Comparison between Boussinesq‐ and Whitham–Boussinesq‐type systems

Abstract

In this paper, we consider a Whitham–Boussinesq‐type system modeling surface water waves of an inviscid incompressible fluid layer. The system describes the evolution with time of surface waves of a liquid layer in the two‐dimensional physical space. Using fixed point argument, we prove that the system is locally well‐posed on the time scale of order scriptO()1false/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$, where ϵ>0$$ \epsilon >0 $$ is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid. We also show that the solution to the Whitham–Boussinesq system approximates the solution of a Boussinesq system on the time scale of order scriptO()1false/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$.

Local well-posedness result for a class of non-local quasi-linear systems and its application to the justification of Whitham-Boussinesq systems

“…The second step is to prove stability estimates and the local well-posedness of the model for sufficently regular data on the relevant timescale. The well-posedness of the aforementioned Whitham-Boussinesq systems has been studied in [6,8,20,5]. In [6], Dinvay diagonalizes linear terms and uses strongly the regularization properties of the operator (G 2 ) −1 , from which stems an existence time of length T ǫµ −1/2 (in dimension d = 1).…”

“…Yet, due to the use of dispersive estimates, these two results do not provide the control of solutions and their derivatives uniformly with respect to µ ∈ (0, 1], as required for the rigorous justification of the system as an asymptotic model for water waves. The precise dependency of the time on which solutions exist and are controlled (in a ball of twice the size of the initial data in the relevant Banach space) is clarified in subsequent works: the result of Tesfahun [20] in dimension d = 2 exhibits a time interval of length T ǫ −2+δ µ 3/2−δ with δ > 0 arbitrarily small, while the result of Deneke, Dufera and Tesfahun [5] in dimension d = 1 exhibits a time interval of length T ǫ −1 µ 1/2 .…”