Speaker：Prof. Xu Xiaojing

        Beijing Noraml University

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

Time：5^th, Dec., 2020, 10:00

Location：实验楼105 (线下)

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.

Speaker Introduction：

许孝精，博士，北京师范大学数学科学学院教授、博士生导师、副院长。 2000年本科和2005年博士毕业于吉林大学。主要从事偏微分方程及其应用方向的研究，重点研究来自流体动力学中的偏微分方程组的适定性。博士学位论文获吉林省优秀博士学位论文，曾获北京师范大学励耘优秀青年教师一等奖。合作编写本科生教材《数学物理方程》（教育部“十一五”国家级规划教材），论文发表在JMPA、SIMA等等国际重要数学期刊。