Blow up solutions to a viscoelastic fluid system and a coupled Navier-Stokes/Phase-Field system in R^2

TL;DR

This paper constructs explicit finite-time blow-up solutions for the Oldroyd-B viscoelastic fluid model and a coupled Navier-Stokes/Phase-Field system in two-dimensional space, highlighting potential singularity formation in these models.

Contribution

It provides the first explicit blow-up solutions for these complex fluid models, advancing understanding of their singularity behavior.

Findings

Explicit blow-up solutions constructed

Finite-time singularity demonstrated

Insights into fluid model stability and singularities

Abstract

We find explicit solutions to both the Oldroyd-B model with infinite Weissenberg number and the coupled Navier-Stokes/Phase-Field system. The solutions blow up in finite time.