Strichartz type estimates of the Airy equation

TL;DR

This paper establishes precise conditions under which certain Strichartz-type estimates hold for the Airy equation, aiding the analysis of low regularity solutions for KdV and stochastic KdV equations.

Contribution

It provides a complete classification of index relationships for Strichartz estimates in the Airy equation context, which was previously unknown.

Findings

Characterization of necessary and sufficient conditions for the inequality

Complete classification of index relationships for the estimates

Applications to low regularity well-posedness of KdV equations

Abstract

In this article, we show the necessary and sufficient conditions for the inequality $\|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}$, where $$\|u\|_{X^{s,b}}:=\|\hat{u}(\tau,\xi)\langle \xi\rangle^s\langle \tau-\xi^3\rangle^b \|_{L_{\tau,\xi}^2}. $$ Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg--de Vries equations and stochastic Korteweg--de Vries equations.