How to use Navier-Stokes equation to model incompressible flow when pressure moves towards infinity?

I have a container of inviscid, incompressible fluid with a piston at one end. It is completely closed, except the fact that I can move the piston(say $x_p$). From definition of Bulk Modulus–

$$
mathcal{B}=-Vfrac{partial P}{partial V}
$$

Since $mathcal{B}=infty$ for any incompressible fluid, so any displacement change to the piston will give rise to infinite pressure.

I am thinking how to model this rise in pressure using Navier-Stokes equation. Let me write the simplified equations-

$$
frac{partial{u}}{partial{x}}=0
frac{partial{u}}{partial{t}} + u frac{partial{u}}{partial{x}}=-frac{1}{rho} frac{partial p}{partial x}
$$

If I give some finite value of displacement to piston, how to show that $p rightarrowinfty$?