Antontsev, S. N.de Oliveira, H. B.2017-04-072017-04-072016-091578-7303AUT: HOL01377http://hdl.handle.net/10400.1/9328In this work, we consider the evolutive problem for the incompressible Navier-Stokes equations with a general diffusion which can be fully anisotropic. The existence of weak solutions is proved for the associated initial problem supplemented with no-slip boundary conditions. We prove also the properties of extinction in a finite time, exponential time decay and power time decay. With this respect, we consider the important case of a forces fields with possible different behavior in distinct directions. Perturbations of the asymptotically stable equilibrium are established as well.engjournal article10.1007/s13398-015-0262-2