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

TL;DR

This paper establishes the global well-posedness of the incompressible Oldroyd-B model in critical Besov spaces without damping, for small initial data, by analyzing frequency-dependent behaviors of the system.

Contribution

It provides a novel proof of global existence in critical Besov spaces for the Oldroyd-B model without damping, using frequency analysis and Green's matrix properties.

Findings

Global well-posedness proven for small initial data

Behavior of Green's matrix analyzed across frequencies

Construction of frequency-dependent energy estimates

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 $\big(u,(-\Delta)^{-\frac12}\mathbb{P}\nabla\cdot\tau\big)$ 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.