GLOBAL WEAK SOLUTIONS TO THE CAMASSA-HOLM EQUATION

The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition u0(x) has end states u±, which has much weaker constraints than that u0(x) ∈ H(R) discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption u− < u+.

1. Introduction. In this paper, we are concerned with the global existence of weak solutions to the Camassa-Holm equation with initial data u(0, x) = u 0 (x) → u ± as x → ±∞, (1.2) which is formally equivalent to a dispersive shallow water equation [1]: where u is the fluid velocity in the x direction(or equivalently the height of water's free surface above a flat bottom). The equation (1.1) is completely integrable (see [15], [8] for the periodic case and [3], [10], [13] for the non-periodic case). A few nonlinear dispersive and wave equations are lucky enough to be completely integrable in the sense that there exist a Lax pair formulation of the equation. This means in particular that they enjoy infinitely many conservation laws. In many cases these conservation laws provide control on high Sobolev norms which seems to be quite an exceptional event. For the general discussion on complete integrability in infinite dimensions, we refer to [19] for a detailed exposition. Quite a number of PDEs have been discovered to be completely integrable. However, most of them still remain obscure due to the lack of any physical significance. The equation which was discovered by Funchssteiner and Fokas [20], has enjoyed such obscurity. But a few years ago, this equation was rederived by Camassa and Holm [6] using an asymptotic expansion directly in the Hamiltanian for Euler's equations in the shallow water regime. They showed that (1.4) is Bi-Hamiltanian, i.e., it can be expressed in Hamiltanian form in two different ways. The novelty of Camassa and Holm's work was that they gave a physical derivation for (1.4) and showed that, for the special case K = 0, (1.4) possessed a solitary waves of the form c exp(−|x − ct|) with discontinuous first derivatives, which they named "peakon"(travelling wave solutions with a corner at their peak). More importantly, the peakons are orbitally stable (cf. [17])which means that the shape of the peakons is stable so that these wave patterns are physically recognizable, moreover, these peakons are solitons (cf. [16], [4]). Another feature of the Camassa-Holm equation is that it can be treated as a generalization of the Benjamin-Bona-Mahoney(BBM) equation or the Korteweg-de Vries(KdV) equation in some sense(See [6]). These three various equations all gave the good and consistent approximation for the full inviscid water wave equation in the small-amplitude and shallow-water regime. However, the Camassa-Holm equation has several important features that distinguish it from the BBM and KdV equations. Namely, while all solutions to BBM and KdV are global, the Camassa-Holm equation has global smooth solutions as well as smooth solutions that blow up (cf. [12]). Moreover, the only way singularities can develop in a solution corresponding to a smooth initial data decaying at infinity is in the form wave breaking: the slope u x becomes unbounded while u stays bounded (cf. [9]) (this elaborate statement can be also found in [2] and [30]). Since the physical significance that (1.4) exhibits, which was discovered by Camassa and Holm, (1.4) has attracted a broad interest from researchers(see [1,3,4,11,15]). Cooper and Shepard [18] derived a variational approximation to the solitary waves of (1.4) for general K. In [7], the numerical solutions of time-dependent form and a discussion of the Camassa-Holm equation as a Hamiltanian system was presented. Boyd [2] derived a perturbation series for general K which converges even at the peak limit and gave three analytical representations for the spatially periodic generalization of the peakon called "Coshoidal wave". In [26], zero curvature formulation are given for the "dual hierarchies" of standard soliton equation hierarchies including the Camassa-Holm equation hierarchy. As pointed out exactly in [14], (1.1) represents the equation for geodesics on the diffeomorphism group. In the paper [30], Xin and Zhang obtained the global-in-time existence of weak solutions to Camassa-Holm equations for the special case K = 0 with initial data u(0, x) = u 0 (x) ∈ H 1 (R). Recently, Bressan and Constantin in [5] obtained the unique global conservative solutions of the Camassa-Holm equation with initial data in H 1 . We observe that the assumption u 0 (x) ∈ H 1 (R) implies u 0 (x) → 0 as x → ±∞, which is a rigorous constraint in applications. The aim of this paper is to establish the global existence of the weak solution to problem (1.1)-(1.2) with initial data u 0 (x) → u ± as x → ±∞, where the initial data with end states u ± has much weaker constraints than that u 0 (x) ∈ H 1 (R). Toward this end, we assume that the limits u ± of the initial data u 0 (x) at x = ±∞ satisfy u − < u + , i.e., a hyperbolic wave is a rarefaction wave. Under this circumstance, we perturb the Cauchy problem (1.1), (1.2) around the rarefaction wave w R (x/t) which satisfies the Riemann problem (2.1). Then we reformulate the Cauchy problem (1.1), (1.2) to a new Cauchy problem (3.1) and prove the global existence of a weak solution to this new problem, and thus a global weak solution of original problem (1.1), (1.2) follows. Moreover, we study the asymptotic behavior of the solution of problem (1.1), (1.2) and show that the solution tends to a rarefaction wave as t → ∞.
Before giving the precise statements of the main results, we introduce the definition of a weak solution to the Cauchy problem (1.1)-(1.2) similarly as in [30]: where C is a positive constant depending only on u + , u − and φ(t, x) satisfies Here σ > 0 is an arbitrary constant and K q is chosen such that K q ∞ 0 (1+y 2 ) −q dy = 1 for each q > 1/2.
(2) u(t, x) satisfies equation (1.1) in the sense of distributions and takes on the initial data pointwise.
The main result of this paper is as follows.
Theorem 1.1 can be regarded as the extension of Theorem 1.2 in [30] as u − = u + = 0. Our main goal is to study the stability of the simple wave to the Camassa-Holm Equations. That is, one shall be interested in the following initial-boundary problem (1.7) Since the solution of (1.7) has a boundary at x = 0, the signs of the characteristic speeds u ± divide the asymptotic state into five cases (c.f. [23]): (1.8) When u − < 0 and u + = 0, we observe that φ(x) = u − e −x is a stationary solution to (1.7). The interesting problem is to study the asymptotic behavior of solutions to the system (1.7), i.e., one can show that the initial-boundary problem (1.7) admits a unique global solution u(t, x) which converges, as t → +∞, i) to the stationary solution or ii) to the rarefaction wave of the Burgers equation. However, the key point is to obtain the existence of the weak solutions. This is the main purpose of this paper. The stability of the simple wave to the Camassa-Holm Equations will be discussed in further work.
The main idea of proving Theorem 1.1 closely follows the method developed by Xin-Zhang in [30]. The major difference is that our assumption on initial datum is not in H 1 (R), not even in L 2 (R). One key observation is that we can perturb the initial datum with a rarefaction wave define by (2.1) because of u − < u + . The difference quantity, v, between the solution u to the original problem and the rarefaction wave φ satisfies another new "shallow-water-like" equation (see (3.1)). The initial datum of this difference converges to zero as |x| → +∞. The crucial element is to show that this new problem has a global weak solution. Note that the rarefaction wave φ is not in L 1 (R) for any t > 0, although it is smooth enough. Some extra efforts have to be made to deal with the complexity of the appearance of φ (when φ ≡ 0 this new problem is nothing but the problem in [30]). We obtain the global existence result by using the vanishing viscosity method as performed in [30]. It turns out that the Young measures play a role in passaging limits of the approximate viscous solutions v ε . A further question can be addressed: is this global weak solution u close to the rarefaction wave φ in some sense? Indeed, the positivity of ∂ x φ and the estimates for the approximation v ε gives us the integrability of v(·, x) for a.e. x. This, combined with the control of ∂ t v, in turn shows that v(t, x), i.e., u(t, x) − φ(t, x) converges to zero as t → +∞ for all x.
The rest of this paper is organized as follows. In section 2 we establish some preliminary estimates for the smooth rarefaction wave φ. In section 3, we reformulate the original problem to a new equivalent Cauchy problem and establish the global existence of this new problem with viscosity. In section 4, we show the existence of a global existence of problem (1.1) and (1.2) and examine the asymptotic behavior of solutions.
Notation: Hereafter, we use C to denote generic constants without any confusion, which may change from line to line. When the dependence of the constant on some index or a function is important, we highlight it in the notation. L p = L p (R)(1 ≤ p ≤ ∞) denotes usual Lebesgue space with the norm and the integral region R will be omitted if it does not cause any confusion. In the double integral, the differential dxdt will be omitted often for the simplicity of the presentation, i.e., the integral

Preliminaries.
To investigate the Cauchy problem (1.1) and (1.2), we first consider the following Riemann problem for non-viscous Burgers equation It is well known( for more details, see Smoller [27] ), when u − < u + , that the solution of the Riemann problem (2.1) is the center rarefaction wave It is clear that the solution w R (x/t) is discontinuous. Using a similar approach applied in [24] and [28], the smooth solution of the Riemann solution w R (t, x) can be approximated by with σ > 0 an arbitrary constant and K q chosen such that for each q > Since φ 0 (x) is monotonically increasing, the method of the characteristic curve allows a unique smooth solution in all time. Then we have the following lemma.
Lemma 2.1. There exists a unique smooth solution φ(t, x) to problem (2.2) which has the following properties by settingũ (ii) For any p with 1 ≤ p ≤ ∞, there exists a constant C p,q depending on p, q such that (iii) There exists a constant C q depending on q such that Proof. The proof of the global existence of solutions to the Cauchy problem of conservation law (2.2) is very routine, which follows from the method of characteristics directly. It only remains to prove the last inequality in (iii) since the rest of estimates have been proved in [24,25,28,32]. Indeed, the method of characteristic curve yields for all time t, where x 0 (t, x) is given by the relation Noting that and where the prime means the derivative with respect to x 0 .
In addition, when x 0 ≥ 1, we have Thus the Cauchy-Schwartz inequality gives the following estimate where we used the variable transformation dx 0 = ∂x0 ∂x dx. Next we are going to estimate the three integrals on the right hand side of (2.12) respectively. To estimate the first integral, we break the integral domain R into two parts in order to use the inequality (2.11). That is Then we proceed to estimate the integrals on the right hand side of (2.13). Indeed, noting that φ 0 (x 0 ) is bounded, we get from (2.8) and (2.10) the last inequality resulting from the fact that R (φ ′ 0 (x 0 )) r dx 0 ≤ C r,q σ r−1ũr , which is clear from (2.8). Now we introduce the variable transformation y = φ ′ 0 (x 0 )t and deduce from (2.10) and (2.11) that Thus the combination of (2.14) and (2.15) completes the estimate for the first integral on the right hand side of (2.12). Next we estimate the second integral on the right hand side of (2.12). In fact, it follows from (2.9) that (2.16) Furthermore, we estimate the third integral on the right hand side of (2.12) by Substitution of (2.14),(2.15),(2.16) and (2.18) into (2.12) yields On the other hand, since where h(ũ) is a function ofũ and satisfies lim u→0 h(ũ) = 0; 3. Global existence for the normalized problem with viscosity. Let v = u − φ, we can recast the Cauchy problem (1.1) to the following reformulated Cauchy problem For the convenience of presentation, we define It is formally equivalent to the following problem where I denotes the identity operator. We plan to obtain a global solution of (3.2) as a weak limit of a viscosity solution approximation, which solves the following viscous problem and η ε denotes the standard mollifier.
Next we are devoted to proving the global existence of solutions to problem (3.3), which consists of a local existence and the a priori estimates. For the sake of simplification, we will drop the superscript ε in v ε (t, x) to denote the solution of (3.3) in the rest of this section if there is no any ambiguity.
Since we can rewrite (3.3) as follows To obtain the global existence, it only remains to derive the a priori estimates, which is given in the following lemma.
where C 1 > 0 is a constant independent of ε and C 2 > 0 is a constant depending on ε.
Proof. We first prove inequality (3.5). To this end, we multiply (3.3) by v and integrate it with respect to x over R to get (3.7) The third term of the left hand side of (3.7) is estimated as Moreover, the second term of the right hand side of (3.7) can be rearranged as Substituting (3.8) and (3.9) into (3.7), we get the following identity Applying Young inequality and Corollary 2.2 yields In addition, noting that φ(t, x) is bounded, we have from Corollary 2.2 that Therefore, substituting (3.12) into (3.11), one has that Hence, the substitution of (3.13) into (3.11) yields (3.14) Integrating (3.14) with respect to t over [0, t], we arrive at Applying Gronwall's inequality to (3.16) gives us the inequality (3.18) To finish the proof of (3.5), it remains to estimate t 0 |∂ 2 x v| 2 dxdτ . Toward this end, we differentiate the first equation of (3.4) with respect to x and get We integrate the above equation to derive that Integrating the above inequality, and applying Corollary 2.2 as well as (3.18), we get the boundedness of t 0 |∂ 2 x v| 2 dxdτ . Together with (3.18), we obtain (3.5). Next, we derive (3.6). Indeed, it is straightforward to deduce from (3.4) that where Then we have By the Gagliardo-Nirenberg inequality, it follows that Substituting (3.22), (3.21) and (3.20) into (3.19), one has where C = C(ε, v(t, ·) L ∞ , h( u)), which can be chosen independent of time due to Now integrating (3.23) with respect to t over [0, t] and applying inequality (3.5) establish inequality (3.6). Hence the proof of Lemma 3.2 is completed.
The combination of Lemma 3.1 and Lemma 3.2 gives the global existence theorem.

Existence of a weak solution.
In this section, we will prove the global existence of weak solution to problem (3.2) using the vanishing viscosity method. It turns out that the difficulty is how to pass limits of the viscous solution v ε . A Young measure argument guarantees the weak limit v which corresponds to a global weak solution to (3.2). As a consequence, u = v + φ is a global weak solution to the Cauchy problem (1.1)-(1.2). Before giving the proof, we first establish some crucial estimates for ∂ x v ε .
Proof. Let q ε = ∂ x v ε . Then it satisfies that for any t > 0, Then we can deduce that

Then the comparison principle gives us
Direct computation tells us that Q ε (t) ≤ 2 t + 4R, which in turn implies our Lemma.
Proof. It is shown in Lemma 3.2 that {v ε } is bounded in L ∞ (R + , H 1 (R)). Let Then it is easy to see that P ε = P 1ε + P 2ε and as well as Here we have used Hölder inequality and Fubini theorem. Now it is easy to deduce from (4.2), Corollary 2.2 and Lemma 3.2 that for all t > 0, where C is a constant independent of ε and t. Then it is clear that for any T > 0 from the above inequality, equation (3.4) and Lemma 3.2. Therefore there exist some function uniformly for any compact set in R + × R.
Proof. We sketch the proof which is comparable to the proof in [30]. The main difference is that the appearance of nontrivial φ in the present case. The proof is divided into six steps.
In the following we will denote Step 2. A similar argument as above applied to E(λ) = λ give us that Hence we get Step 3. Define where R > 0 and χ A denotes the characteristic function of the set A. Since Q R (λ) is convex, we have It can be shown that q(t, x) ⇀ q 0 (x) = ∂ x v 0 (x) as t → 0 + in L 2 (R). Then However, the energy estimate gives us lim t→0 + (q(t, x)) 2 dx ≤ q 2 (t, x)dx ≤ (q 0 (x)) 2 dx.
The proof of Lemma 4.4. is finished.
Now we are in a position to prove the main results, i.e., Theorem 1.1. From Lemma 4.2 and Lemma 4.4, we deduce that ∂ x v ε → ∂ x v as ε → 0 + in L 2 loc (R + × R), which implies u(t, x) = v(t, x) + φ(t, x) is the desired global weak solution to the Cauchy problem (1.1)-(1.2). The proof is standard, see, e.g., [30]. So we omit the details. To finish the proof, it remains to investigate the asymptotic behavior of the solution v(t, x). Recalling (3.5), we have

Fubini's Theorem gives us that
Hence there is a Lebesgue measure zero set N 1 ⊂ R such that v(·, x) ∈ L 2 (R) for all x ∈ R \ N 1 . (4.9) On the other hand, it follows from the proof of Lemma 4.3 that ∂ x P (t, ·) L 2 ≤ C for all t > 0. This, (3.2 ′ ) and the estimates for φ, v gives us that ∂ t v ∈ L ∞ ([0, ∞); L 2 (R)). We may assume that sup t∈R + (∂ t v(t, x)) 2 dx < +∞.