Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Abstract

We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of approximate Riemann invariants for a Whitham–Boussinesq system and is adapted to unidirectional waves. The second one is based on a generalisation of Birkhoff’s normal form algorithm for almost smooth Hamiltonians and is adapted to bidirectional propagation. In both cases we clarify the improved accuracy on the fully dispersive Whitham model with respect to the long wave Korteweg–de Vries approximation.

Long time well-posedness of Whitham–Boussinesq systems

“…On the other hand, due to the improved dispersion relation of (1.1), Emerald [22] was able to decouple the parameters (ε, μ) and prove an error estimate between the Whitham equation and the water wave system with a precision O(μεt) for 0 t ε −1 in the shallow water regime: R SW = {(ε, μ) : 0 μ 1, 0 ε 1}.…”

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

“…For instance, in the case of the Whitham equation, the local well-posedness in the relevant time scale follow by classical arguments on hyperbolic systems. The consistency of the water waves equations with this model has been recently proved in [22] at the order of precision O(µε) in the unidirectional case, but the method supposes well-prepared initial conditions. In the bidirectional case, the author proved an order of precision O(µε + ε 2 ) and doesn't suppose well-prepared initial conditions.…”

Analysis on continuity of the solution map for the Whitham equation in Besov spaces

“…There are extensive mathematical research results for the Whitham‐type equations. For example, based on two different approaches, Emerald [8] rigorously derived the Whitham equations from the water‐wave equations in the shallow water regime. Whitham [26, 27] conjectured that there would be a highest, cusped, traveling‐wave solution for Equation (1.1), and Ehrnström and Wahlén [5] found this wave and showed several properties of it.…”