Global Solution to the Incompressible Oldroyd-B Model in Hybrid Besov Spaces

Abstract

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant $\om\in (0,1)$.  It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in $\dot{H}^s\cap\dot{B}^{\fr{d}{2}}_{2,1}$ with $-\fr{d}{2}<s<\fr{d}{2}-1$.  In particular, if $d\ge3,$ and taking $s=0$, then $\dot{H}^0\cap\dot{B}^{\fr{d}{2}}_{2,1}\approx B^{\fr{d}{2}}_{2,1}$. Since $B^{s}_{2,\infty}\hookrightarrow B^{\fr{d}{2}}_{2,1}, s>\fr{d}{2}$, this result extends the work by Chen and Miao [Nonlinear Anal.,{68}(2008), 1928--1939].