Suppose I have a sequence of functions $f_n$ converging to $f$ in the Sobolev space $H^s(\mathbb{R}^3)$. If $J_n u$ denotes smooth mollification, i.e., for some smooth function $\rho\ge 0$ with compact support and $\int_{\mathbb{R}^3} \rho \,dx =1,$ $$J_n u (x) = n^{-3} \int_{\mathbb{R}^3} \rho\left(\frac{x-y}{n}\right)u(y) dy,$$ will we have that $J_n f_n \to f$ in $H^s$? I know that for $u\in H^s$, $J_n u \to u$ in $H^s$, and writing down the definition didn't help much since you end up with a term involving $(f_n(y) - f(x))$.
The reason I ask is the following: let's say I have a sequence of functions $v^\epsilon (t,x)$ that converges to a function $v \in C([0,T]; C^2(\mathbb{R}^3))$ (note that they first obtain convergence in $C([0,T]; H^{m'}(\mathbb{R}^3))$, where $m'$ is less than some fixed $m>3/2 + 2$).
The book I'm reading asserts that then $$J_\epsilon^2 \Delta v^\epsilon - PJ_\epsilon [ (J_\epsilon v^\epsilon)\cdot \nabla (J_\epsilon v^\epsilon)] \to \Delta v- P(v\cdot \nabla v)$$
in the space $C([0,T]; C(\mathbb{R}^3))$ where $P$ is the Leray projector. An answer in the affirmative to my first question would allow me to understand the book's assertion to my actual confusion.
The important property is that $J_n$ is a $1$-Lipschitz operator on $L^2$ for each derivative, so it's an $1$-Lipschitz operator on $H^s.$ This means $$\|J_nf_n-f\|_{H^s}\leq \|J_nf_n-J_nf\|_{H^s}+\|J_nf-f\|_{H^s} \leq \|f_n-f\|_{H^s}+\|J_nf-f\|_{H^s}\to 0.$$
More generally this follows from Arzelà–Ascoli.