We need to introduce the subsets of bounded energy in F 0 . Note that the total energy is equal to H(∞) − H(−∞) = H L ∞ as H(−∞) = 0 and H is increasing, see Definition 2.1.
The set F M is also conserved by relabeling as, for any f ∈ G, H • f L ∞ = H L ∞ . The ball B M is included in F M but the reverse inclusion does not hold. However, as the next lemma shows, when we restrict ourselves to F 0 , the sets F 0 ∩ F M and F 0 ∩ B M are in fact equivalent. Proof. Since y ξ + H ξ = 1, H ξ ≥ 0, y ξ ≥ 0, we get 0 ≤ H ξ ≤ 1 and 0 ≤ y ξ ≤ 1.
Finally, we have to show that U L 2 ≤ C(M ). Therefore observe that by (2.12c), R U 2 y ξ dξ ≤ R H ξ dξ ≤ M . This together with the fact that X ∈ F 0 yields Thus it is left to estimate U L ∞ , which can be done as follows, where we used that U ξ (ξ) = 0, when y ξ (ξ) = 0 by (2.12c). For almost every ξ such that y ξ (ξ) > 0, we have where the infimum is taken over all the sequences {X n } N n=0 ∈ F M 0 which satisfy X 0 = X α and X N = X β .
We can now prove our main stability theorem.
Theorem 3.10. Given T > 0 and M > 0, there exists a constant C M which depends only on M and T such that, for any X α , X β ∈ F M 0 and t ∈ [0, T ], we have Proof. By the definition of d M , for any ε such that 0 < ε ≤ 1 there exists a sequence Let us denote X t n = S t (X n ), g t n = y t n + H t n ,X t n =S t X n = Π(X t n ) = X t n • (g t n ) −1 . By Lemma 2.5, we have g t n ∈ G κ for some κ which depends only on M and T . The sequence {X t n } has endpoints given byS t (X α ) andS t (X β ). Since X n ∈ F M and the set F M is preserved by the flow of the equation and relabeling, we have To use the stability result (2.14), we have to bound X n−1 • f n−1 and X n . By Lemma 3.8, there existsM such that X ≤M for any X ∈ F M 0 . Hence, X n ≤M as X n ∈ F M 0 . Since f n−1 is a priori arbitrary, it may seem difficult to bound X n • f n−1 , and it is important to note here that the relabeling invariant pseudosemimetricJ, see (1.14), would not provide us with a bound on this term and the following estimates in fact motivate the Definition 3.1. Indeed, by (3.1), we obtain (3.12) which yields as X α , X β ∈ F M 0 . Therefore, by the triangle inequality, X n−1 • f n ≤ 5M + 2 so that X n−1 • f n and X n are bounded by a constant depending only on M . Thus, we can use (2.14) and get from (3.13) that X t n−1 • f t n−1 −X t n ≤ C M X n−1 • f n−1 − X n , where from now on C M denotes some constant dependent on M and T . Similarly forf t n = g t n •f n • (g t n−1 ) −1 , we get that The result follows by letting ε tend to zero.

From Lagrangian to Eulerian coordinates
We now introduce a second set of coordinates, the so-called Eulerian coordinates. Therefore let us first consider X = (y, U, H) ∈ F. We can define Eulerian coordinates as in [13] and also obtain the same mappings between Eulerian and Lagrangian coordinates (see also Figure 2). For completeness we will state the results here.  (i) u ∈ H 1 (R), and (ii) µ is a positive Radon measure whose absolutely continuous part µ ac satisfies (4.1) µ ac = u 2 + u 2 x . We can define a mapping, denoted by L, from D to F 0 :  Then (y, U, H) ∈ F 0 , and we denote by L : D → F 0 the map which to any (u, µ) associates X ∈ F 0 .
Thus from any initial data (u 0 , µ 0 ) ∈ D, we can construct a solution of (2.11) in F with initial data X 0 = L(u 0 , µ 0 ) ∈ F 0 . It remains to go back to the original variables, which is the purpose of the mapping M defined as follows:  u(x) = U (ξ) for any ξ such that x = y(ξ), belongs to D. We denote by M : F 0 → D the map which to any X in F 0 associates (u, µ).
In fact, M can be seen as a map from F/G → D, as any two elements belonging to the same equivalence class in F are mapped to the same element in D (cf. [13]). Moreover, identifying elements belonging to the same equivalence class, the mappings L and M are invertible and We will now use these mappings for defining also a Lipschitz metric on D.  Next we show that T t is a Lipschitz continuous semigroup by introducing a metric on D. Using the map L we can transport the topology from F 0 to D.  d D ((u, µ), (ũ,μ)) = d(L(u, µ), L(ũ,μ)).
The Lipschitz stability of the semigroup T t follows then naturally from Theorem 3.10. It holds on sets of bounded energy, which are given as follows.
Proof. Let X n = (y n , U n , H n ) = L(u n , (u 2 n +u 2 nx )dx) and X = (y, U, H) = L(u, (u 2 + u 2 x )dx). Then as in the proof of [13, Proposition 5.1] one can show that (5.2) X n → X in E.
Proposition 5.2. Let (u n , µ n ) be a sequence in D that converges to (u, µ) in D. Then Proof. Let X n = (y n , U n , H n ) = L(u n , µ n ) and X = (y, U, H) = L(u, µ) . By the definition of the metric d D , we have lim n→∞ d(X n , X) = 0. Using (3.9), we immediately obtain that (5.4) X n → X in L ∞ (R).
The rest can be proved as in [13,Proposition 5.2].