Improved existence time for the Whitham equation and a Whitham-Boussinesq system

Abstract

In this paper, we investigate the time of existence of the solutions to two full dispersion models derived from the water waves equations in the shallow water regime: the Whitham equation and a Whitham-Boussinesq system in dimension one and two. The regime is characterized by the nonlinearity parameter $\epsilon\in(0,1]$ and the shallow water parameter $\mu\in(0,1]$. We extend the lifespan of the solution beyond the hyperbolic time $\epsilon^{-1}$. More precisely, we establish well-posedness on the timescale of order $\mu^{\frac{1}{4}^-}\epsilon^{(-\frac{5}{4})^+}$ in the one-dimensional case, and of order $\mu^{\frac{1}{4}^-}\epsilon^{(-\frac{3}{2})^+}$ in dimension two. We emphasize that for the two-dimensional case, we obtain a time of existence of order $\epsilon^{-\frac54}$ in the long wave regime $\mu \sim \epsilon$. This kind of result seems to be new, even for the Boussinesq systems. The proofs combine energy methods with Strichartz estimates. Here, a key ingredient is to obtain new refined Strichartz estimates that include the small parameter $\mu$. These techniques are robust and could be adapted to improve the lifespan of solutions for other equations and systems of the same form.