Sharp rates of decay and global-in-time stability of large solutions to the three-dimensional incompressible phan-thien-tanner system of polymeric flows

This paper develops the sharp rates of decay and the global-in-time stability of global solutions with general initial data to the three-dimensional incompressible Phan-Thien-Tanner (PTT) system. The PTT system has shown great advantages in the dynamic properties of polymer fluids. This paper is devoted to showing the sharp upper and lower bounds of decay rates of large solutions themselves and their spatial derivatives (including highest-order spatial derivatives of veloc-ity) for the PTT system. The other result is that large solutions to the PTT system are stable. We show that any perturbed solution will remain close to the reference solution if initially they are close to each other. The proof is based on time-weighted energy estimates, Fourier time-splitting, and semigroup methods. As a byproduct, our results can be applied to the simpler case of the Oldroyd-B system.