In this paper, we prove two results about the blow-up criterion of the three-dimensional incompressible Navier-Stokes equation in the Sobolev space . H 5∕2 . The first one improves the result of Cortissoz et al. The second deals with the relationship of the blow up in . H 5∕2 and some critical spaces. Fourier analysis and standard techniques are used.

This document presents a priori estimates related to statistical moments and integrability properties for solutions of systems of monatomic gas mixtures modelled with the homogeneous Boltzmann equation with long range interactions for hard potentials. We detail the conditions for the generation and propagation of polynomial and exponential moments, and the integrability in Lebesgue spaces.