ON THE CAUCHY PROBLEM FOR A HIGHER-ORDER µ -CAMASSA-HOLM EQUATION

. In this paper, we study the Cauchy problem of a higher-order µ -Camassa-Holm equation. We ﬁrst establish the Green’s function of ( µ − ∂ 2 x + ∂ 4 x ) − 1 and local well-posedness for the equation in Sobolev spaces H s ( S ), s > 72 . Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in H s ( S ), s ≥ 4. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump disconti- nuities in the third derivatives.


School of Mathematical & Statistical Sciences, University of Texas-Rio Grande Valley
Texas 78539, USA

(Communicated by Adrian Constantin)
Abstract. In this paper, we study the Cauchy problem of a higher-order µ-Camassa-Holm equation. We first establish the Green's function of (µ − ∂ 2 x + ∂ 4 x ) −1 and local well-posedness for the equation in Sobolev spaces H s (S), s > 7 2 . Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in H s (S), s ≥ 4. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

Introduction. The Camassa-Holm equation
was introduced in [2] to model the unidirectional propagation of shallow water waves over a flat bottom. u(t, x) represents the fluid velocity at time t and in the spatial direction x. It is a re-expression of the geodesic flow both on the diffeomorphism group of the circle [14] and on the Bott-Virasoro group [31]. Eq.(1.1) has a bi-Hamiltonian structure [25] and is completely integrable [2,9]. Moreover, it has been extended to an entire integrable hierarchy including both negative and positive flows and shown to admit algebro-geometric solutions on a symplectic submanifold [36]. The Cauchy problem of (1.1), in particular its well-posedness, blow-up behavior and global existence, have been well-studied both on the real line and on the circle, e.g., [1,8,10,11,12,13,16,17,18,19,26,27,33,43]. Eq.(1.1) with weakly dissipative term was studied in [42].

FENG WANG, FENGQUAN LI AND ZHIJUN QIAO
Equation (1.1) has been recently generalized into some µ-versions and higher order forms. Khesin et al. in [30] introduced a µ-version of Camassa-Holm equation as follows where u(t, x) is a time-dependent function on the unit circle S = R/Z and µ(u) = S udx denotes its mean. This equation describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal with external magnetic filed and self-interaction. Moreover, Eq.(1.2) is also an Euler equation on D s (S) and it describes the geodesic flow on D s (S) with the right-invariant metric given at the identity by the inner product [30] f, g µ = µ(f )µ(g) + S f (x)g (x)dx . In [30,32], the authors showed that Eq.(1.2) is bi-Hamiltonian and admits both cusped and smooth travelling wave solutions which are natural candidates for solitons. The orbit stability of periodic peakons was studied in [3]. A weakly dissipative µ-Camassa-Holm equation was studied in [34].
For the higher order Camassa-Holm equation, [5,15] considered the following equation which describes exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. In [5], Coclite et al. established the existence of global weak solutions and presented some invariant spaces under the action of the equation. Tian et al. [40] investigated the global existence of strong solutions to Equation (1.3) with k = 2. Ding and Lv [21] studied the existence of global conservative solutions to (1.3). Recently, Coclite and Ruvo [7] showed the convergence of the solution to (1.3). Ding et al. [20,22] discussed traveling solutions of (1.3) and their evolution properties.
In this paper, we will consider a µ-version of (1.3) with k = 2 as follows m t + 2mu x + m x u = 0, m = (µ − ∂ 2 x + ∂ 4 x )u, (1.4) where u(t, x) is a time-dependent spatially periodic function on the unit circle S = R/Z and µ(u) = S udx denotes its mean. We first give the Green's function of the operator (µ − ∂ 2 x + ∂ 4 x ) −1 and local well-posedness of (1.4). Then we show the global existence of strong solutions to (1.4). Next, for any T 0 > 0 and s ≥ 4, we prove that the data-to-solution map is Hölder continuous from any bounded subset of H s (S) into C([0, T 0 ]; H r (S)) with 0 ≤ r < s, but is not uniformly continuous from any bounded subset of H s (S) into C([0, T 0 ]; H s (S)). Motivated by the recent work [6], we establish the existence of global weak solutions in H 2 (S) without using an Oleȋnik-type estimate (see [4,43]), which is not easy to be verified in numerical experiment. Lastly, we show the existence of single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.
We noticed that Mclachlan and Zhang [35] have studied another higher-order Camassa-Holm equation as follows which is derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to the H k metric. A µ-version of (1.5) with k = 2, first proposed in [24], was very recently studied in our recent paper [41], in which we also established the Green's function of the operator (µ − ∂ 2 x ) −2 and showed it admits single peakon solutions, but they are completely different from the results in the present paper.
The rest of the paper is organized as follows. In Section 2, the Green's function of the operator (µ − ∂ 2 x + ∂ 4 x ) −1 and local well-posedness for (1.4) with initial data in H s (S), s > 7 2 , are established. In Section 3, we show the global existence of strong solutions. The Hölder continuity and non-uniform continuity of solution map for the equation are established in Section 4. In Section 5, we show the global existence of weak solutions. The existence of single peakon solutions is proved in Section 6.

2.
Preliminaries. In this section, we will give the Green's function of the operator For a periodic function g on the circle S = R/Z, we have where δ 0 (k) = 1, k = 0, 0, k = 0.

FENG WANG, FENGQUAN LI AND ZHIJUN QIAO
Since the Green's functions of (µ − ∂ 2 the Green's function g(x) is given by and is extended periodically to the real line, that is where [x] denotes the largest integer part of x. The graph of g 1 (x) − g µ (x) can be seen in Fig.3 in [32]. Note that µ(g) = 1.

the following identity holds
For any s ∈ R, H s (S) is defined by the Sobolev space of periodic functions where the pseudodifferential operator

We can check that
x is an isomorphism between H s (S) and H s−4 (S). Moreover, when w ∈ H r+j−4 (S) for j = 0, 1, 2, 3, we have On the other hand, integrating both sides of Eq.(2.3) over S with respect to x, we obtain d dt µ(u) = 0.
Then it follows that µ(u) = µ(u 0 ) := µ 0 . Thus, Eq.(2.3) can be rewritten as Applying the Kato's theorem [28], one may follow the similar argument as in [34] to obtain the following local well-posedness result for Eq.(2.4). Moreover, the solution depends continuously on the initial data, and T is independent of s.
Since u(t, ·) ∈ H s (S) ⊂ C 2 (S) for s > 7 2 , and S u x dx = 0, Corollary 1 and (2.5) imply that It then follows that which implies that Hence, we get This completes the proof of the lemma.
3. Global existence of strong solution. In this section, we present the global existence of strong solution to Eq.(2.4). Firstly, we will give some useful lemmas.
Lemma 3.1. (see [29]) If r > 0, then H r (S) ∩ L ∞ (S) is an algebra. Moreover, where c r is a positive constant depending only on r.
Lemma 3.2. (see [29]) If r > 0, then where Λ r = (1 − ∂ 2 x ) r/2 and c r is a positive constant depending only on r. Lemma 3.3. (see [23,38]) If f ∈ H s (S) with s > 3 2 , then there exists a constant c > 0 such that for any g ∈ L 2 (S) we have in which for each ε ∈ (0, 1], the operator J ε is the Friedrichs mollifier defined by

1)
where j ε (x) = 1 ε j( x ε ) and j(x) is a nonnegative, even, smooth bump function supported in the interval . Now we establish the global existence of strong solution to Eq.(2.4). 7 2 , and let T be the maximal existence time of the solution u to Eq.(2.4) with the initial data u 0 . Then there exists a constant c > 0 depending on s and u 0 such that Proof. Note that the product uu x only has the regularity of H s−1 (S) when u ∈ H s (S). To deal with this problem, we will consider the following modified equation , then multiplying the resulting equation by Λ s J ε u and integrating with respect to x ∈ S, we obtain In what follows next we use the fact that Λ s and J ε commute and that J ε satisfies the properties Let us estimate the first term of the right hand side of (3.3).
where we have used Lemma 3.2 with r = s and Lemma 3.3.
Here and in what follows, we use " " to denote inequality up to a positive constant. Furthermore, we estimate the second term of the right hand side of (3.3) in the following way , where we have used Lemma 3.1 and (2.1).
Since u(t, ·) ∈ H s (S) ⊂ C 2 (S) for s > 7 2 , and S u xx dx = 0, Corollary 1 implies that Thus, Combining (3.4) and (3.5) and using Lemma 2.3, we have Letting ε → 0, we get , where c is a constant depending on s and u 0 . An application of Gronwall's inequality yields , which completes the proof of the theorem.
Note that in Lemma 2.3 we have Since u ∈ H 5 (S) ⊂ C 4 (S) and S u xxx dx = 0, Corollary 1 implies that On the other hand, we can deduce the following estimate from Eq.(2.4) which ensures that the solution u does not blow up in finite time, that is, T = ∞. This completes the proof of Theorem 3.5.
4. Non-uniform dependence on initial data. In this section, we will first give an estimate of the solution size in time interval [0, T 0 ] for any fixed T 0 > 0, and then we show that, for any s ≥ 4, the data-to-solution map is Hölder continuous from any bounded subset of H s (S) into C([0, T 0 ]; H r (S)) with 0 ≤ r < s, but is not uniformly continuous from any bounded subset of H s (S) into C([0, T 0 ]; H s (S)).
Similar as the proofs of in [41], we can obtain the following Lemma 4.1 and Theorem 4.2.
Then, for any fixed T 0 > 0, we have , for all u(0), w(0) ∈ B(0, h) and u(t), w(t) the solutions corresponding to the initial data u(0), w(0), respectively. The constant c depends on s, r, T 0 and h.
Next, we prove that the data-to-solution map is not uniformly continuous. Firstly, we will recall some useful lemmas. [27]) Let σ, α ∈ R. If n ∈ Z + and n 1, then Relation is also true if cos(nx − α) is replaced by sin(nx − α).
Lemma 4.5. (see [27]) If r > 1 2 , then there exists a constant c r > 0 depending only on r such that where ω is in a bounded subset of R and n ∈ Z + . Now we compute the error of the approximate solutions. Note that and Since A −1 µ commutes with ∂ x , we have By (2.1) and Lemma 4.3, for n 1, we have Thus, for the error F := 5 i=1 F i of the approximate solution, we have the following estimate.
Lemma 4.6. If ω is bounded, then for n 1, we have In particular, if s > 1+σ 2 , then where r s > 0 and

4.2.
Difference between approximate and actual solutions. Let u ω,n be the solution of Eq.(2.4) with initial data given by the approximate solution u ω,n evaluated at time zero. That is, u ω,n solves the following Cauchy problem By Theorem 2.1, we know that u ω,n is the unique solution of (4.2) and exists globally in time. To estimate the difference between the approximate and actual solutions, we let v = u ω,n − u ω,n , then for t > 0 and x ∈ S, v satisfies the following Cauchy problem Proof. Applying Λ σ to both sides of (4.3), multiplying the resulting equation by Λ σ v and integrating it with respect to x, we obtain By Hölder inequality, we know For G 2 , we have where we have used integrating by parts, the Sobolev imbedding theorem and Lemma 4.4.

4.3.
Non-uniform dependence. The following theorem is our main result in this section.  Since 2s − σ ≥ s ≥ 4, applying Lemma 4.7 and the interpolation inequality  Therefore, by (4.4) and (4.5), we know which completes the proof.

Global existence of weak solution.
In this section, we establish the existence of global weak solution in H 2 (S). Firstly, the Cauchy problem (2.4) can be rewritten as follows The main result of this section is as follows.
Theorem 5.2. Let p > 2. For any u 0 ∈ H 2 (S) satisfying ∂ 2 x u 0 ∈ L p (S), the Cauchy problem (5.1) has an admissible global weak solution in the sense of Definition 5.1.
Then (5.5) holds for all 0 ≤ t < T . By Corollary 1, (5.4) and (5.5), we obtain that ). This in turn implies that Next, we prove T = ∞, that is, the solution u ε exists globally for each ε > 0. Similar to the proof of Theorems 3.4-3.5, we can show that the H s (S)-norm of u ε (t, ·) does not blow up on [0, T ) if ∂ 3 x u ε (t, ·) L ∞ (S) < ∞ holds. By Corollary 1, x u ε (t, ·) L 2 (S) , so we only need to derive an a priori estimate on ∂ 4 x u ε L 2 (S) . Applying the operator A µ to Eq.(5.3), then multiplying both sides by A µ u ε and integrating over S with respect to x, we get By Gronwall's inequality and the above estimates of u ε L ∞ (S) and ∂ x u ε L ∞ (S) , we know, for each ε > 0 and t ∈ [0, T ), there exists a constant C(ε, u 0 ) > 0 depending on ε and u 0 such that ∂ 4 x u ε (t, ·) L 2 (S) ≤ C(ε, u 0 ). Thus, we have T = ∞, which completes the proof of the lemma.

Precompactness.
In this subsection, we are ready to obtain the necessary compactness of the viscous approximation solutions u ε (t, x).
For convenience, we denote P ε = P 1,ε + P 2,ε , where P 1,ε , P 2,ε are defined by Lemma 5.4. Assume u 0 ∈ H 2 (S). For each t ≥ 0 and ε > 0, the following inequalities hold Here and in what follows, we use C 0 to denote a generic positive constant, independent of ε, which may change from line to line.
Next we turn to estimates of time derivatives.
Lemma 5.5. Assume u 0 ∈ H 2 (S). For each T, t > 0 and 0 < ε < 1, the following inequalities hold Proof. By the first equation of (5.3) and Lemmas 5.3-5.4, we have Differentiating the first equation of (5.3) with respect to x, one obtains Moreover, differentiating the first equation of (5.3) with respect to x two times, we have

FENG WANG, FENGQUAN LI AND ZHIJUN QIAO
and then By the definition of P 1,ε , we know By Lemmas 5.3-5.4, (5.6), (5.7) and Young's inequality, we get T 0 which completes the proof of Lemma 5.5.
Lemma 5.6. Let u 0 ∈ H 2 (S) and ∂ 2 x u 0 ∈ L p (S) for some p > 2. Then the following inequality holds Multiplying the above equation by pq ε |q ε | p−2 , we have

By Lemmas 5.3-5.4 and Young's inequality, we know
The Gronwall inequality implies the desired result.
Here and in what follows, we use overbars to denote weak limits in spaces to be understood from the context. Lemma 5.8. The following inequality holds in the sense of distributions Proof. Taking ξ ∈ C 2 (R) convex and multiplying (5.8) by ξ (q ε k ), we have . In particular, we can use the entropy q → (q + ) 2 /2 and get ∂ t Letting k → ∞, we have The rest of the proof is the same as in [41].