Refined bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations

TL;DR

This paper improves well-posedness results for periodic dispersive equations with general nonlinearities by using refined bilinear estimates, establishing optimal regularity conditions, and applying these to generalized KdV equations.

Contribution

It introduces refined bilinear estimates replacing Strichartz estimates, leading to optimal well-posedness results for a broad class of periodic dispersive equations.

Findings

Unconditional local well-posedness in $H^s$ for $s \\ge 1-\\frac{\alpha}{4}$

Optimal regularity condition for generalized KdV ($\alpha=2$)

Global existence for $\alpha$ in $[4/3,2]$

Abstract

We improve our previous result [L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same 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 whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in $H^s(\mathbb{T}) $ for $ s\ge 1-\frac{\alpha}{4} $ with $ s>1/2 $. It is worth noting that this result is optimal in the case $\alpha=2$ (generalized KdV equation) in view of the restriction $ s>1/2 $ for the continuous injection of $ H^s(\mathbb{T}) $ into $ L^\infty(\mathbb{T}) $. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani in the context of Schr\"odinger equations on a compact manifold. Finally, the main theorem yields global existence results for $ \alpha \in [4/3,2] $.