I have the system formed by the equations of conservation of mass and momentum: $$\mathrm{div}(\vec{v}) = 0 \tag1$$ $$\triangle\,\vec{v}=\vec{grad}\,p \tag2$$ Providing also their respective boundary conditions.
I have solved this system noting that because of the linearity and the commutativity of the Laplacian, in the second equation, one can take its divergence to obtain: $$\triangle\,p=0 \tag3$$ Subjected to some additional boundary conditions.
My question is:
Can I assert that the solution given by equation $(2)$ once calculated the pressure distribution from $(3)$ fulfills $(1)$?
Thank you in advance for your answers.