given u and v as two vectors and T as a second-order tensor,
we know that the laplacian of a product of two vectors satisfies: $$\nabla^{2}(\textbf{u}\cdot\textbf{v})=\nabla^{2}\textbf{u}\cdot\textbf{v}+2\nabla\textbf{u}:\nabla\textbf{v}+\textbf{u}\cdot\nabla^{2}\textbf{v}$$ where $\cdot$ represent inner product and $:$ represent double inner product.
My question is what about the laplacian of a product of vector and tensor?
$$\nabla^{2}(\textbf{u}\cdot\textbf{T})=\ ??$$
$$\nabla^{2}(\textbf{T}\cdot\textbf{u})=\ ??$$