Bilinear Strichartz estimates for the KdV equation on the torus

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.

The revised version is resubmitted. The prior version of this paper was written to explore the counterexample of $L^p$ Strichartz estimates for $p=8$, but our analyses were methodologically flawed, the conclusions were incorrect