# Non-unique pressure for Navier-Stokes for incompressible fluid

The Navier-Stokes system for incompressible fluid says \begin{aligned} \dot{\mathbf{u}} + (\mathbf{u}\cdot\nabla)\mathbf{u}-\nu\Delta\mathbf{u}+\nabla{p} &= 0,\\ \nabla\cdot\mathbf{u}&=0. \end{aligned} The non-slip boundary condition for the velocity $\mathbf{u}$ says $\mathbf{u}|_{\partial\Omega}=0.$ But the pressure boundary condition is unclear. Most references I found use Neumann boundary condition which is $\nabla{p}\cdot\mathbf{n}=0$ on $\partial\Omega$ ($\mathbf{n}$ is normal vector of $\partial\Omega$). As consequence, $p$ can be determined up to an arbitrary additive constant! Does this make any sense in physics? How can I determine the constant?

There is also an initial value condition for $p$. We can add an arbitrary function $q(t)$ satisfying $q(0)=0$ to $p(\mathbf{x},t).$ That is, if $p(\mathbf{x},t)$ is a solution, so is $p(\mathbf{x},t)+q(t).$

no matter how deep it is in the water because it is the change in the pressure that determines the flow. So yes, $p$ being defined to an additive constant makes complete sense.
I don't know that you can solve for the constant because there are a multitude of gauges, $\phi$, that can satisfy the $\nabla\phi=0$ requirement in the NS equations.
Note that the B.C. in $\nabla p\cdot n$ you mention is only an artifact of some numerical techniques where the pressure equation is not solved directly (Chorin method).