Global existence and optimal decay of solutions to the incompressible Oldroyd-B model with only stress tensor dissipation and without damping mechanism

Abstract

We study the $d$-dimensional ($d\geq2$) incompressible Oldroyd-B model with only stress tensor diffusion and without velocity dissipation as well as the damping mechanism on the stress tensor. Firstly, based upon some new observations on the model, we develope the pure energy argument (independent of spectral analysis) in general $L^p$ framework, and present a small initial data global existence and uniqueness of solutions to the model. Our results yield that the coupling and interaction of the velocity and the non-Newtonian stress actually enhances the regularity of the system. Later, by adding some additional $L^2$ type conditions on the low frequencies of the initial data $(u_0,\tau_0)$, %but without any more smallness restrictions, we obtain the optimal time-decay rates of the global solution $(u,\tau)$. Our result solves the problem proposed in Wang, Wu, Xu and Zhong \cite{Wang-Wu-Xu-Zhong} ({\it J. Funct. Anal.}, 282 (2022), 109332.).