Extended lifespan of the fractional BBM equation

Abstract

For $0<\alpha <1$ and with initial data $||u_0|{|}_{H^{N+\frac{\alpha }{2}}}=\epsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation ${\partial }_{t}u+{\partial }_{x}u+u{\partial }_{x}u+|\mathrm{D}{|}^{\alpha }{\partial }_{t}u=0$, 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 [Ehrnström, Wang, 2018], in which the authors obtain an enhanced existence time for the fractional KdV equation.