玻尔兹曼方程和流体方程中的渐进极限和边界层分析问题

本项目围绕玻尔兹曼方程和流体方程的以下几个重要方面进行研究：（1）、有界区域内玻尔兹曼方程的DiPerna-Lions重整化解到不可压缩纳维-斯托克斯的流体极限收敛，特别是边界的几何性质对初始层和边界层的相互作用的影响；（2）、周期边界和全空间条件下从平衡态附近玻尔兹曼方程的经典解到弱可压缩纳维-斯托克斯的渐进极限问题；（3）、在解析解的范畴下，半平面上的玻尔兹曼方程到不可压缩欧拉方程的极限问题，这个过程涉及到非线性的普朗克边界层方程；（4）、有界区域内非截断碰撞核的具有麦克斯韦反射边界条件的玻尔兹曼方程和朗道方程的整体重整化解的存在性问题；（5）、在最佳物理尺度下从玻尔兹曼方程的DiPerna-Lions重整化解到声波方程的弱解极限问题；（6）、有界区域和半空间的线性玻尔兹曼方程的高阶声波展开问题；（7）、在周期边界和全空间条件下，具一般初值非等熵可压缩纳维-斯托克斯方程的不可压缩极限。

This proposal is about the following important research problems on the Boltzmann and fluid equations: (1),In bounded domain,the strong convergence justifications from the DiPerna-Lions renormalized solutions of the Boltzmann equations to incompressible Navier-Stokes equations with various boundary conditions.In particular the effect from the geometry of the boundary to the interactions between the initial layer and boundary layers; (2), For the periodic condition or whole space, the justification of the limit from the classical solutions of the Boltzmann equation around the equilibruim to the weakly compressible Navier-Stokes equations; (3), In the context of analytic solutions, the convergence from Boltzmann equation to incompressible Euler equations on half space. This process involves the nonlinear Prandtl equations; (4),The existence problem on the global renormalized solutions of the non-cutoff Boltzmann equation and Landau equations; (5), For the optimal physical scaling, the convergence from the DiPerna-Lions solutions of the Boltzmann equation to the acoustic system; (6), For the bounded domain or half space, the higher order acoustic expansion from the linear Boltzmann equation with Maxwell reflection boundary condition; (7),For the periodic boundary condition or whole space, the compressible incompressible limit of the nonisentropic compressible Navier-Stokes equations.