The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

Abstract

This paper is a continuation of a previous work by two of the Authors [36] on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered in [36] in particular we prove a long time well-posedness result for a delicate "strongly dispersive" Boussinesq system.where τ ≥ 0 is the surface tension parameter (Bond number).Recall also [6] that (1.2) is linearly well-posed whenand when a = c, b ≥ 0, d ≥ 0. An important step to justify rigorously (1.2) as an asymptotic model for water waves is to establish the well-posedness of the Cauchy problem on time scales of order 1/ǫ, with uniform bounds in suitable Sobolev spaces, the error estimate being then (see [5,29]).This step has been established in [36] (see also [34]) for most of Boussinesq systems with and without surface tension. The idea in [36] is to find an appropriate symmetrization of the system and this is not a straightforward task since one cannot obviously use the classical symmetrizer of the underlying Saint-Venant (shallow water) hyperbolic system. This will be reviewed in the first section of this paper. A complete proof of cases that were not fully developed in [36] will be given in Section 4. 1 Introducing surface tension enlarges the range of physically admissible parameters (a,b,c,d) and so even a local theory 2 for a few linearly well -posed systems is still missing, for instance the cases b = d = 0, a < 0, c = 0 and b = d = 0, a = 0, c < 0). Both cases will be considered here but the later leads to serious difficulties and the long time existence for it is the main result of the present paper.Note also that the (linearly well-posed) "exceptional KdV-KdV" case b = d = 0, a = c > 0 which is studied in [30] leading to well-posedness on time scales of order 1/ √ ǫ in Sobolev spaces H s (R 2 ), s > 3/2 which are larger than the "hyperbolic" one H s (R 2 ), s > 2 is not covered neither in [36] nor in the present paper so that a long time existence is still open in this case 3 .An important mathematical issue concerning Boussinesq systems (1.2) is that despite they describe the same dynamics of water waves, their mathematical properties are rather different, due essentially to their different linear dispersion relations. Of course those dispersion relations all coincide in the long wave limit but there are quite different in the short wave limit. A convenient way to classify the system is according to the order of the Fourier multiplier operator given by the eigenvalues of the linearized operator (see [6]). The order can be −1, 0, 1, 2 or 3. The two last cases are referred to as the strongly dispersive ones.1 Due to the large number of cases to be considered, we chosed in [36] to give complete proofs for a limited number of cases.2 That is not taking care of the dependence of the lifespan of the solution with respect to ǫ 3 However, the case b = d = 0, a < 0, c < 0 that can only occur with a strong surface tension is covered by the theory in [36].

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

“…The reason for such a strategy is that r the two unknowns, ζ and w, are controlled in different functional spaces, one being continuously embedded in the other but to the price of a non uniform constant (see Lemma 3), and the inclusion being strict; r the most singular term of the system, namely, the one that involves the operator of highest order, comes from the surface tension component, and couples the two unknowns (it appears as an off-diagonal component of the quasilinearized system). This is why, one cannot use standard energy methods in Sobolev-based functional spaces, as commutator estimates fail to control all coupling terms; see also the discussion in [47].…”

Discrete energy estimates for the $abcd$-systems

“…where τ ≥ 0 is the Bond number which characterizes the surface tension parameter, see [19] or [21]. In [12], models taking into account more general topographies of the bottom of the channel are obtained.…”

The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space

“…al. [45,46] studied in great detail the long time existence problem by focusing in the small physical parameter ε appearing from the asymptotic expansions. They showed well-posedness (on a time interval of order 1/ε) for (1.1).…”