The relativistic Navier–Stokes equations following from the first‐order theory of Eckart are derived. Exact analytic solutions, which are the generalizations of the well‐known nonrelativistic solutions for Couette flow, are found. In particular, it is shown that even in the case when the dissipative coefficients are constant the temperature distribution influences the velocity distribution. The relativistic generalization of the classical Busemann theorem is found for the case of dissipative coefficients varying with temperature. For almost every type of these flows the velocity and temperature profiles are flattened as compared with respective nonrelativistic profiles. ©1996 American Institute of Physics.