Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

Abstract

The aim of this paper is to study the existence and uniqueness of solutions of an initial-boundary value problem for a viscoelastic two-phase material with capillarity in one space dimension. Therein, the capillarity is modelled via a nonlocal interaction potential. The proof relies on uniform energy estimates for a family of difference approximations: with these estimates at hand we show the existence of a global weak solution. Then, by means of a nontrivial variant of the arguments in [2], uniqueness and optimal regularity are proven. The results of this paper also apply to interaction potentials with non-vanishing negative part and constitute a base for an analysis of the time-asymptotic behaviour.