# Mini-Workshop on Nonlinear Partial Differential Equations and Fluid Mechanics

演讲人：李竞，闫伟，黄祥娣，罗天文

讲座时间：2017-11-30 14:30:00

讲座地点：信息楼257会议室

讲座内容

14:30 - 15:10 李竞 教授 (中科院数学与系统科学研究院)

*On the global strong solutions of the compressible Navier-Stokes equations with large data*

15:10 - 15:50 闫伟 教授(北京应用物理与计算数学研究所)

*Blow up of smooth solutions to compressible Navier-Stokes equations*

15:50 - 16:10 茶歇 (Tea Break)

16:10 - 16:50 黄祥娣 教授(中科院数学与系统科学研究院)

*Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier–Stokes System with Vacuum and Large Oscillations*

16:50 - 17:30 罗天文 教授(清华大学)

* Some Results on The Three-Dimensional Prandtl Equations*

**On the global strong solutions of the compressible Navier-Stokes equations with large data**

Jing Li

Chinese Academy of Sciences

**Abstract.** In this talk, we will present some results on the global existence of strong and weak solutions of the compressible Navier-Stokes equations with large data. In particular, both the time-independent upper bound of the density and the large-time behavior of the strong and weak solutions are also obtained.

**Blow up of smooth solutions to compressible Navier-Stokes equations**

Wei Yan

Institute of Applied Physics and Computational Mathematics, Beijing

**Abstract. **In this talk, we talk about the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see definition in the paper). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy. Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions.

**Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier–Stokes System with Vacuum and Large Oscillations**

Xiangdi Huang

Chinese Academy of Sciences

**Abstract.** For the three-dimensional full compressible Navier–Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid, we establish the global existence and uniqueness of classical solutions with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish.

Moreover, for the initial data, which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from a vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.

**Some Results on The Three-Dimensional Prandtl Equations**

Tianwen Luo

Tsinghua University

**Abstract. **We obtain a class of weak solutions of the three-dimensional Prandtl equations, which are related to the secondary flows in the three-dimensional boundary layers. This is a joint work with Prof. Zhouping Xin.

演讲人简介

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