Global weak conservative solutions of the μ-Camassa–Holm equation

In this paper we consider global weak conservative solutions of a μ-Camassa–Holm equation. By employing a technique of change of variables on the Lagrangian variable, we successfully construct global weak conservative solutions of the μ-Camassa–Holm equation.


Introduction
In this paper, we are concerned with the Cauchy problem of the following periodic μ-Camassa-Holm (μCH) equation [1]: where u(t, x) is a real-valued spatially periodic function and μ(u) = S 1 u(t, x) dx denotes its mean. This equation was introduced in [1] as an integrable equation arising in the study of the diffeomorphism group of the circle, and it can be viewed as a natural generalization of the famous Camassa-Holm (CH) equation [2] u tu txx + 3uu x -2u x u xxuu xxx = 0.
Moreover, μCH equation (1) has some properties similar to CH equation (2). For instance, like CH equation (2), μCH equation (1) is also completely integrable in the sense that it has a Lax pair representation, a bi-Hamiltonian structure, an infinite sequence of conservation laws, and peaked soliton solutions (called peakons). μCH equation (1) also arises geometrically as equations for geodesic flow in the context of the diffeomorphism group of the circle Diff(S 1 ) endowed with a right-invariant Riemannian metric induced by the μ inner product u, v = μ(u)μ(v) Recently, well-posedness and wave breaking of the CH equation have been studied extensively. More precisely, it was shown in [3,4] that the CH equation has a unique solution in C([0, T), H s (R)) with s > 3 2 . It was proved in [5] that the CH equation is locally well posed in Besov spaces C([0, T), B s p,r (R)) with s > max( 3 2 , 1 + 1 p ), 1 ≤ p ≤ +∞, 1 ≤ r < +∞. It was shown in [6][7][8] that the strong solutions of CH equation will blow up in finite time when the slope of initial data was bounded by a negative quantity. Also, well-posedness and wave breaking of the μCH equation were studied and some results about the local and global existence and uniqueness in Sobolev spaces were established in [1]. Moreover, the μCH equation admits a similar phenomenon as the CH equation, that is, the solutions break down in finite time when the initial data satisfy some proper conditions. In particular, it was shown in [9] that μCH equation (1) also admits peakons u( , and φ is extended periodically to the real line. Chen, Lenells, and Liu [10] showed that the above periodic peakons are orbitally stable. In order to study solutions of CH equation (2) after the time of break-down, Bressan and Constantin [11,12] introduce some new variables to resolve the singularities formed by wave breaking, and they successfully obtain a semigroup of global solutions by returning to the original variables. Holden and Raynaud [13,14] also study solutions of CH equation (2) after the formation of singularities, but they introduce a set of different new variables, and this can establish a bijective map between Eulerian and Lagrangian coordinates for CH equation (2). Motivated by the works of Lenells [15], who constructed global weak conservative solutions of the so-called periodic Hunter-Saxton (HS) equation Lee [16] constructs global weak conservative solutions of CH equation (2) by using transformation of variables on the Lagrangian variable η: It is interesting that the author obtains the same global results for CH equation (2) as those presented by Bressan and Constantin [11] and also improves the corresponding results on the persistence of the smoothness of Lagrangian trajectories presented in [17]. More recently, following closely the ideas used in [12,14], Tiğlay [18] studies the periodic Cauchy problem for μCH equation (1) and proves the existence and uniqueness of conservative weak solutions.
In this paper, we investigate the global weak conservative solutions of μCH equation (1). We closely follow the main ideas of Lee [16] in the study of conservative weak solutions of CH equation (2). We show that the wave breaking singularities of μCH equation (1) can be removed by introducing the same transformation (4) about the new Lagrangian variable ρ, even though the structure of the μCH equation is in some sense different from the CH equation and the HS equation. Also, we obtain the global results of the μCH equation due to Tiğlay [18] in much simpler way.
The paper is organized as follows. In Sect. 2, we use the Lagrangian variable ρ defined by (4) to change μCH equation (1) into an abstract ODE system which admits global solutions. In Sect. 3, we construct global weak conservative solutions of the original μCH equation by using this global solution in ρ variables. Moreover, by putting together all the previous results obtained, we give and prove the main result of the paper. Finally, some conclusions and discussions are presented in Sect. 4.

The setting
Firstly, the Cauchy problem for periodic μCH equation (1) can be rewritten in the weak form By taking methods analogous to those in [16], we define the following flow equation in terms of Lagrangian variable η: then Cauchy problem (5) can be written in η variable as follows: where and the kernel g(x) of the elliptic operator Λ -1 is given by g(x) = 1 2 (x 2 -|x|) + 13 12 . Now our main aim is to write Eq. (7) in terms of the new variable ρ defined by (4). For this purpose, we take full advantage of the conserved quantities μ(u) of the μCH equation to finish our work. More precisely, note that μ(u) = 1 0 u(t, x) dx = 1 0 u 0 (x) dx is a constant determined by the initial condition u 0 , so in view of (4) and (6) we have By carrying out exactly similar arguments as those given in [16], we can determine η t in terms of μ, ρ, and ρ t . That is, Also, notice that ρ = √ η x and η tx = 2ρρ t , then we have Next, we begin to rewrite Eq. (7) in ρ variables. By differentiating Eq. (7) with respect to x, we have where we have applied the chain rule and the identity Λ -1 (-∂ 2 x ) = 1μ. For convenience, we denote that so Eq. (10) becomes since η ttx - Thus, we can derive the following Cauchy problem with respect to ρ variables: If we assume that ρ and ρ t are just functions in L 2 (S 1 ) satisfying Eq. (13), then this equation can be used to describe the integral curve of the vector field (ρ, ρ t )). According to the smoothness of f with respect to ρ and ρ t variables, we can conclude that the solutions of (13) are locally well posed.
Proof Firstly, note that It is easy to see that G is smooth with respect to ρ t since it is linear in ρ t . On the other hand, we have and so we can observe that F is also smooth with respect to ρ t . Thus, it implies that the vector field f is smooth with respect to ρ t . Similarly, we can check that the vector field f is smooth with respect to ρ.
Note that G is periodic on S 1 if and only if 1 0 ρ 2 dx = 0. The following proposition shows that we need to restrict ρ to be on the unit sphere U := {ρ ∈ L 2 (S 1 ) : ρ L 2 = 1} if we want to obtain the global well-posedness of Cauchy problem (13).

Proposition 2.2
Let f (ρ, ρ t ) be the vector field on TL 2 (S 1 ) defined by (13). If ρ ∈ U , then This proves that 1 0 ρρ t dx = 0 for all time since the integral is initially zero.
It is well known that theḢ 1 energy 1 0 u 2 x dx is a conserved quantity for μCH equation (1). Next, we show that theḢ 1 energy for the corresponding ρ equation is also conserved.

Proposition 2.4 If ρ ∈ U, then the flow described by (13) exists for all time t.
Proof In view of Hölder's inequality, we have It is easily seen that |2μG -F| is uniformly bounded since ρ L 2 = 1 and ρ t L 2 is uniformly bounded by the energy conservation. This implies that the right-hand side of (13) is uniformly bounded in (ρ, ρ t ). Thus, by Wintner's theorem from the ODE theory [19], we can conclude that the solutions of Cauchy problem (13) exist for all time.

Global weak solutions of the μCH equation
We want to construct global weak solutions for the original μCH equation (5) from the global solution ρ. To do this, we first prove the following important lemma.
Lemma 3.1 Let G be defined by (9).
Proof Note that Denote By differentiating G with respect to t, we have So we only need to show that H(t, 0) + c (t) = 0. However, we have = -H(t, 0), since 1 0 ρ 2 dx = 1 and Φ is periodic with respect to x. Therefore, we have G t = -H. Define From Lemma 3.1, we can see that -H(t, 0) = c (t), where c(t) is defined by (17). So the relation ∂K ∂t = G holds true between K and G. Next, we will show that (K, G) satisfies the second order equation (7).
Following [15,16], we decompose S 1 by A := x ∈ S 1 : K x > 0, i.e., ρ(t, x) > 0 , and meas(Z) = 0. Performing the same argument as in [16], we can obtain that the set N has a measure zero for almost all time. And then we have the following lemma. Proof The proof is very similar to that of Lemma 8 in [16], and hence is omitted.
Lemma 3.2 shows that ρ = 0 is satisfied only on a set of measure zero for almost all time. Notice that K x = ρ 2 , we can derive that K is generically a homeomorphism except for a set of measure zero. In other words, K -1 is well defined for almost all time.
Proof It suffices to show that G t = -H is equivalent to Eq. (7) in the weak sense. Note that K is a diffeomorphism on the set A, so we have In view of that meas(K(A)) = 1, we can observe that the first integral of the above is equivalent to the right-hand side of Eq. (7). Moreover, we have N ρ 2 t dy = 0 for almost all time t from Lemma 3.2. Therefore, for almost all time t, (K, G) satisfies Eq. (18).
Next, we check that the velocity field u satisfying flow equation (6) is well defined in H 1 for all time. More precisely, we can obtain the following proposition. It should be remarked here that the proofs for this proposition are omitted since it is exactly the same argument as in [16]. (7) in the Lagrangian form. Then the velocity field u ∈ C(R + , H 1 (S 1 )) ∩ Lip(R + , L 2 (S 1 )) is well defined by

Proposition 3.4 Let (K, G) be a weak solution of μCH equation
Now, we are in a position to state the main result of this paper. (19) is the global weak solution to the Cauchy problem for periodic μCH equation (5). Moreover, this solution satisfies the following property: S 1 u 2 x (t, x) dx = S 1 (u 0 (x)) 2 dx.

Theorem 3.5 The function u defined by
Proof We want to show that the function u defined by (19) satisfies Eq. (5) in the weak sense. That is, it suffices to prove that for all ϕ ∈ C ∞ 0 (S 1 × R + ), where p = Λ -1 (2μ(u)u + 1 2 u 2 x ). By virtue of (19), we have Note that K t = G and K tx = G x , so we have Thus, in view of that ϕ has a compact support, we can get On the other hand, from Proposition 3.3 we know that • K for almost all t. Also, note that u = G • K -1 and u x = G x K x • K -1 . Therefore, we have So we complete the proof of (20).
Next, we show that S 1 u 2 x (t, x) dx is a conserved quantity for μCH equation (5). Note that K x = ρ 2 and G x = 2ρρ t , then is conserved by virtue of Proposition 2.3. This proves that S 1 u 2 x (t, x) dx = S 1 (u 0 (x)) 2 dx.

Conclusions
In this paper, we successfully construct global weak conservative solutions of the μCH equation by applying the so-called Lagrangian change of variable technique. We believe that this technique could be extended to study other generalized μ-type equations with different nonlocal nonlinearities which have similar remarkable properties as the μCH equation. More recently, by virtue of an isometric embedding of the group of diffeomorphism group, Gallouët and Vialard [20] have rewritten the CH equation as an incompressible Euler equation on the cone manifold. Following this geometric point of view developed in [20], we think that the μCH equation can be also embedded in the incompressible Euler equation on a non-compact Riemannian manifold. We leave these for future research.