Enhanced lifespan of solutions to Whitham-Boussinesq system with surface tension

Abstract

We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter $0<ε\le 1$, a shallow water parameter $0<μ\le 1$ and a nonnegative surface tension parameter $β$. In this paper, we establish well-posedness of the associated Cauchy problem for $β>1/3$ on a time scale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1} )^{4/3}$ in one dimension and of order $( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1})^{2-}$ in two dimensions, where $β_\ast= \min(|3β-1|, 1/2)$, assuming a non-cavitation condition with a parameter $h_0>0$. In addition, we established well-posedness for the classical Whitham equation with either $β=0$ or $β>1/3$ on the timescale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} ε^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.