Global well-posedness of the BCL system with viscosity

Abstract

The BCL system, a kind of equations governing the motion of the free surface of water waves in R 3 , is studied. Some results on the global existence, uniqueness and regularity of solutions to such system with small initial data are obtained.

Modeling shallow water waves

“…This has been proved for the "standard" Boussinesq system (34) in [171,7], where a weak solution is constructed using a parabolic regularization of the mass conservation equation, mimicking the hyperbolic theory; the solution is then proved to be regular and unique. For the general abcd systems (37) in dimension d = 1, global well posedness has been proved in some specific cases using the particular structure of the equations, such as the Bona-Smith system (a = −1/3, b = 0, c = −1/3, d = 1/3) [23] and the Hamiltonian cases (b = d > 0, a ≤ 0, c < 0) [20]; for this latter system, the two-dimensional case has been treated in [90]. When b = d < 0 refined scattering results in the energy space have also been proved [115,114].…”

“…quite obviously, enstrophy (or, equivalently, turbulent energy), is created in the vicinity of wave breaking, where the gradient of the velocity becomes important; the mechanical energy of the wave is consequently decreased. This mechanism restores the local conservation of the total energy (89). However, in a second step, the small scale dissipation of the total energy must be taken into account; there should therefore be a dissipation mechanism D such that…”

Long Time Existence for a Strongly Dispersive Boussinesq System

“…Remark 1.1. The global well-posedness of Boussinesq systems has been only established in a few cases, including the one-dimensional case a = c = b = 0, d > 0 that can be viewed as a dispersive perturbation of the hyperbolic Saint-Venant (shallow water) system, see [5,27,22], and the Hamiltonian cases b = d > 0, a ≤ 0, c < 0, see [9] for the one-dimensional case and [12] for the two-dimensional case. We also refer to [16,17] for scattering results in the energy space for those one-dimensional Hamiltonian systems when b = d > 0.…”

Long time existence for the Boussinesq -- Full Dispersion systems

“…Section 3 is devoted to the proof of Theorem 1.1 which involves the symmetrization techniques used in our previous work [30] (see also [31] on the Boussinesq (abcd) systems). In Section 4, we prove Theorem 1.2 by adapting the proof of a similar result for the Hamiltonian Boussinesq systems (see [9,16]). Finally an Appendix is devoted to the proof of the equivalence of norms (3.8), (3.37) and (3.54).…”