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 

 
 
 
 
 S
 y
 m
 
 n
 
 (
 
 R
 
 )
 
 \mathbf {Sym}_n(\mathbb {R})
 

. This allows us to establish new a priori estimates for inviscid gases flowing in the whole space 

 
 
 
 R
 
 d
 
 \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.