Minimizing movements solutions for a monotone model of droplet motion

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 $L_t^{\infty }{C}_x^{1,1/2-}$-regularity of solutions.

22 pages, 1 figure