Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism

Abstract

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in ${\mathbb{R}}^d$ for the small initial data. Our proof is based on the observation that the behaviors of Green's matrix to the system of $(u,(-\Delta {)}^{-\frac{1}{2}}\mathbb{P}\nabla \cdot \tau )$ as well as the effects of $\tau$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.