Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$
Luc Molinet, Tomoyuki Tanaka
TL;DR
This work addresses the Cauchy problem for one-dimensional dispersive equations on ${\mathbb R}$ with a high-frequency dispersive symbol $i|\xi|^{\alpha}\xi$ for $1\le\alpha\le 2$ and a real analytic nonlinearity $f$ through the form $\partial_x f(u)$. The authors develop improved refined linear and bilinear Strichartz estimates on ${\mathbb R}$, enabling unconditional local well-posedness in $H^s({\mathbb R})$ with sharp thresholds $s\ge s(\alpha)$ where $s(\alpha)=(5-2\alpha)/4$ for $1\le\alpha<\tfrac32$ and $s(\alpha)=\tfrac12$ for $\alpha\ge\tfrac32$, together with global existence for $\alpha\in[\tfrac54,2]$ under small data or appropriate energy growth conditions. The analysis integrates an energy method with time-localized refined estimates on frequency interactions, aided by a frequency envelope framework and Christ–Kiselev type arguments to handle nonlinearities. The results unify local and global theory for generalized KdV and related models, providing optimal LWP thresholds in many regimes and establishing unconditional uniqueness. The techniques are likely to influence a broader class of dispersive models with analytic nonlinearities and nonlocal dispersive operators.
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 |ξ|^αξ$ with $ 1 \le α\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{5-2α}{4} $ whenever $ 1\le α<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $α\in [\frac{3}{2},2] $. This result is optimal in the case $α\ge \frac{3}{2}$ 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 $ α\in [\frac{5}{4},2] $.