气体运动学方程解的大时间行为

Compressible Navier-Stokes equations and Boltzmann equation are both fundamental equations of gas motion, which have the deep physical background and application value. They are closely related and asymptotic equivalent to each other. In fact, the Chapman-Enskog expansion in second order of Boltzmann equation is compressible Navier-Stokes equations. The study on the large-time behavior of solutions for the above equations is a hot issue in partial differential equation, many famous mathematicians have made great contribution in this direction. However, there are still many unsolved difficulty mathematical problems. The applicant has made some progress as follows: 1. We obtain the stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system under large perturbation；2. We prove the time-asymptotic stability of a superposition of two viscous shock waves for one-dimensional Boltzmann equation under the general initial perturbation without the zero total macroscopic mass condition. ..This program is mainly contributed to the following two topics: 1. Large-time behavior of solutions to the inflow problem for the compressible Navier-Stokes system under large initial perturbation. 2. The stability of solutions for the Boltzmann equation under large initial perturbation.

可压缩Navier-Stokes方程和Boltzmann方程是气体运动学的基本方程，均有重要的物理背景和实际意义。二者密切相关，渐近等价。事实上，Boltzmann方程的二阶Chapman-Enskog展开正是可压缩Navier-Stokes方程。关于其解的大时间行为的研究一直是非线性偏微分方程的热点问题，很多著名数学家均做出过重要贡献。但是，仍然有很多未解决的数学难题。申请人完成了以下工作：1. 在大扰动情形下，证明了可压缩Navier-Stokes方程粘性接触间断波复合疏散波的稳定性；2. 在初始质量非零的情况下，证明了Boltzmann方程两个粘性激波的渐近稳定性。. 本项目将进一步研究: 1. 在初始扰动大的情形下，可压缩Navier-Stokes方程内流问题解的大时间行为；2. 在初始扰动大的情形下，Boltzmann方程基本波的稳定性。

可压缩Navier-Stokes方程和Boltzmann方程是气体运动学的基本方程，均有重要的物理背景和实际意义。二者密切相关，渐近等价。事实上，Boltzmann方程的二阶Chapman-Enskog展开正是可压缩Navier-Stokes方程。关于其解的大时间行为的研究一直是非线性偏微分方程的热点问题，很多著名数学家均做出过重要贡献。但是，仍然有很多未解决的数学难题。本项目主要研究: 1. 在初始扰动大的情形下，可压缩Navier-Stokes方程初边值问题解的大时间行为；2. 在初始扰动大的情形下，Boltzmann方程基本波的稳定性。具体来说，对于可压缩Navier-Stokes方程，我们首先证明了对于一般气体内流问题边界层解的存在性，并在初始扰动小的条件下证明了其稳定性，发表在SIAM J. Math. Anal.。对于理想多方气体，我们证明了边界层解复合疏散波和粘性接触间断波在大扰动条件下的稳定性，发表在Nonlinearity。另外，我们证明了可压缩Navier-Stokes方程平面疏散波在三维无穷长平直管道内的渐近稳定性，发表在Arch. Rational Mech. Anal.。对于动力学方程，我们证明了双极Vlasov-Poisson-Boltzmann方程复合波的渐近稳定性，发表在SIAM J. Math. Anal.。关于Boltzmann方程平面疏散波在三维无穷长平直管道内的渐近稳定性，这一结果发表在Kinet. Relat. Models。