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

TL;DR

This paper extends the known lifespan of solutions to the Whitham equation and a Whitham-Boussinesq system in shallow water regimes, using advanced energy and Strichartz estimates to achieve longer existence times.

Contribution

It provides new lower bounds on the existence time of solutions for these water wave models, especially in two dimensions, with techniques adaptable to similar equations.

Findings

Extended solution lifespan beyond hyperbolic time scale.

Derived refined Strichartz estimates incorporating small parameters.

Achieved order $rac{5}{4}$ in time of existence in 2D long wave regime.

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.