  We extend the scope of our recent Compensated Integrability theory, by
exploiting the multi-linearity of the determinant map over ${\bf
Sym}_n(\mathbb{R})$. This allows us to establish new {\em a priori} estimates
for inviscid gases flowing in the whole space ${\mathbb R}^d$. Notably, we
estimate the defect measure (Boltzman equation) or weighted spacial
correlations of the velocity field (Euler system). As usual, our bounds involve
only the total mass and energy of the flow.
