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 $-1<\alpha<1$ , $\alpha\neq0$. 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).
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2018
- DOI:
- 10.48550/arXiv.1804.06297
- arXiv:
- arXiv:1804.06297
- Bibcode:
- 2018arXiv180406297E
- Keywords:
-
- Mathematics - Analysis of PDEs;
- 76B03;
- 35S30;
- 76B15
- E-Print:
- SIAM J. Math. Anal.(2019)