I'm struggling to prove the well-definedness of distribution.
Let $\vec{b} \in \mathcal{D}^\prime (\mathbb{R}^n)^n$. For $N>0$, define a smooth cutoff function $\psi_N (x)=\psi (x/N)$, where $$ \psi \in C^\infty_0 (\mathbb{R}^n),\quad \psi (x)=1\quad \text{if } |x|\leq \frac{1}{2},\quad \psi(x)=0 \quad \text{if } |x|>1.$$
Define $f_N =\Delta^{-1} \mathrm{div}(\psi_N \vec{b})$. Then I want to gurantee $f_N$ is well-defined in terms of distributions.
How can I show it?