Existence and uniqueness of the global solution to the Navier-Stokes equations

A proof is given of the global existence and uniqueness of a weak solution to Navier–Stokes equations in unbounded exterior domains

Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

The aim of this work is to prove an existence and uniqueness result of Kato-Fujita type for the Navier-Stokes equations, in vorticity form, in $2-D$ and $3-D$, perturbed by a gradient type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by V. Barbu and the second named author in \cite{b1}, that treats the stochastic $3D$-Navier-Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in \cite{b1}, but the transformation is different and adapted to our gradient type noise. Then global unique existence results are proved for the transformed equation, while for the original stochastic Navier-Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time

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]