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

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.