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

TL;DR

This paper proves local well-posedness for the Navier--Stokes equations with a 90-degree contact angle in a bounded cylinder, using an $L^p$-$L^q$ framework for optimal initial data regularity.

Contribution

It introduces an $L^p$-$L^q$ approach to establish local well-posedness for viscous fluids with contact angles, improving upon previous methods.

Findings

Establishes local well-posedness for small initial data.

Uses an $L^p$-$L^q$ framework for optimal regularity.

Provides a new analysis for free boundary Navier--Stokes problems.

Abstract

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.