Bilinear Strichartz estimates for the KdV equation on the torus

TL;DR

This paper investigates bilinear Strichartz estimates for the periodic KdV equation, providing counterexamples for certain $L^p$ estimates and establishing new estimates that reveal smoothing effects, with applications to the quartic generalized KdV.

Contribution

It introduces new bilinear Strichartz estimates for the periodic KdV, including counterexamples and frequency region analysis, bridging results between the torus and real line.

Findings

Counterexample for false $L^p$ Strichartz estimates at $p=8$

Bilinear estimates in different frequency regions showing smoothing effects

Asymptotic matching of estimates between torus and real line as period increases

Abstract

In this paper, we consider the bilinear Strichartz estimates for the periodic KdV equation. We give a concrete counterexample to the false $L^p$ Strichartz estimates for $p=8$, at least for a subset of the range of Lebesgue exponents $p$.Moreover, we prove the bilinear Strichartz estimate in different frequency regions, which represents a kind of a smoothing effects. When the period tends to infinity, the frequency localized version of Strichartz estimate on the torus matches to the one in the real line setting. In addition, the quartic generalized KdV equation is considered as an application.