Navier-Stokes方程组及相关复杂流体力学模型的若干数学问题

In our previous NSFC program, we completed 25 papers, including 21 SCI papers. The applicant was awarded by the Program for New Century Excellent Talents in University. After a suitable randomization, we construct the local unique strong solution for incompressible NS equations with a large set of initial data in L^2. We consider the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier-Stokes equations with a density-dependent viscosity coefficient, where the initial data are only small in the energy-norm. . Moreover, we give a description of the long time behavior of the solution, study the propagation of singularities in solutions, and show that if there is a vacuum domain initially, then the vacuum domain will exist for all time, and vanishes as time goes to infinity. And we also study the same problem for the 2D viscous liquid-gas two-phase flow model. We consider the boundary layer effect as the zero shear viscosity limit for the Navier–Stokes equations of compressible flows with density-dependent viscosity coefficient and cylindrical symmetry. In this project, we will continue to study the fluid or complex fluid equations by exploiting the geometry, modern analysis, and stochastic analysis methods. We will study the relationship between the local (global) existence of solutions and some properties of the system, such as the nonlinearity of the equations, the regularity and decay property of the initial data, etc. Searching a class of large initial data, such that the incompressible NS system or the related complex fluid systems are global wellposed. Continue to use the concentrated compactness methods to study the breakdown or the long time behavior of the solutions. Continue to study the relationship between the viscosity term and the global wellposedness of the Cauchy problem of the incompressible NS equations and other related equations. Continue to study the random data Cauchy theory of the fluid dynamic systems.

申请人主持的青年科学基金项目按计划顺利进行，完成论文25篇，其中发表（录用）SCI论文21篇；申请人入选教育部“新世纪优秀人才支持计划”；通过概率化初值方法证明了不可压缩NS方程组关于一大类L^2初值是局部适定的；得到了第二粘性系数依赖于密度的高维NS方程组关于小能量初值的整体解存在性和长时间性态、真空发展的估计、奇性发展分析，并研究了相应的气固两相粘性流体运动方程组；研究了粘性依赖于密度的柱面对称可压缩NS方程组的边界层问题等。在本项目中我们将继续应用几何与现代分析技术、随机分析方法等来研究流体或复杂流体力学方程组的适定性问题。探讨方程的非线性程度、初值正则性和解的衰减性对解存在性的深层影响；寻找新的一类大初值使得NS系统和相关模型具有整体适定性；继续用集中紧原理研究解的长时间性态和解的破裂性质；研究粘性项对系统的适定性的影响；利用概率化初值的方法研究流体力学方程组等。

本项目按计划顺利完成，发表录用SCI论文15 篇。我们研究了有重要意义的几类方程，如三维轴对称不可压缩Navier-Stokes 方程组，变密度不可压缩Navier-Stokes方程组，二维不可压缩推广的Boussinesq系统，描述复杂流体的粘弹性流体力学方程组等。应用现代分析技术等研究了系统的适定性问题；探讨了方程的非线性程度、初值正则性和解的衰减性对解存在性的深层影响等。得到了变密度不可压缩Navier-Stokes方程组柯西问题关于一类大初值的整体适定性。研究了三维轴对称不可压缩Navier-Stokes方程组的柯西问题，得到了旋转速度$u^\theta$在临界空间中的正则性指标，证明了关于小初始旋转速度的整体适定性。并进一步研究了变密度系统关于小初始旋转速度的整体适定性，以及速度关于时间的衰减率估计。还研究了描述复杂流体的一类粘弹性流体力学方程组Oldroyd-B模型，其中耦合系数不小，证明了系统关于一类大振荡初始速度的整体适定性。还研究了复杂流体中的一类具有各向异性的双曲-抛物耦合系统，利用法形式的思想和流体的不可压缩性质，提供了一种简便的能量估计方法，得到了系统的适定性。