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arXiv · 1902.06336

Extended lifespan of the fractional BBM equation

Dag Nilsson
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Abstract

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 $\partial_tu+\partial_xu+u\partial_xu+\vert\mathrm{D}\vert^\alpha\partial_tu=0$, can be extended beyond the hyperbolic existence time $\frac{1}{\varepsilon}$, to $\frac{1}{\varepsilon^2}$. 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.

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Dag Nilsson. 2019-02-17. Extended lifespan of the fractional BBM equation. https://arxiv.org/abs/1902.06336

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