Identity for laplacian of a vector dotted with a vector

[dakg]

Homework Statement

I have $\int \nabla^2 \vec{u} \cdot \vec{v} dV$ where u and v are velocities integrated over a volume. I want to perform integration by parts so that the derivative orders are the same. This is the Galerkin method.

Homework Equations

The Attempt at a Solution

I have found identities involving $\nabla \vec{u}$ and $\nabla \vec{v}$ as a tensor scalar product and I have tried to work out a product rule:
$\nabla \cdot (\vec{v} \cdot \nabla \vec{u}) = \nabla \vec{u} : \nabla \vec{v} = \nabla^2 \vec{u} \cdot \vec{v}$.

I am having trouble figuring out if this is correct. I know i have scalars on the right hand side. On the left hand side I have the divergence of a vector dotted with a tensor, which I think will lead to a scalar.

Any help is most appreciated.

Thank you,
dakg

 

[dakg]

Is it Green's First Identity that I need? Does it hold for vectors?