# Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
## 2 Problem Setting and Discretization  
The motion of incompressible newtonian fluids is governed by the well known Navier-Stokes equations, a system of partial differential equations that arises from the conservation of mass and momentum. In the general non-stationary case, they take the form  
\[\begin{cases}\rho\frac{\partial\mathbf{u}}{\partial t}-\mu\Delta\mathbf{u}+\rho (\mathbf{u}\cdot\
abla)\mathbf{u}+\
abla p=\mathbf{f},&\mathbf{x}\in\Omega, t>0\\ \
abla\cdot\mathbf{u}=0,&\mathbf{x}\in\Omega,t>0\end{cases}\]  
where \(\Omega\subset\mathbb{R}^{3}\) is the domain in which the motion evolves; \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)\) is the velocity field; \(p=p(\mathbf{x},t)\) is the pressure field; \(\rho\) and \(\mu\) are the fluid density and dynamic viscosity; \(\mathbf{f}\) is a forcing term.  
The first equation is usually used in a different form: using a unit density, we obtain the following version of the Navier-Stokes equations  
\[\begin{cases}\frac{\partial\mathbf{u}}{\partial t}-\
u\Delta\mathbf{u}+(\mathbf{ u}\cdot\
abla)\mathbf{u}+\
abla p=\mathbf{f},&\mathbf{x}\in\Omega,t>0\\ \
abla\cdot\mathbf{u}=0,&\mathbf{x}\in\Omega,t>0\end{cases} \tag{1}\]  
where now \(\
u=\mu/\rho\) is the kinematic viscosity, \(p\) is the density-scaled pressure field and \(\mathbf{f}\) is a forcing term per unit mass.  
The first of the two equations imposes the conservation of momentum; the term \(\
u\Delta\mathbf{u}\) takes into account the diffusive processes, while \((\mathbf{u}\cdot\
abla)\mathbf{u}\) describes the convective processes. The equation \(\
abla\cdot\mathbf{u}=0\) imposes the incompressibility of the fluid, i.e., the density \(\rho\) is a constant, both in space and time.  
For the problem to be well posed, Equations (1) need some initial condition  
\[\mathbf{u}(\mathbf{x},0)=\mathbf{u}_{0}(\mathbf{x})\quad\forall\mathbf{x}\in\Omega\]  
and boundary conditions, e.g., \(\forall t>0\)  
\[\begin{cases}\mathbf{u}(\mathbf{x},t)=\phi(\mathbf{x},t),&\forall\mathbf{x} \in\Gamma_{D}\\ \Big{(}\
u\frac{\partial\mathbf{u}}{\partial\mathbf{n}}-p\mathbf{n}\Big{)}( \mathbf{x},t)=\psi(\mathbf{x},t),&\forall\mathbf{x}\in\Gamma_{N}\end{cases}\]  
where \(\mathbf{u}_{0}\), \(\phi\) and \(\psi\) are given functions, \(\Gamma_{D}\) and \(\Gamma_{N}\) form a partition of the boundary of \(\Omega\) and \(\mathbf{n}\) is the outward-facing unit normal to \(\partial\Omega\).  
The first kind of boundary condition is said of Dirichlet type, and \(\Gamma_{D}\) will be addressed as the Dirichlet portion of the boundary, while the second kind is said of Neumann type, and \(\Gamma_{N}\) will be called the Neumann portion of the boundary.  
The Navier-Stokes Equations (1) are nonlinear, due to the term \((\mathbf{u}\cdot\
abla)\mathbf{u}\); moreover, there are no general results about existence, regularity and uniqueness of the solution, in particular in three dimensions. In fact, this is one of the most important open problems in mathematics and one of the Millennium problems.  
In the following, we will always consider the stationary Navier-Stokes equations, i.e., Equation (1) without the time derivative of \(\mathbf{u}\).