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

Abstract

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

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• The wave breaking for Whitham-type equations revisited
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• Enhanced existence time of solutions to evolution equations of Whitham type
• Highest Cusped Waves for the Burgers-Hilbert equation
• Periodic Hölder waves in a class of negative-order dispersive equations
• Stability of traveling waves for the Burgers-Hilbert equation
• One-sided Hölder regularity of global weak solutions of negative order dispersive equations