Local well-posedness of incompressible viscous fluids in bounded cylinders with ${90}^{\circ }$-contact angle

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.