Kind of things that I always find in books and I never remember.
I am looking for a simpler/expanded form of the LHS of the following poisson equation in the case $ \nabla \cdot \underline v =0$
$$ \rho\nabla \cdot ( (\underline v \cdot \nabla) \underline v) = - \Delta p $$
Your equation is not the usual Poisson equation... By rewritting the left hand side in cartesian coordinates, you get :
$$\nabla \cdot ((v\cdot \nabla)v )= \sum_{j=1}^n \sum_{i=1}^n \frac{\partial}{\partial x_j}\left(v_i \frac{\partial v_j}{\partial x_i} \right)$$
$$= \sum_{j=1}^n \sum_{i=1}^n \frac{\partial v_i}{\partial x_j}\frac{\partial v_j}{\partial x_i} + v_i\frac{\partial v_j}{\partial x_i \partial x_j} $$
$$=\sum_{j=1}^n \sum_{i=1}^n \frac{\partial v_i}{\partial x_j}\frac{\partial v_j}{\partial x_i} + \sum_{i=1}^n v_i \frac{\partial} {\partial x_i } \left( \sum_{j=1}^n \frac{\partial v_j}{\partial x_j} \right) $$
Now, by rewritting $\nabla\cdot v = 0$ in cartesian coordinate, you get $\sum_{j=1}^n \frac{\partial v_j}{\partial x_j} = 0$
So this means that
$$\nabla \cdot ((v\cdot \nabla)v )=\sum_{j=1}^n \sum_{i=1}^n \frac{\partial v_i}{\partial x_j}\frac{\partial v_j}{\partial x_i}$$
Edit : you can note this with the double dot product
$$ \sum_{j=1}^n \sum_{i=1}^n \frac{\partial v_i}{\partial x_j}\frac{\partial v_j}{\partial x_i} = \nabla v : (\nabla v)^t$$