On the lack of compactness and existence of maximizers for some   Airy-Strichartz inequalities
  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. 
Luiz Gustavo Farah and Henrique Versieux
math.AP