# On a model for a sliding droplet:Well-posedness and stability of translating circular solutions

## TL;DR

This paper analyzes a mathematical model for a viscous droplet sliding down an inclined plane, establishing conditions for well-posedness and demonstrating the stability of circular translating solutions under certain slope and boundary conditions.

## Contribution

It provides a rigorous analysis of the well-posedness and stability of the model, identifying critical slopes and conditions for the stability of circular solutions.

## Key findings

- Well-posedness holds for slopes below a critical value.
- Translating circular solutions are asymptotically stable under small inclines.
- Loss of well-posedness occurs when the linearization is no longer parabolic.

## Abstract

In this paper the model for a highly viscous droplet sliding down an inclined plane is analyzed. It is shown that, provided the slope is not too steep, the corresponding moving boundary problem possesses classical solutions. Well-posedness is lost when the relevant linearization ceases to be parabolic. This occurs above a critical incline which depends on the shape of the initial wetted region as well as on the liquid's mass. It is also shown that translating circular solutions are asymptotically stable if the kinematic boundary condition is given by an affine functionand provided the incline is small.