The lack of compactness in the Sobolev-Strichartz inequalities

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.