  In this article, the author investigates flow lines of the classical Willmore
flow, which start to move in a smooth parametrization of a Hopf-torus in
$\mathbb{S}^3$. We prove that any such flow line of the Willmore flow exists
globally, in particular does not develop any singularities, and subconverges to
some smooth Willmore-Hopf-torus in every $C^{m}$-norm. Moreover, if in addition
the Willmore energy of the initial immersion $F_0$ is required to be smaller
than or equal to the threshold $\frac{8\pi^2}{\sqrt{2}}$, then the unique flow
line of the Willmore flow, starting to move in $F_0$, converges fully to a
conformally transformed Clifford torus in every $C^{m}$-norm, up to time
dependent, smooth reparametrizations. Key instruments for the proofs are the
equivariance of the Hopf-fibration $\pi:\mathbb{S}^3 \longrightarrow
\mathbb{S}^2$ w.r.t. the effect of the $L^2$-gradient of the Willmore energy
applied to smooth Hopf-tori in $\mathbb{S}^3$ and to smooth closed regular
curves in $\mathbb{S}^2$, a particular version of the Lojasiewicz-Simon
gradient inequality, and a well-known classification and description of smooth,
arc-length parametrized solutions of the Euler-Lagrange equation of the elastic
energy functional in terms of Jacobi Elliptic Functions and Elliptic Integrals,
dating back to the 80s.
