So for the left side of the top equation I take it as the partial derivative of the vector U overtime then add vector U then take the dot product of the gradient of vector U then add one over the density times the gradient of pressure if that were the Gist?
For the right side of the equation I was thinking that the nabla dot nabla is meaning the Laplacian operator but I don't feel sure. Is that the Laplacian operator there?
For the bottom equation I take it as that the divergence of you should equal 0. Is that the Gist?
Right on all counts. By definition the Laplacian is the divergence of the gradient, so $\nabla^2 \vec{u}=\nabla \cdot \nabla \vec{u}$.
$\nabla \cdot \vec{u}=0 $ does mean the divergence of $\vec{u}$ is zero. This in turn implies $\vec{u}=\nabla \times \vec{A}$ for some vector field $\vec{A}$.
If I remember correctly there's an additional equation in play for conservation of mass: $\nabla \cdot (\rho \vec{u})+\frac{\partial \rho}{\partial t}=0$.
That implies $\nabla \rho \cdot \vec{u}+\rho \nabla \cdot\vec{u}+\frac{\partial \rho}{\partial t}=0$. Since the middle term is $0$, if the density is constant in time, it's also constant in space.