  We consider a free boundary problem of the Navier--Stokes equations in the
three-dimensional Euclidean space with moving contact line, where the
90$^\circ$-contact angle condition is posed. We show that for given $T > 0$ the
problem is local well-posed on $(0, T)$ provided that the initial data are
small. In contrast to the strategy in Wilke (2013), we study the transformed
problem in an $L^p$-in-time and $L^q$-in-space setting, which yields the
optimal regularity of the initial data.
