• Navier Stokes equations
• non uniqueness
• energy conservation

In this thesis we have mainly studied the convex integration technique and its consequences on the Navier-Stokes equations on the 3 dimensional torus.
After studying the essential properties of smooth Navier-Stokes equations solutions, we have worked on the convex integration scheme, to find weak solutions (low regularity solutions) to the Navier Stokes equations with a priori prescribed kinetic energy, using intermittent jets (a mathematical object). The previous construction leads to the non-uniqueness of Navier Stokes solutions into the class of weak solutions. We have investigated the natural question on how many smooth solutions there are into the class of weak solutions proving a new result which asserts that smooth solutions are a nowhere dense set into the class of weak solutions. Some interesting consequences of this result are taken into account.
Finally, we have discussed viscous eddies, a mathematical object introduced in the last years, useful for the convex integration procedure. We provided a simple convex integration scheme with viscous eddies.