Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by
$$ L f (x) = b(x) f '(x) $$
with domain $D(L) := C^1(\mathbb T;\mathbb R)$. We assume here $b\in C^1(\mathbb T;\mathbb R)$.
In the case $b(x)\neq 0$ for every $x$, we have that $L$ is closed (use theorem for exchanging uniform convergence and derivation) and generates a strongly continuous semigroup $T_t$ on $C(\mathbb T;\mathbb R)$ (by using that $b(x) f'(x) -f(x) = g(x)$ has a $C^1$ solution $f$ for every continuous $g$).
In general ($b(x)$ possibly having zeros) $(L,D(L))$ is no longer closed.
Does its closure still generate a strongly continuous semigroup?
(In case, can I characterise the domain of the closure more explicitly?)