A study of constrained Navier-Stokes equations and related problems

Fundamental questions in the theory of partial differential equations are that of existence and uniqueness of the solution. In this thesis we address these questions corresponding to two models governing the dynamics of incompressible fluids, both being the modification of classical Navier-Stokes equations: constrained Navier-Stokes equations and tamed Navier-Stokes equations.

The former being Navier-Stokes equations with a constraint on the L^2 norm of the solution considered on a two-dimensional domain with periodic boundary conditions. We prove existence of the unique global-in-time solution in deterministic setting and establish existence of a pathwise unique strong solution under the impact of a stochastic forcing.

The tamed Navier-Stokes equations were introduced by Röckner and Zhang [75], to study the properties of solutions of the 3D Navier-Stokes equations. We use three new ideas to prove the existence of a strong solution and existence of invariant measures: approximating equation on an infinite dimensional space in contrast to classical Faedo-Galerkin approximation; tightness criterion related to the Dubinsky's compactness theorem introduced recently by Brzeźniak and Motyl [23]; and lastly proving the existence of invariant measures based on continuity and compactness in the weak topologies [62].