报告题目：Some well-posedness results for the multi-dimensional viscoelastic flows

报告人：张挺教授，浙江大学

报告时间：2024年5月16日（周四）9:00

报告地点：20-200

报告摘要：In this talk, we mainly focus on the multi-dimensional viscoelastic flows of Oldroyd type. First, considering a system of equations related to the incompressible viscoelastic fluids of Oldroyd–B type, we obtain the existence and uniqueness of the global solution, and the pointwise estimates of solutions. Then, considering a system of equations related to the compressible viscoelastic fluids of Oldroyd–B type with the general pressure law, $P′(\bar{\rho})+\alpha>0$, with $\alpha >0$ being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data $u_0$ and the low frequency part of $\rho_0$, $\tau_0$ are small enough compared to the viscosity coefficients. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces.  At Last, considering the multi-dimensional compressible Oldroyd-B model, which is derived by Barrett, Lu, and Suli (Comm. Math. Sci. 2017) through the micro-macro analysis of the compressible Navier-Stokes-Fokker-Planck system in the case of Hookean bead-spring chains. We would provide a unified method to study the system with the background polymer number density $\eta_\infty\geq0$, including the vanishing case and the nonvanishing case, and establish the global-in-time existence of the strong solution for the associated Cauchy problem when the initial data are small in the critical Besov spaces.   

报告人简介：张挺，理学博士，浙江大学数学科学学院教授，博士生导师，入选国家**计划“**支持计划”，教育部“**支持计划”,浙江省**青年科学基金项目获得者，曾获教育部自然科学奖二等奖(第三完成人)，浙江省教学成果奖一等奖（6/8）。主要研究方向是偏微分方程及其应用。考虑了有重要物理背景的一类粘性依赖于密度的Navier-Stokes方程的自由边界问题。当密度为零时，粘性系数会退化为零，使问题产生了本质的困难。考虑不同情况，如密度是否连续、有无外力影响、有无外压强影响等，研究了一维系统或球面对称系统的整体（局部）适定性、解的渐近性态和收敛率估计等问题。利用调和分析方法，在各向异性的Sobolev-Besov空间中，研究了粘性是各向异性的三维Navier-Stokes方程组的整体（局部）适定性问题。利用概率论方法，探讨不可压缩 Navier-Stokes方程组在低正则性空间中的适定性问题等。在《Arch. Rational Mech. Anal.》、《Commun. Math. Phys.》、《Math. Ann.》等杂志上发表文章一百多篇。