Local well-posedness of incompressible viscous fluids in bounded cylinders with 90°-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^∘-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. We study the transformed problem in an L^p-in-time and L^q-in-space setting with 2<p<∞ and 3<q<∞ satisfying 2/p+3/q<1, which provides a weaker setting than the one that was allowed by Wilke (2020) whose case was p=q>5.