Long time well-posedness of Whitham–Boussinesq systems

Abstract

Consideration is given to three different full dispersion Boussinesq systems arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water. We prove that, under a non-cavitation condition on the initial data, these three systems are well-posed on a time scale of order O ( 1 ε ) , where ɛ is a small parameter measuring the weak nonlinearity of the waves. For one of the systems, this result is new even for short time. The two other systems involve surface tension, and for one of them, the non-cavitation condition has to be sharpened when the surface tension is small. The proof relies on suitable symmetrizers and the classical theory of hyperbolic systems. However, we have to track the small parameters carefully in the commutator estimates to get the long time well-posedness. Finally, combining our results with the recent ones of Emerald provide a full justification of these systems as water wave models in a larger range of regimes than the classical (a, b, c, d)-Boussinesq systems.

Long Time Well-Posedness and Full Justification of a Whitham-Green-Naghdi System

“…In particular, one can read [14] for a similar argument in the case of the classical Green-Naghdi system. Lastly, the reader might also find it useful to read the detailed proof, using these methods, in the case of the Benjamin-Ono equation in [32], and likewise in the case of Whitham-Boussinesq systems demonstrated in [36].…”

“…As mentioned above, at this stage in the proof, the argument is classical and the details can be found in e.g. [3,32,36]. □ we obtain the following system…”

“…When nonflat bottoms are considered, it has been proved in [14] to be consistent with the water waves with a precision O(µ(ε + β)). With respect to the second point of the justification, it has been proved for a large class of Whitham-Boussinesq systems with flat bottoms [36,23], to be well-posed on the time scale O( 1 ε ). Lastly, we also mention earlier results on the local-well posedness on a fixed time scale given in [9,10,12].…”

Well-Posedness for the Extended Schrödinger–Benjamin–Ono System

“…Then, the proof of the existence and the uniqueness is a combination of the parabolic regularization method with the energy estimates obtained in Propositions 3.1 and 3.2 and taking into account Remarks 3.1 and 3.2. We refer to the proof of Theorem 1.6 in [26] for the details in a similar setting.…”

“…The continuous dependence and persistence is obtained by applying the Bona-Smith approximation method. We refer to [4,12,20,26] for the details.…”