| The first part of this thesis is devoted to a regularity criterion for solutions of the Incompressible Navier-Stokes equations in terms of regularity of the solutions along the streamlines. More precisely, we prove that we can ensure the full regularity of a given suitable weak solution provided we have good control on the second derivative of the velocity along the direction of the streamlines of the fluid.;In the second part of this thesis, we will show that the global regularity of a suitable weak solution u for the incompressible Navier-Stokes equations holds under the condition that |u|5/(log(1+| u|)) is integrable in space time variables. |
