Find divergence free field with compact support

Question

I search a divergence free vector field which is smooth and compactly supported. Take any divergence free vector field on some compact set and extend it by 0 on the outside. The vector field has compact support. This means that the convolution has compact support. Let $F$ be the name of the vector field. Is the divergence of the convolution the same as the convolution of the divergence? I think this should solve the problem. If we know \begin{align} \int_{\mathbb{R}^n}(F\ast \varphi_{\varepsilon}) \nabla \varphi dx=\int_{\mathbb{R}^n} \text{div}(F \ast\varphi_{\varepsilon}) \varphi dx=\int_{\mathbb{R}^n} (\text{div}(F)\ast \varphi_{\varepsilon}) \varphi dx=0 \end{align} where $\varphi_{\varepsilon}$ is the standard mollifier. This means for any weakly divergence free vector field with compact support we can find a smooth vector field with compact support right?