Abstract: We considered the first initial boundary value problem for a class of anisotropic non-Newtonian  fluid equations in a three-dimensional smooth bounded region Ω. Firstly, we constructed the approximate solution of the problem by using Galerkin method, and then obtained the uniform estimates of the approximate solution by using the energy estimation method. Finally, we proved the existence of the weak solution by using the compact method and the monotone method.

Key words: anisotropy, non-Newtonian fluid, micropolar fluid, weak solution, existence