= Solution
The componentwise <product rule for the Laplacian> and the harmonicity of $\boldsymbol\Phi$ give
$$
\nabla^2(\mathbf x\mathbin{\cdot}\boldsymbol\Phi)
=2\nabla\mathbin{\cdot}\boldsymbol\Phi
+\mathbf x\mathbin{\cdot}\nabla^2\boldsymbol\Phi
=2\nabla\mathbin{\cdot}\boldsymbol\Phi.
$$
Hence
$$
\begin{aligned}
\nabla\mathbin{\cdot}\mathbf u
&=\nabla^2(\mathbf x\mathbin{\cdot}\boldsymbol\Phi+\chi)
-2\nabla\mathbin{\cdot}\boldsymbol\Phi\\
&=2\nabla\mathbin{\cdot}\boldsymbol\Phi+0
-2\nabla\mathbin{\cdot}\boldsymbol\Phi=0.
\end{aligned}
$$
Thus $\boxed{\nabla\mathbin{\cdot}\mathbf u=0}$.
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