可压缩Navier-Stokes方程和Boltzmann方程解的渐近行为

Compressible Navier-Stokes equations and Boltzmann equation, which have deep physical background and practical significance, are the fundamental equations in the mathematical theory of fluid dynamics. The theoretical study on the asymptotic behavior of solutions for compressible Navier-Stokes equations and Boltzmann equation has been one of the hot spots of research in the field of PDE. In fact,.the hydrodynamic limit of Boltzmann equation is one of the fundamental part of the famous Hilbert’s sixth problem, “Mathematical treatment of the axioms of physics”. On the other hand, the vanishing viscosity limit of initial boundary problem of Navier-Stokes equations, which is closely related to the boundary layer, is a very difficult problem. Up to now, there are many important progresses in these areas. Especially, the applicant and his coworkers have proved the hydrodynamic limit of Boltzmann equation when the Riemann solution of Euler equations is superposition of shock wave, rarefaction wave and contact discontinuity. But many problems are still open. For example, most of the known results are achieved in the framework of Riemann solution. When the initial data is consist by three piecewise constants (interacting data), then the elementary waves may interact at some finite time.. This program is mainly contributed to the study of the asymptotic behavior of solutions for compressible Navier-Stokes equations and Boltzmann equation. Specially, we will study: 1. the asymptotic behavior of the solutions of Cauchy problem with interacting data for one-dimensional compressible Navier-Stokes equations and Boltzmann equation; 2. the vanishing viscosity limit of multi-dimensional compressible Navier-Stokes equations with Navier slip boundary conditions in the general domain.

可压缩Navier-Stokes方程和Boltzmann方程是流体力学中的基本方程，有着很强的物理背景和实际意义，对其解的渐近行为的研究一直以来都是偏微分方程中的研究热点。 事实上，Boltzmann方程的流体动力学极限是著名的Hilbert第六问题的核心内容之一。另一方面，由于边界层的出现，Navier-Stokes方程初边值问题解的粘性极限也非常具有挑战性。目前为止，流体极限方面的研究吸引了众多学者的关注，取得了很多研究成果，但是仍有许多未解决的数学难题。本项目将主要研究：1. 在允许波发生碰撞的情形下的一维可压缩Navier-Stokes方程和Boltzmann方程Cauchy问题解的渐近行为，即Navier-Stokes方程解的粘性极限和Boltzmann方程解的流体动力学极限；2.高维可压缩Navier-Stokes方程初边值问题解的粘性极限。

可压缩Navier-Stokes方程是流体力学中的基本模型，用来描述粘性流体的运动。Boltzmann方程描述了“稀薄气体”的运动规律，是统计力学中的基本方程。对可压缩Navier-Stokes方程和Boltzmann方程的研究一直以来都是偏微分方程中的研究热点。本项目主要研究可压缩Navier-Stokes方程和Boltzmann方程解的渐近行为。首先，我们证明了碰撞激波情形下一维等熵可压缩Navier-Stokes方程解的消失粘性极限；对于无穷远处状态为两个不同常数时，证明了一维可压缩Navier-Stokes方程在一般初值情形的低马赫数极限，还发现了新的波现象；我们还首次严格地从一维Boltzmann方程得到了Korteweg理论。相关论文分别发表在Sci. China Math、Advances in Mathematics、Quarterly of Applied Mathematics上。其次，我们还证明了一般区域中高维可压缩Navier-Stokes方程的Navier-slip类型初边值问题的解的消失粘性极限，相关论文发表在SIAM J. Math. Anal.、 Arch. Rational Mech. Anal.、Math. Models Methods Appl.Sci.上。最后，除了项目中提出的研究目标，我们还证明了Boltzmann方程的一类大初值解的整体适定性并得到了解的大时间衰减速率，相关论文发表在 Arch. Rational Mech. Anal.上。