The boundary-value problem for the generalized Navier-Stokes equations with anisotropic diffusion is considered in this work. For this problem, we prove the existence of weak solutions in the sense that solutions and test functions are considered in the same admissible function space. We also prove the existence of very weak solutions, i.e., solutions for which the test functions have more regularity. By exploiting several examples we show, in the case of dimension $3$, that these existence results improve its isotropic versions in almost all directions or for particular choices of all the diffusion coefficients.
"Analysis of the existence for the steady Navier-Stokes equations with anisotropic diffusion." Adv. Differential Equations 19 (5/6) 441 - 472, May/June 2014.