On the lack of compactness and existence of maximizers for some Airy-Strichartz inequalities

TL;DR

This paper develops a linear profile decomposition for the Airy equation in certain Sobolev spaces and uses it to prove the existence of maximizers for related Strichartz inequalities, addressing issues of compactness.

Contribution

It introduces a new profile decomposition for the Airy equation and demonstrates the existence of maximizers for associated Strichartz inequalities, filling a gap in understanding their compactness properties.

Findings

Established a linear profile decomposition for the Airy equation.

Proved the existence of maximizers for Airy-Strichartz inequalities.

Addressed the lack of compactness in these inequalities.

Abstract

This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb{R})$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.