  We study one-parameter curves on the universal Teichm\"uller space $T$ and on
the homogeneous space $M=\Diff S^1/\Rot S^1$ embedded into $T$. As a result, we
deduce evolution equations for conformal maps that admit quasiconformal
extensions and, in particular, such that the associated quasidisks are bounded
by smooth Jordan curves. Some applications to Hele-Shaw flows of viscous fluids
are given.
