# Boundary condition for Pressure in Navier-Stokes equation

I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf

I could not really understand the description. Could someone explain a little bit more?

It says, "For the lid driven cavity problem this means that homogeneous Neumann boundary conditions are prescribe everywhere. This implies in particular that the pressure $P$ is only defined up to a constant, which is fine, since only the gradient of $P$ enters the momentum equation."

What does it mean "only defined up to a constant"? By solving the Poisson equation, you obtain the values of $P$ and can calculate its gradient. The constant means this pressure value? And if it's not "only up to constant", what do we get? derivative or integral?

When we say that the pressure is only defined "up to a constant", what we mean is this: If $u,p$ is a solution to the Navier-Stokes equations, then $u,p+c$ for any constant $c$ is also a solution. In other words, the solution is not unique, but the non-uniqueness has a very particular structure: two solutions $u,p$ and $u,p'$ can only differ in a way where $p-p'=const$.
For the lid driven cavity problem, we apply the Dirichlet boundary condition (or, no-slip boundary condition) to the velocity field $\tt U$, that is, $\tt U = U_{\tt Lid}$ on the moving Lid and $\tt U=0$ on the rest of the boundary. Based on this, we should have ${\tt U}^{n+1} = {\tt U}^{n}$ on the boundaries. Hence, if we have $\tt U^{n+1} = U^{n} - \Delta t \nabla P ^{n+1}$ everywhere (as shown in the article) then $$\tt \nabla P ^{n+1} = 0$$ must be hold on the boundaries (this is so-called the homogeneous Neumann boundary conditions). In addition, if a pressure field $\tt P$ satisfies the momentum equations then $\tt P + const$ also satisfies.