Speaker：Prof. Xiaojing Xu

          Beijing Noraml University

Title: Sharp decay estimates for Oldroyd-B model with only fractional stress tensor diffusion

Time：5th, Dec., 2020, 10:00

Location：Conference Room 105 at Experiment Building at Haiyun Campus

Abstract:

Precise large-time behavior of physical quantities plays a crucial role in understanding many physical phenomena. For partial differential equation (PDE) models with full dissipation, powerful methods such as the Fourier-splitting technique have been developed. However, these methods may not apply to PDE systems with only partial dissipation. In this talk, I shall offer new ideas on how to obtain precise large-time decay estimates on a partially dissipated system. We examine the d-dimensional incompressible Oldroyd-B model without velocity dissipation and with only fractional diffusive stress. The discovery here is that the coupling and interaction of the velocity and the non-Newtonian stress actually enhances the regularity and the stability of the system. Without the stress, the Sobolev norms of the velocity could grow rather rapidly in time, let alone decay at explicit rates. Making use of the interaction, we deduce a system of damped wave equations obeyed by the velocity and the Leray projection of the divergence of the stress. By constructing a suitable Lyapunov functional, we are able to control the growth in the derivatives and extract explicit decay rates. The optimal decay rates are established by representing the wave equations in an integral form and applying a bootstrapping argument.