A Note on the Generalized Camassa-Holm Equation YunWu

We study the generalized Camassa-Holm equation which contains the Camassa-Holm (CH) equation and Novikov equation as special cases with the periodic boundary condition. We get a blow-up scenario and obtain the global existence of strong and weak solutions under suitable assumptions, respectively. Then, we construct the periodic peaked solutions and apply them to prove the ill-posedness inH with s < 3/2.


Introduction
In this paper we study the global strong and weak solutions to the generalized Camassa-Holm equation with periodic boundary condition: (0, ) =  0 () ,  ∈ T, where  ≥ 1,  ∈ N, and T = R/2Z.When  = 1, (1) reduces to the the well-known CH equation: The CH equation was derived independently by Fokas and Fuchssteiner in [1] and by Camassa and Holm in [2].
Fokas and Fuchssteiner derived (3) in studying completely integrable generalizations of the KdV equation with bi-Hamiltonian structures, while Camassa and Holm proposed (3) to describe the unidirectional propagation of shallow water waves over a flat bottom.As shown in [2], the CH equation is completely integrable and possesses an infinite number of conservation laws.Moreover, the CH equation is such an equation that exhibits both phenomena of soliton interaction (peaked soliton solutions) and wave breaking (the solution remains bounded while its slope becomes unbounded in finite time [3]), while the KdV equation does not model breaking waves [4].In fact, wave breaking is one of the most intriguing long-standing problems of water wave theory [5].The essential feature of CH should be pointed out: the fact that the traveling waves have a peak at their crest is exactly like for the waves of greatest height solutions of the governing equations for water waves (see [6][7][8] for the details).
From a mathematical point of view the Camassa-Holm equation is well studied and a series of achievements had been made.Constantin [9] and Misiołek [10] investigated the Cauchy problem for the periodic Camassa-Holm equation.Constantin et al. [3,[11][12][13][14] studied the wave breaking of the Cauchy problem for the CH equation.Recently, Jiang et al. gave a new and direct proof for McKean's theorem in [15].Xin and Zhang [16] proved that (3) has global weak solutions for initial data in  1 (R).Bressan and Constantin developed a new approach to the analysis of the CH equation and proved the existence of the global conservative and dissipative solutions in [17,18].Holden and Raynaud [19,20] also obtained the global conservative and dissipative solutions.The large time behavior of the CH equation was firstly established in [21].In [22], Himonas et al. studied the persistence properties and infinite propagation speed for the CH equation.
In 2009, Novikov [23] found a new integrable equation: It is derived that (4) possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities and admits exact peaked solutions (, ) = ±√ −|−| with  > 0 [24,25], as 2 Journal of Function Spaces well as the explicit formulas for multipeakon solutions [25,26].By using the Littlewood-Paley decomposition and Kato's theory, the well-posedness of the Novikov equation has been studied in Besov spaces   , (R) and in the Sobolev space   (R) (see [27,28]).Wu and Yin [29] established some results on the existence and uniqueness of global weak solutions to the Novikov equation.Jiang and Ni [30] established some results about blow-up phenomena of the strong solution to the Cauchy problem for (4).For the periodic boundary condition case, Tı g lay [31] proved that for  > 5/2 the periodic Novikov equation is locally well-posed in   (T).Later the range of regularity index of local well-posedness was extended to  > 3/2 in [32]; furthermore, it is shown that the solution maps for both periodic boundary value problem and Cauchy problem of the Novikov equation are not uniformly continuous from any bounded subset in   into ([0, ];   ).When  < 3/2, Grayshan [33] proved that the properties of the solution map for (4) are not (globally) uniformly continuous in Sobolev spaces   .For the nonuniform dependence and ill-posedness results in Besov spaces, we refer to [34][35][36][37].
In this paper we consider the generalized CH equation (1) with  ≥ 3. Let  =  −   , then (1) takes the form of a quasilinear evolution equation of hyperbolic type: Applying the operator (1 −  2  ) −1 , (1) can be expressed as the following nonlocal form: From the above equation, we can view (1) as a nonlocal perturbation of the Burgers-type equation: We recall the following results in [38] for (1) and (2).
In this paper we use the following notations.We use ≲ to denote estimates that hold up to some universal constant which may change from line to line but whose meaning is clear from the context. ≈  stands for  ≲  and  ≳ .All function spaces are over T and we drop T in all function spaces if there is no ambiguity.For linear operators  and , we denote [, ] =  − .The Fourier transform of the function () is defined by f() = F() = ∫ T  − (),  ∈ Z.The inverse Fourier transform is given by The operator   = (1 −  2  ) /2 for any real number  is defined by D () = (1 +  2 ) /2 f().  (T) is the standard Soblev space on T whose norm is defined by For each  ∈ (0, 1],   stands for the Friedrichs mollifier defined by where * stands for the convolution.Here   () = F −1 ( ĵ()) and () is a Schwartz function satisfying 0 ≤ ĵ() ≤ 1 for all the  ∈ R and ĵ() = 1 for any  ∈ [−1, 1].
We will also use another mollifier.Define where the constant  > 0 is chosen so that ∫ R () = 1.
For  ∈ (0, 1], we set   () = (1/)((1/)).To define the mollifier J , we first let  Ω be the characteristic function on Ω ⊂ R and When  < 3/2, the solution map is not uniformly continuous, and we establish the ill-posedness as follows.We refer to [33,36,39] for the ill-posedness results for the CH equation, Novikov equation, and the b-family equation.
Theorem 5.If  < 3/2, then (1) and ( 2) are ill-posed in   in the sense that the solution map is not uniformly continuous from   into ([0, ];   ) for any  > 0.More precisely, there exist two sequences of weak solutions   and V  in   of ( 1) and ( 2) such that, for any  ∈ [0, ], there hold the following estimates: where  1 (),  2 () only depend on .
We outline the rest of the paper.In the next section, we give some preliminaries.We deal with the blow-up criterion and prove Theorems 2 and 3 in Section 3. In Section 4, we study the global weak solution and prove Theorem 4. We demonstrate Theorem 5 in Section 5.

Blow-Up Criterion and Global Existence of Strong Solutions
The aim of this section is to prove Theorems 2 and 3 which show that the solution blows up only when the slope of the wave blows up and the solution exists globally if  0 =  0 − 0 does not change sign.Rewrite (1) as the following form: First, we have the following lemma.
Lemma 10.Let  > 3/2, 0 <  ≤ −1, and (, ) is a solution to ( 1) and ( 2) with  0 ∈   .Then there exists a constant   such that Proof.Since the term −  ( −1  2  ) +  −1     is only in  −2 , we cannot apply   to either side of (26) when  > −2.So we apply the operator     to (26), multiply both sides of the resulting equation by 2    , and integrate over T to obtain where we used  2  = 1 −  2 and We now estimate   , 1 ≤  ≤ 4, respectively.For  1 , we first note that   is self-adjoint, then commute the operator   with   , and use ( 21) and ( 22) to get By using the Cauchy-Schwarz inequality, Lemmas 6, 7, and 8, integration by parts, and ( 23), we have Therefore In the same way,  2 can be estimated as For  3 , we use the Cauchy-Schwarz inequality, Lemma 7, and (23) For the estimate for  4 , we have Similar to (34), Hence we obtain that Combining ( 32), ( 33), (34), and (37) yields Integrating both sides with respect to  results in Let  tend to 0; we get (27) and therefore complete the proof of this lemma.
Remark 11.Take   =  + 1 in ( 27); then for   ∈ (1, ], we have This estimate will be also used in the proofs of Lemma 20. Let  be the solution to problem (1) and ( 2).We define  ⋆  0 to be its lifespan, Then the following alternative property holds: Proof of Theorem 2. From (1) we can deduce that which implies that Taking  =  − 1 in Lemma 10 and using ‖  ()‖  ∞ is finite, then ‖()‖   is bounded and the case (ii) in (42) would not occur, which implies that  can be extended beyond  ⋆  0 .On the other hand, if  does not blow up, then ‖()‖   is bounded on [0, ] for any  > 0.
. Thus we complete the proof.
Remark 12. Actually, we can prove a more precise blow-up criterion for sufficiently regular solutions to (1); that is, if  0 ∈   , with  ≥ 2, then the solution (, ) blows up if and only if lim inf In fact, multiplying both sides of ( 5) by  =  −   and integrating over T, we have where  > 0. By Gronwall's inequality, we obtain that the  2 norm of  is bounded on [0,  ⋆  0 ) which is equivalent to the boundedness of ‖‖  2 since ‖‖  2 ≤ ‖‖  2 ≤ 2‖‖  2 .On the other hand, since  ≥ 2 and  −1 is an algebra, we know that Remark 13.The new blow-up criterion (46) is better than the one obtained in Theorem 2, which is quite common for nonlinear hyperbolic PDE (see [5,44]).For the Camassa-Holm and related equations, the blow-up criterion is often written as lim inf which is different from (46).In fact, if  blows up and  −1 < 0, then   → +∞.
Thus   is positive and (, ) is an increasing diffeomorphism of R before the blow-up time.
We note that the Green's function of   2) satisfies Proof.We discuss the following results for the case  ≥ 3; the lemma follows by using a simple density argument.By (53) and the positivity of  T , we know  keeps the sign of  0 , and hence  =  T *  keeps the sign of  0 .Therefore, employing (59), we obtain that, for That is, if  0 does not change sign, then |  | ≤ ||.By  1 →  ∞ , we obtain the desired estimate.
Proof of Theorem 3. Combining Lemma 17 and Theorem 2, we have Theorem 3.

Global Weak Solutions
In this section, we prove that (1) and ( 2) have a unique global weak solution in lower-order Sobolev space   , 1 <  ≤ 3/2.First, we establish some estimates for the strong solutions to (1) with  > 3/2.
To show the existence of weak solution to (1) and ( 2) in lower-order Sobolev space   with 1 <  ≤ 3/2, we will consider the following problem first: where () is given in (17) and J is the mollifier introduced in ( 11)- (13).It follows from Proposition 1 that, for each  > 0, there exists a   > 0 such that the above problem has a unique solution   (, ) ∈  ∞ ([0,   ];  ∞ ).
Proof.We first note that, by the construction of J , if  ≥ 0 (or  ≤ 0), then J  ≥ 0 (or J  ≤ 0).Thus, if (1 −  2  ) 0 does not change sign, so does Hence Theorem 3 yields that   (, ) is a global solution.

Existence of Global Weak
Solution.Now we prove the existence of a global weak solution to problem (16).
Proof of Theorem 4. With the aid of Propositions 21 and 22, we complete the proof of Theorem 4.

Existence of Periodic Peaked Solutions
where   is the periodic Dirac delta function at  =  (mod 2).Thus Direct computation shows that Therefore, we have Using (80), we can compute that Similarly, Putting these results together, we see that Therefore, when  < 3/2, we have Hence (, ) =  1/ sech  cosh  ∈   with  < 3/2.

Proof of Theorem 5.
By Proposition 23, we know that (1) has two sequences of periodic peakon (weak) solutions: where   ,   ∈ R are constants velocity which will be specified later.By (88), we know Note  < 3/2; when  = 0, we have where
Remark 24.From Theorems 2 and 5, we see  = 3/2 is the critical index of regularity for well-posedness in Sobolev space   for (1) and ( 2).