article

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

Authors: Alexander Dressel and Christian Rohde

Issue: Volume 57 (2008), Issue 2, 717-756

Abstract:

$The aim of this paper is to study the existence and uniqueness ofsolutions 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 G. Andrews, \emph{On the existence of solutions to the equation $u_{tt} = u_{xxt} + \sigma(u_{x})_{x}$} (j. Differential Equations \textbf{35} (1980), 200--231), uniqueness and optimal regularity are proven. The results of this paper alsoapply to interaction potentials with non-vanishing negative part and constitute a base for an analysis of the time-asymptotic behaviour.$