Macroscopic limit of the kinetic Bloch equation

This work concerns the existence of solution of the kinetic spinor Boltzmann equation as well as the asymptotic behavior of such solution when \begin{document}$ \varepsilon \to 0 $\end{document} , that is when the time relaxation of the spin-flip collisions is very small in comparison to the time relaxation parameter of the collisions with no spin reversal. Due to the lack of regularity of the weak solution, the switching term \begin{document}$ H_\varepsilon\times M_\varepsilon $\end{document} is not stable under the weak convergences. Hence we establish new estimates of the solutions in a weighted Sobolev space of order 3.