Let $\Omega \subset \mathbb{R}^n$ be a bounded open set and let $f(\cdot,\cdot)$ and $g(\cdot,\cdot)$ be functions from $[0,T]\times \Omega$ into $\mathbb{R}$.
Suppose that $f \in C^1([0,T];H^s(\Omega))$ and $g \in C^1([0,T];H^1(\Omega))$.
The multiplcation $(fg)(t,x) = f(t,x)g(t,x)$ is sensible, and we can consider $(fg)(t) := f(t,\cdot)g(t,\cdot)$ so that $(fg)\colon [0,T] \to L^1(\Omega)$.
Is it true that $fg$ is in $C^1([0,T];L^1(\Omega))$, and does the product rule $$(fg)'(t) = f'(t)g(t) + f(t)g'(t)$$ hold, in what sense?