On some nonlinear models for the Navier-Stokes equations

We consider in this thesis two nonlinear models for the incompressible Navier-Stokes system. Firstly, we lay the basis for the regularity analysis of those models by establishing various results such as epsilon-regularity theorems (e.g. a version of the Caffarelli-Kohn-Nirenberg theorem), dimension analysis of potential singular sets, partial regularity results in critical cases and various Liouville type theorems. Secondly, by relying on the previous points and by taking advantage of the locality of our models, we were able to establish some new regularity results such as a higher integrability for the gradient of some particular solutions, regularity in the case of spherical symmetry and a boundary regularity result which is known to be false for the Navier-Stokes system. Thirdly, we construct two candidates for non-uniqueness for the Navier-Stokes system as singular limits of solutions of our models; we put a particular emphasis on the analysis of this convergence. Finally, we provide at the end of this work a set of tools to tackle some questions related to establishing energy identity and proving Liouville type theorems for equations of fluids mechanics.