Mixed determinants, Compensated Integrability and new {\em a priori} estimates in Gas dynamics

We extend the scope of our recent Compensated Integrability theory, by exploiting the multi-linearity of the determinant map over ${\mathbf{S}\mathbf{y}\mathbf{m}}_n(\mathbb{R})$. 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.