Mixed determinants, compensated integrability, and new 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 $\mathbf {Sym}_n(\mathbb {R})$. This allows us to establish new 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.