  We offer a complete description of $THH(E(2))$ under the assumption that the
Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an
$E_\infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2)
\rightarrow THH(E(2))\rightarrow \overline{THH}(E(2))$ and describe
$\overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring
spectrum. We state general results about the $K(i)$-local behaviour of
$THH(E(n))$ for all $n$ and $0 \leq i \leq n$. In particular, we compute
$K(i)_*THH(E(n))$.
