# Global existence of weak solutions to viscoelastic phase separation: Part I Regular Case

## TL;DR

This paper proves the existence of weak solutions for a complex viscoelastic phase separation model in two dimensions, combining Cahn-Hilliard and Peterlin-Navier-Stokes equations, and demonstrates solution behavior via finite element simulations.

## Contribution

It establishes the mathematical existence of weak solutions for a coupled viscoelastic phase separation model with polynomial potential in 2D.

## Key findings

- Demonstrates complex solution behavior during spinodal decomposition
- Validates the model using Lagrange-Galerkin finite element method
- Shows transient polymeric network structures in simulations

## Abstract

We prove the existence of weak solutions to a viscoelastic phase separation problem in two space dimensions. The mathematical model consists of a Cahn-Hilliard-type equation for two-phase flows and the Peterlin-Navier-Stokes equations for viscoelastic fluids. We focus on the case of a polynomial-like potential and suitably bounded coefficient functions. Using the Lagrange-Galerkin finite element method complex behavior of solution for spinodal decomposition including transient polymeric network structures is demonstrated.