Global existence in critical spaces for density-dependent incompressible viscoelastic fluids

TL;DR

This paper establishes local and global well-posedness results for density-dependent incompressible viscoelastic fluids in critical spaces, using approximation methods and uniform estimates.

Contribution

It introduces a novel approach combining Friedrichs method and hybrid Besov spaces to prove global existence for these complex fluid models.

Findings

Global existence is proven under small data conditions.

Uniform estimates for linearized models are obtained.

Approximation via ODE sequences facilitates the analysis.

Abstract

In this paper we consider the local and global well-posedness to the density-dependent incompressible viscoelastic fluids. We first study some linear models associated to the incompressible viscoelastic system. Then we approximate the system by a sequence of ordinary differential equations, by means of the Friedrichs method. Some uniform estimates for those solutions will be obtained. Using compactness arguments, we will get the local existence up to extracting a subsequence by means of Ascoli's lemma. With the help of small data conditions and hybird Besov spaces, we finally derive the global existence.