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

TL;DR

This paper improves bilinear estimates and establishes well-posedness for a class of generalized KdV equations on the real line, extending the range of initial data regularity and enabling global solutions for certain parameters.

Contribution

It introduces enhanced linear and bilinear estimates that improve well-posedness results for generalized KdV equations with dispersive operators of order between 1 and 2.

Findings

Unconditional local well-posedness in H^s for s ≥ (5-2α)/4 when 1 ≤ α < 3/2.

Well-posedness in H^s for s > 1/2 when α ∈ [3/2, 2].

Global existence of solutions for α in [5/4, 2].

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 |\xi|^\alpha \xi $ with $ 1 \le \alpha\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\alpha}{4} $ whenever $ 1\le \alpha<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $\alpha\in [\frac{3}{2},2] $. This result is optimal in the case $\alpha\ge \frac{3}{2}$ 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 $ \alpha \in [\frac{5}{4},2] $.