On the validity of Chapman-Enskog expansions for shock waves with small   strength
  We justify a Chapman-Enskog expansion for discontinuous solutions of hyperbolic conservation laws containing shock waves with small strength. Precisely, we establish pointwise uniform estimates for the difference between the traveling waves of a relaxation model and the traveling waves of the corresponding diffusive equations determined by a Chapman-Enskog expansion procedure to first- or second-order. 
Nabil Bedjaoui, Christian Klingenberg, and Philippe G. LeFloch