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$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$.
with a loss of 3d/4 or d/4 -derivatives in the case β = 0 or β = 1, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter β measures surface tension effects. As an application we prove low regularity well-posedness for a Whitham-Boussinesq type system in R d , d 2. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in R and R 2 .
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