A NOTE ON THE CONVERGENCE OF THE SOLUTIONS OF THE CAMASSA-HOLM EQUATION TO THE ENTROPY ONES OF A SCALAR CONSERVATION LAW

. We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive eﬀects. We prove that as the diﬀusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.


Introduction. The nonlinear evolution equation
is known as the Camassa-Holm equation (see [3]). (1) models the propagation of unidirectional water waves of moderate amplitude over a flat bottom. The unknown u(t, x) represents the fluid velocity at time t in the horizontal direction x (see [3,21,33,34]).
In [24,25,26], the authors derived (1) (in a more general form), as an equation describing finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods.
The Camassa-Holm equation goes beyond the Korteweg-de Vries (KdV) and the Benjamin-Bona-Mahony (BBM) ones in the sense that (1) appears as a water-wave equation at quadratic order in an asymptotic expansion for unidirectional shallow water waves modeled by the incompressible Euler equations, whereas the KdV and BBM equations appear at first order in this asymptotic expansion (see [3,34]). Two of the many differences between the KdV and BBM equations and (1) are the following. The KdV and BBM equations admit analytic travelling waves and within the travelling waves for (1) there are the peakons, that present a peak at their crest, being similar to the Stokes waves of greatest height [5,14,15,18,19]. Moreover, (1) experiences breaking waves [5,20], namely solutions that remain bounded but whose slope becomes unbounded in finite time, that is not the case for the KdV and BBM equations.
From a mathematical point of view, the Camassa-Holm equation is well studied. Local well-posedness results are proved in [16,27,36,38] It is also known that there exist global solutions for a certain class of initial data and solutions that blow up in finite time for a large class of initial data (see [13,16,17]). Existence and uniqueness results for global weak solutions of (1) are proven in [1,2,17,7,8,12,22,28,29,30,40,41]. The convergence of finite difference schemes has been proved in [9,10].
We are interested in the no high frequency limit, i.e., we send α → 0 in (1). In this way we pass from (1) to the scalar conservation law We augment (1) with the initial condition We study the dispersion-diffusion for (1). Therefore, following [11], we fix ε, α and consider the following fourth order approximation (see [6]) where u ε, α, 0 is a C ∞ approximation of u 0 such that u ε, α, 0 → u 0 in L p loc (R), 1 ≤ p < 4, as ε, α → 0, for every ε, α and some constant C 0 independent on ε, α.
On the flux f , we assume that it is a C 2 function satisfying for some constants k 0 , k 1 > 0, and the genuinely nonlinear condition In [11], under the assumptions (3), (5), (6), (7), (9), (10), and choosing the convergence of the solution of (1) to a distributional solution of (2) is proven. Moreover, following [23], under the assumption the dissipation of energy is proven. In [31], under the assumption (12), the convergence of the solution of (1) to the unique entropy solution of (2) is proven. In other to do this, the author used the technique of the kinetic methods, which is introduced in [32].
The paper is organized as follows. In Section 2, we prove several a priori estimates on (4). Those play a key role in the proof of our main result, that is given in Section 3.
3. Proof of Theorem 1.1. In this section, we prove Theorem 1.1. In other to do this, the following technical lemma is needed [37].
Proof of Theorem 1.1.