Extended lifespan of the fractional BBM equation

For $0<\alpha <1$ and with initial data $\vert\vert u_0\vert\vert_{H^{N+\frac{\alpha}{2}}}=\varepsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation ${\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_{x}u+u{\mathrm{\partial }}_{x}u+\mathrm{\mid }\mathrm{D}{\mathrm{\mid }}^{\alpha }{\mathrm{\partial }}_{t}u=0$, can be extended beyond the hyperbolic existence time $\frac{1}{\epsilon }$, to $\frac{1}{{\epsilon }^2}$. For the proof we use a modified energy, based on a normal form transformation as in [Hunter, Ifrim, Tataru, Wong, 2015]. In addition we employ ideas and techniques from [Ehrnstr"om, Wang, 2018], in which the authors obtain an enhanced existence time for the fractional KdV equation.