Well-posedness in Critical Spaces for the Density-dependent Incompressible Viscoelastic Fluid System

TL;DR

This paper proves local and global well-posedness for the density-dependent incompressible viscoelastic fluid system in Besov spaces, using fixed point methods under small initial data assumptions.

Contribution

It establishes the well-posedness of the system in critical Besov spaces and extends results to global solutions with small initial data.

Findings

Local well-posedness in Besov spaces for small initial data.

Existence of global solutions under small initial conditions.

Use of Schauder-Tychonoff fixed point argument for proof.

Abstract

We are concerned with the well-posedness of the density-dependent incompressible viscoelastic fluid system. By Schauder-Tychonoff fixed point argument, when $\|{1}/{\rho_0}-1\|_{\dot{B}_{p,1}^{{N}/{p}}}$ is small, local well-posedness is showed to hold in Besov space. Furthermore, provided the initial data $(1/\rho_0 -1,\upsilon_0,U_0-I)$ is small under certain norm, we also get the existence of the global solution.