Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant $\text{om}\in (0,1)$. It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in ${\dot{H}}^s\cap {\dot{B}}_{2,1}^{\text{fr}d2}$ with $-\text{fr}d2<s<\text{fr}d2-1$. In particular, if $d\ge 3,$ and taking $s=0$, then ${\dot{H}}^0\cap {\dot{B}}_{2,1}^{\text{fr}d2}\approx B_{2,1}^{\text{fr}d2}$. Since $B_{2,\mathrm{\infty }}^s\hookrightarrow B_{2,1}^{\text{fr}d2},s>\text{fr}d2$, this result extends the work by Chen and Miao [Nonlinear Anal.,{68}(2008), 1928–1939].