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arXiv · 1804.06297

Enhanced existence time of solutions to the fractional Korteweg-de Vries equation

Mats Ehrnström·Yuexun Wang
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Abstract

We consider the fractional Korteweg-de Vries equation $u_t + u u_x - |D|^\alpha u_x = 0$ in the range of $-1<\alpha<1$ , $\alpha\neq0$. Using basic Fourier techniques in combination with the modified energy method we extend the existence time of classical solutions with initial data of size $\varepsilon$ from $\frac{1}{\varepsilon}$ to a time scale of $\frac{1}{\varepsilon^2}$. This analysis, which is carried out in Sobolev space $H^N(\mathbb{R})$, $N \geq 3$, answers positively a question posed by Linares, Pilod and Saut (SIAM J. Math. Anal. 46 (2014), no. 2, 1505-1537).

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Mats Ehrnström, Yuexun Wang. 2018-04-17. Enhanced existence time of solutions to the fractional Korteweg-de Vries equation. https://doi.org/10.1137/19m1237867

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