The lack of compactness in the Sobolev-Strichartz inequalities

TL;DR

This paper introduces a new method to decompose bounded sequences in Sobolev spaces into dispersive profiles, addressing the lack of compactness in Sobolev-Strichartz inequalities with a broad applicability to various propagators.

Contribution

It develops a general decomposition technique for bounded sequences in Sobolev spaces, differing from previous methods and applicable to a wide range of propagators, including matrix-valued ones.

Findings

Decomposition of bounded sequences into dispersive profiles.

Applicable to a broad class of propagators.

Addresses the lack of compactness in Sobolev-Strichartz inequalities.

Abstract

We provide a general method to decompose any bounded sequence in $\dot H^s$ into linear dispersive profiles generated by an abstract propagator, with a rest which is small in the associated Strichartz norms. The argument is quite different from the one proposed by Bahouri-G\'erard and Keraani in the cases of the wave and Schr\"odinger equations, and is adaptable to a large class of propagators, including those which are matrix-valued.