The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

and Applied Analysis 3 2. Main Results Firstly, we give some notation. The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ ×R is denoted byC∞ 0 . L L R 1 ≤ p < ∞ is the space of all measurable functions h such that ‖h‖pLp ∫ R |h t, x |pdx < ∞. We define L∞ L∞ R with the standard norm ‖h‖L∞ infm e 0supx∈R\e|h t, x |. For any real number s, H H R denotes the Sobolev space with the norm defined by


Introduction
Camassa and Holm 1 used the Hamiltonian method to derive a completely integrable wave equation 1.1 by retaining two terms that are usually neglected in the small amplitude, shallow water limit.Its alternative derivation as a model for water waves can be found in Constantin and Lannes 2 and Johnson 3 .Equation 1.1 also models wave current interaction 4 , while Dai 5 derived it as a model in elasticity see Constantin and Strauss 6 .Moreover, it was pointed out in Lakshmanan 7 that the Camassa-Holm equation 1.1 could be relevant to the modeling of tsunami waves see Constantin and Johnson 8 .
In fact, a huge amount of work has been carried out to investigate the dynamic properties of 1.1 .For k 0, 1.1 has traveling wave solutions of the form c e −|x−ct| , called peakons, which capture the main feature of the exact traveling wave solutions of greatest height of the governing equations see 9-11 .For k > 0, its solitary waves are stable solitons 6, 11 .It was shown in 12-14 that the inverse spectral or scattering approach was a powerful tool to handle Camassa-Holm equation.Equation 1.1 is a completely integrable infinite-dimensional Hamiltonian system in the sense that for a large class of initial data, the flow is equivalent to a linear flow at constant speed 15 .It should be emphasized that 1.1 gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group see 16,17 , and this geometric illustration leads to a proof that the Least Action Principle holds.It is worthwhile to mention that Xin and Zhang 18 proved that the global existence of the weak solution in the energy space H 1 R without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some conditions on the solution 19 .Coclite et al. 20 extended the analysis presented in 18, 19 and obtained many useful dynamic properties to other equations also see 21-24 .Li and Olver 25 established the local well-posedness in the Sobolev space H s R with s > 3/2 for 1.1 and gave conditions on the initial data that lead to finite time blowup of certain solutions.It was shown in Constantin and Escher 26 that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time.After wave breaking, the solution can be continued uniquely either as a global conservative weak solution 21 or a global dissipative solution 22 .For peakons, these possibilities are explicitly illustrated in the paper 27 .For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water models, the reader is referred to 10, 28-32 and the references therein.
In this paper, motivated by the work in 25, 33 , we study the following generalized Camassa-Holm equation The main tasks of this paper are two-fold.Firstly, by using the Kato theorem for abstract differential equations, we establish the local existence and uniqueness of solutions for 1.2 with any β and arbitrary positive integer N in space C 0, T , H s R C 1 0, T , H s−1 R with s > 3/2.Secondly, it is shown that the existence of weak solutions in lower order Sobolev space H s R with 1 ≤ s ≤ 3/2.The ideas of proving the second result come from those presented in Li and Olver 25 .

Main Results
Firstly, we give some notation.
The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ ×R is denoted by inf m e 0 sup x∈R\e |h t, x |.For any real number s, H s H s R denotes the Sobolev space with the norm defined by where h t, ξ R e −ixξ h t, x dx.For T > 0 and nonnegative number s, C 0, T ; H s R denotes the Frechet space of all continuous H s -valued functions on 0, T .We set In order to study the existence of solutions for 1.2 , we consider its Cauchy problem in the form which is equivalent to

2.3
Now, we state our main results.
Then there exists a T > 0 such that 1.2 subject to initial value u 0 x has a weak solution u t, x ∈ L 2 0, T , H s in the sense of distribution and u x ∈ L ∞ 0, T × R .

Local Well-Posedness
We consider the abstract quasilinear evolution equation Let X and Y be Hilbert spaces such that Y is continuously and densely embedded in X, and let Q : Y → X be a topological isomorphism.Let L Y, X be the space of all bounded linear operators from Y to X.If X Y , we denote this space by L X .We state the following conditions in which ρ 1 , ρ 2 , ρ 3 , and ρ 4 are constants depending on max{ y Y , z Y }.
and A y ∈ G X, 1, β i.e., A y is quasi-m-accretive , uniformly on bounded sets in Y .
ii QA y Q −1 A y B y , where B y ∈ L X is bounded, uniformly on bounded sets in Y .Moreover, iii f : Y → Y extends to a map from X into X is bounded on bounded sets in Y , and satisfies

3.4
Kato Theorem (see [35]) Assume that i , ii , and iii hold.If v 0 ∈ Y , there is a maximal T > 0 depending only on v 0 Y , and a unique solution v to problem 3.1 such that and Q Λ.In order to prove Theorem 2.1, we only need to check that A u and f u satisfy assumptions i -iii .

Lemma 3.1. The operator
3.9 Proofs of the above Lemmas 3.1-3.3can be found in 29 or 31 .
Lemma 3.4 see 35 .Let r and q be real numbers such that −r < q ≤ r.Then

3.10
Lemma 3.5.Let u, z ∈ H s with s > 3/2, then f u is bounded on bounded sets in H s and satisfies

3.12
Proof.Using the algebra property of the space H s 0 with s 0 > 1/2, we have 13 from which we obtain 3.11 .
3.14 which completes the proof of 3.12 .
Proof of Theorem 2.1.Using the Kato Theorem, Lemmas 3.1-3.3,and 3.5, we know that system 2.2 or problem 2.3 has a unique solution

Existence of Weak Solutions
For s ≥ 2, using the first equation of system 2.2 derives where c is a constant depending only on r.
Lemma 4.3.Let s ≥ 2 and the function u t, x is a solution of problem 2.2 and the initial data u 0 x ∈ H s R .Then the following inequality holds For q ∈ 0, s − 1 , there is a constant c, which only depends on m, N, k, a, and β, such that

4.6
For q ∈ 0, s − 1 , there is a constant c, which only depends on m, N, k, a, and β, such that x dx and 4.2 derives 4.5 .Using ∂ 2 x −Λ 2 1 and the Parseval equality gives rise to For q ∈ 0, s − 1 , applying Λ q u Λ q to both sides of the first equation of system 2.3 and integrating with respect to x by parts, we have the identity 1 2

4.9
Abstract and Applied Analysis We will estimate the terms on the right-hand side of 4.9 separately.For the first term, by using the Cauchy-Schwartz inequality and Lemmas 4.1 and 4.2, we have

4.10
Using the above estimate to the second term yields For the third term, using the Cauchy-Schwartz inequality and Lemma 4.1, we obtain

4.12
For the last term in 4.9 , using Lemma 4.1 repeatedly results in

4.13
It follows from 4.9 to 4.13 that there exists a constant c depending only on m, N and the coefficients of 1.2 such that 1 2 Integrating both sides of the above inequality with respect to t results in inequality 4.6 .
To estimate the norm of u t , we apply the operator 1 − ∂ 2 x −1 to both sides of the first equation of system 2.3 to obtain the equation

4.15
Applying Λ q u t Λ q to both sides of 4.15 for q ∈ 0, s − 1 gives rise to x β∂ x u x N dτ.

4.16
For the right-hand side of 4.16 , we have
It follows from Theorem 2.1 that for each ε the Cauchy problem Lemma 4.4.Under the assumptions of problem 4.23 , the following estimates hold for any ε with 0 < ε < 1/4 and s > 0

4.24
where c 1 is a constant independent of ε.
The proof of this Lemma can be found in Lai and Wu 33 .

4.25
Letting p > 0 be an integer and multiplying the above equation by u x 2p 1 and then integrating the resulting equation with respect to x yield the equality

4.26
Applying the H ölder's inequality yields where , integrating both sides of the inequality 4.28 with respect to t and taking the limit as p → ∞ result in the estimate Using the algebra property of H s 0 R with s 0 > 1/2 yields u ε H 1/2 means that there exists a sufficiently small δ > 0 such that in which Lemma 4.3 is used.Therefore, we get From 4.30 and 4.32 , one has

4.33
From Lemma 4.4, it follows from the contraction mapping principle that there is a T > 0 such that the equation has a unique solution W ∈ C 0, T .Using the Theorem presented at page 51 in 25 or Theorem 2 in Section 1.1 presented in 37 yields that there are constants T > 0 and c > 0 independent of ε such that u x L ∞ ≤ W t for arbitrary t ∈ 0, T , which leads to the conclusion of Lemma 4.5.Using Lemmas 4.3 and 4.5, notation u ε u and Gronwall's inequality results in the inequalities u ε H q ≤ C T e C T , u εt H r ≤ C T e C T ,

4.35
where q ∈ 0, s , r ∈ 0, s − 1 and C T depends on T .It follows from Aubin's compactness theorem that there is a subsequence of {u ε }, denoted by {u ε n }, such that {u ε n } and their temporal derivatives {u ε n t } are weakly convergent to a function u t, x and its derivative u t in L 2 0, T , H s and L 2 0, T , H s−1 , respectively.Moreover, for any real number R 1 > 0, {u ε n } is convergent to the function u strongly in the space L 2 0, T , H q −R 1 , R 1 for q ∈ 0, s and {u ε n t } converges to u t strongly in the space L 2 0, T , H r −R 1 , R 1 for r ∈ 0, s − 1 .Thus, we can prove the existence of a weak solution to 2.2 .
Proof of Theorem 2.2.From Lemma 4.5, we know that {u ε n x } ε n → 0 is bounded in the space L ∞ .Thus, the sequences {u ε n } and {u ε n x } are weakly convergent to u and u x in L 2 0, T , H r −R, R for any r ∈ 0, s − 1 , respectively.Therefore, u satisfies the equation 4.36 with u 0, x u 0 x and g ∈ C ∞ 0 .Since X L 1 0, T × R is a separable Banach space and {u ε n x } is a bounded sequence in the dual space X * L ∞ 0, T × R of X, there exists a subsequence of {u ε n x }, still denoted by {u ε n x }, weakly star convergent to a function v in L ∞ 0, T × R .It derives from the {u ε n x } weakly convergent to u x in L 2 0, T × R that u x v almost everywhere.Thus, we obtain u x ∈ L ∞ 0, T × R .

Lemma 4 . 5 .
If u 0 x ∈ H s R with s ∈ 1, 3/2 such that u 0x L ∞ < ∞.Let u ε0 bedefined as in system 4.23 .Then there exist two positive constants T and c, which are independent of ε, such that the solution u ε of problem 4.23 satisfies u εx L ∞ ≤ c for any t ∈ 0, T .Proof.Using notation u u ε and differentiating both sides of the first equation of problem 4.23 or 4.15 with respect to x give rise to