Logarithmically Improved Blow-up Criteria for a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model

TL;DR

This paper develops logarithmically improved criteria for predicting blow-up in a coupled Navier-Stokes and phase field model describing vesicle membrane deformation in viscous flows, using vorticity in Besov spaces.

Contribution

It introduces novel blow-up criteria based on vorticity in Besov spaces for a complex fluid-structure interaction model, improving previous understanding of solution regularity.

Findings

Established two logarithmically improved blow-up criteria.

Criteria depend solely on vorticity in homogeneous Besov spaces.

Results apply to three-dimensional vesicle-fluid interaction models.

Abstract

In this paper, we study a hydrodynamical system modeling the deformation of vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier-Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces.