Improved existence time for a Boussinesq system

Abstract

We consider a strongly dispersive Boussinesq-type system arising as a model for the propagation of weakly nonlinear water waves. The system is characterized by a nonlinearity parameter $0<\epsilon \le 1$ and a shallow water parameter $0<\mu \le 1$. In the Boussinesq regime $\mu \sim \epsilon$, Saut, Wang and Xu \cite{SWX2017} used a hyperbolic approach to prove the well-posedness of the system on a time scale of order $1/\epsilon$, in both one and two dimensions, under a non-cavitation condition. In this paper, we establish well-posedness on a time scale of order $({\mu }^{1/4}{\epsilon }^{-1}{h}_0{)}^{4/3}$ in one dimension and of order $({\mu }^{1/4}{\epsilon }^{-1}{h}_0{)}^{2-}$ in two dimensions, assuming a non-cavitation condition with a parameter $h_0>0$. Our results show quantitatively how the lifespan depends on the balance between nonlinearity and dispersion in the regimes $\epsilon \lesssim \mu$ and $\mu \ll \epsilon$. They also make explicit how the existence time deteriorates as $h_0\to 0$, that is, as the wave approaches cavitation. Our proofs combine dispersive and Strichartz estimates with energy estimates.

Withdrawn by the authors because this version was submitted before both authors had approved the final manuscript