可压Navier-Stokes方程及相关流体动力学方程研究

项目名称： 可压Navier-Stokes方程及相关流体动力学方程研究

项目编号： No.10871134

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 电工技术

项目作者： 李海梁

作者单位： 首都师范大学

项目金额： 30万元

中文摘要： 本项目研究可压Navier-Stokes方程及相关流体动力学方程的定性性态，如适定性、正则性、渐近行为等，取得了多项结果。比如，对粘性系数依赖密度的一维可压Navier-Stokes 方程的初边值问题，当初始值分片光滑并含真空时，证明了整体弱解的存在性、正则性、真空的有限时间内消失现象、及其对解的爆破、渐近性态的影响；建立了一维自由界面的动力学特征和扩张速率；建立了高维空间自由界面问题整体弱解的存在性和正则性、流体运动的Lagrange结构性质、自由界面的动力学特征、以及高维真空的形成、消失机制。对高维单极可压缩Navier-Stokes-Poisson方程，证明了整体强解在Besov或Sobolev空间的整体适定性、以及在能量空间的时间最优衰减速率，建立了因电场的影响所导致动量衰减变慢的结果。对高维双极可压缩Navier-Stokes-Poisson方程组，证明了整体经典解的时间最优衰减速率，阐明了内蕴电场、以及不同电荷间的相互作用对电荷运输的本质影响；等等。发表相关SCI论文19篇，包括Comm. Math. Phys., Arch. Ration. Mech. Anal.,等。

中文关键词： 可压缩Navier-Stokes方程；适定性；渐近行为

英文摘要： In this project we have investigated the well-posedness, the regularities, and the asymptotics of the solutions to the compressible Navier-Stokes equations and the related hydrodynamical models. First, we dealt with the compressible Navier-Stokes equations with density-dependent viscosity coefficients (CNS), and established the global existence of weak solutions and the finite time vanishing of the vacuum state to the initial boundary value problem of the CNS in one-dimension for piecewise regular initial data with finite vacuum included. We further proved the global well-posedness of weak solution to the free boundary problem for the one-dimensional CNS with the stress free boundary condition, and obtained the regularities of the solution and the long time expending rate of the free boundary. We also established the global existence，the regularities and the Lagrange transport properties of the multi-dimensional spherically symmetric weak solutions to the free boundary problem of the CNS with the stress free boundary condition, and investigated the regularities and the time-expanding rates of the free boundary and the finite time vanishing or long time formation of vacuum state. In addition, we proved the global well-posedness of global strong solution to the CNS equations with additional term such as rotation or capillary, etc. Then, we obtained the global well-posedness of strong solution to the Cauchy problem for multi-dimensional compressible unipolar Navier-Stokes-Poisson equations in Sobolev space and Besov space respectively when the initial data is a small perturbation of the constant state. In particular, we established the optimal time-decay rates of the solution in L^2 norm, and showed that the optimal rate of the momentum was less than the rate of the compressible Navier-Stokes (CNS) equations due to the influence of electric filed governed by the Poisson equation. We further established the optimal time-decay rates of the global solution in L^2 norm to the compressible bipolar Navier-Stokes-Poisson equations, especially, we proved that the optimal decay rate of each individual momentum is slower than the rate of the CNS equations, but the decay rate of total momentum of two momentum terms was faster than each individual momentum and almost the same as the CNS equations, because of the influence of electric filed and the interplay between the two carriers. These revealed the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.

英文关键词： Compressible Navier-Stokes Equations; Well-posedness; Asymptotics