Minimizing movements solutions for a monotone model of droplet motion

TL;DR

This paper analyzes the uniqueness and regularity of minimizing movements solutions for a droplet model with piecewise monotone forcing, revealing conditions for unique evolution and establishing regularity results.

Contribution

It provides a classification of solutions, reduces the evolution to elliptic problems, and proves regularity, advancing understanding of droplet motion models.

Findings

Solutions evolve uniquely on monotone intervals

Branching non-uniqueness occurs at jumps and monotonicity changes

Established $L^ Infty_tC^{1,1/2-}_x$-regularity of solutions

Abstract

We study the uniqueness and regularity of minimizing movements solutions of a droplet model in the case of piecewise monotone forcing. We show that such solutions evolve uniquely on each interval of monotonicity, but branching non-uniqueness may occur where jumps and monotonicity changes coincide. This classification of minimizing movements solutions allows us to reduce the quasi-static evolution to a finite sequence of elliptic problems and establish $L^\infty_tC^{1,1/2-}_x$-regularity of solutions.