I am trying to understand the Navier-Stokes equation for incompressible fluid flow. $\newcommand{\vect}[1]{{\bf #1}}$ $$ \frac{\partial \vect{u}}{\partial t} + (\vect{u} \cdot \nabla)\vect{u} - \nu\nabla^2\vect{u} = -\nabla \frac{p}{\rho_0} + \vect{g} \tag{1} $$
where
- $\vect{u}$ is the flow vector field
- $t$ is time
- $\nu$ is the viscosity of the fluid
- $p$ is the pressure (a map from $\mathbb{R}^3 \mapsto \mathbb{R})$
- $\vect{g}$ is some external force (a vector)
- $\rho$ is the density of the fluid.
Additionally the mass conservation condition $$ \nabla \cdot \vect{u} = 0\label{2}\tag{2} $$ holds.
According to wikipedia,
The solution of the equations is a flow velocity. It is a vector field - to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time.
Is the pressure given? Or are we also solving for the pressure?