The existence of global weak solutions to the shallow water wave model with moderate amplitude

Abstract: The existence of global weak solutions to the shallow water model with moderate amplitude, which is firstly introduced in Constantin and Lannes’s work (2009), is investigated in the space C([0,∞) × )⋂ L((0,∞); H( )) without the sign condition on the initial value by employing the limit technique of viscous approximation. A new one-sided lower bound and the higher integrability estimate act a key role in our analysis.


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In this paper, we use the limit technique of viscous approximation to prove the existence of global weak solutions for a shallow water wave model with moderate amplitude. It is shown from the proof of main Theorem that the weak solutions are stable when a regularizing term vanishes. The method is so effective and can be applied to solve some control problems and economic model. Moreover, the shallow water wave model with moderate amplitude we investigate captures breaking wave, which is a major interest in shallow water wave. Overall, the results we obtain can be applied in many hydrodynamic problems. Wang, Cogent Mathematics (2016) and also claimed that if the maximal existence time is finite, then blow-up occurs in form of wave breaking. In Duruk Mutlubas (2013), the local well-posedness of system (1) is proved for initial data in H s with s > 3 2 using Kato's semigroup method for quasi-linear equations. Orbital stability and existence of solitary waves for system (1) was obtained in Duruk Mutlubas and Geyer (2013), Geyer (2012). Mi and Mu (2013) investigated the local well-posedness of system (1) in Besov space using Littlewood-Paley decomposition and transport equation theory, and proposed that if initial data u 0 is analytic its solutions are analytic. Moreover, persistence properties on strong solutions were also presented (see Mi & Mu, 2013).
One of the close relatives of the first equation of problem (1) is the rod wave equation (Dai, 1998;Dai & Huo, 2000) where ∈ and u = u(t, x) stands for the radial stretch relative to a prestressed state in non-dimensional variables. Equation (2) is a model for finite-length and small-amplitude axial-radial deformation waves in the cylindrical compressible hyperelastic rods. Since Equation (2) was derived by Dai (1998), Dai and Huo (2000, many works have been carried out to investigate its dynamic properties. In Constantin and Strauss (2000), Constantin and Strauss studied the Cauchy problem of the rod equation on the line (nonperiodic case), where the local well-posedness and blow-up solutions were discussed. Moreover, they also proved the stability of solitary waves for the equation (see Constantin & Strauss, 2000). Later, Yin (2003Yin ( ,2004 and Hu and Yin (2010) discussed the smooth solitary waves and blow-up solutions. Zhou (2006), the precise blow-up scenario and several blow-up results of strong solutions to the rod equation on the circle (periodic case) were presented. For other techniques to study the problems relating to various dynamic properties of other shallow water wave equations, the reader is referred to Coclite, Holden, and Karlsen (2005), Yan, Li, and Zhang (2014), Fu, Liu, and Qu (2012), Guo and Wang (2014), Himonas, Misiolek, Ponce, and Zhou (2007), Holden and Raynaud (2009), Li and Olver (2000), Qu, Fu, and Liu (2014), Lai (2013) and the reference therein. Xin and Zhang (2000) use the limit method of viscous approximations to analyze the existence of global weak solutions for Equation (2) with = 1 (Namely, Camassa-Holm equation). Motivated by the desire to extend the works (Xin & Zhang, 2000), the objective of this paper was to establish the existence of global weak solutions for the system (1) in the space C under the assumption u 0 (x) ∈ H 1 ( ). Following the idea in Xin and Zhang (2000), the limit method of viscous approximations is employed to establish the existence of the global weak solution for system(1). In our analysis, a new one-sided lower bound (see Lemma 3.4) and the higher integrability estimate (see Lemma 3.3), which ensure that weak convergence of q is equal to strong convergence, play a crucial role in establishing the existence of global weak solutions.
The rest of this paper is as follows. The main result is presented in Section 2. In Section 3, we state the viscous problem and give a corresponding result. Strong compactness of the derivative of viscous approximations is obtained in Section 4. Section 5 completes the proof of the main result.

Using the Green function
* f for all f ∈ L 2 , and G * (u − 12 u xx ) = u, where we denote by * the convolution. Then we can rewrite system(1) as follows which is also equivalent to the elliptic-hyperbolic system Now we give the definition of a weak solution to the Cauchy problem (3) or (4). (4) in the sense of distributions and takes on the initial value pointwise.
The existence of global weak solutions to the Cauchy problem (4) will be established by proving compactness of a sequence of smooth functions {u } >0 solving the following viscous problem The main result of present paper is collected in following theorem.
Theorem 2.2 Assume that u 0 (x) ∈ H 1 ( ). Then the Cauchy problem (4) has a global weak solution u (t, x) in the sense of Definition 2.1. In addition, there is a positive constant C = C(∥ u 0 ∥ H 1 ( ) ), independent of , such that
In fact, suitably choosing the mollifier, we have Differentiating the first equation of problem (5) with respect to variable x and letting q (t, The starting point of our analysis is the following well-posedness result for problem (5).

The first equation of (5) is rewritten as
Multiplying (11) where u = u (t, x) is the unique solution of (5) and Proof In the proof of this lemma, we will use the identity and then, we get x . (22) (20) Using the Tonelli theorem and the Hölder inequality, it holds Making use of (27) and (28), we complete the proof of (15).
Lemma 3.3 Let 0 < < 1, T > 0, and a, b ∈ , a < b. Then there exists a positive constant C 1 depending only on ‖u 0 ‖ H 1 ( ) , , T, a and b, but independent of , such that where u = u (t, x) is the unique solution of (5).
Proof The proof of Lemma 3.3 is similar to that of Lemma 4.1 in Xin and Zhang (2000). Here, we omit its proof. □ Lemma 3.4 For an arbitrary T > 0, the following estimate on the first-order spatial derivative holds Proof Using (9), we get Let f = f (t) be the solution of Since f = f (t) is a supersolution of the parabolic equation (36) with initial value u ,0 , due to the comparison principle for parabolic equations, we get Consider the function F(t) = 4 7 t + √ 4C 7 , observing that dF dt + 7 4 F 2 − C = 2 t √ 4C 7 > 0 for any t > 0 and using the comparison principle for ordinary differential equations, we have f (t) ≤ F(t) for all t > 0. It completes the proof. □ Lemma 3.5 There exists a sequence { j } j∈N tending to zero and a function is the unique solution of (5).