Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$

Abstract

We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R}$, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $i|\hat{I}¾{|}^{\hat{I}}\pm \hat{I}¾$ with $1\le \hat{I}\pm \le 2$, and that the nonlinear term is of the form ${\partial }_{x}f(u)$ where $f$ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R})$ for $s\ge \frac{5-2\hat{I}\pm }{4}$ whenever $1\le \hat{I}\pm <\frac{3}{2}$, and for $s>\frac{1}{2}$ whenever $\hat{I}\pm \in [\frac{3}{2},2]$. This result is optimal in the case $\hat{I}\pm \ge \frac{3}{2}$ in view of the restriction $s>\frac{1}{2}$ required for the continuous embedding $H^s(\mathbb{R})\hookrightarrow L^{\infty }(\mathbb{R})$. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R}$. Our local well-posedness results enable us to derive global existence of solutions for $\hat{I}\pm \in [\frac{5}{4},2]$.

50 pages