Density scaling invariance of incompressible Euler flow ($\rho \mapsto C\rho ,p\mapsto Cp$)

A homogeneous fluid in a prescribed gravitational field obeys $D\mathbf{u}/Dt=-\nabla (p/\rho )+\mathbf{g}$ and $\nabla \cdot \mathbf{u}=0$. Multiplying both constant mass density and pressure by the same positive factor preserves the velocity solution, provided initial and boundary data remain compatible. Thus velocity-only observations do not determine density in this model. Independently prescribed pressure gradients or the fluid's own gravity can break this invariance.