Convex Integration and the Navier-Stokes Equations

This work is devoted to applying the convex integration technique that has been recently developed in fluid dynamics to the incompressible Navier-Stoke equations in dimensions $d \geq 3$. The main results include the existence of stationary weak solutions in dimensions $d \geq 4$ proved in Chapter 2 and in $3D$ which is proved in Chapter 3 where we also construct weak solutions in $3D$ whose energy profiles are discontinuous on some dense sets of positive Lebesgue measure in time.