关于Euler和Navier-Stokes方程斯托克斯方程的一些已知结果由不同的作者得到。二维Navier-斯托克斯方程解的存在性和光滑性是人们长期以来所知的。Leray证明了三维空间中的Navier-斯托克斯方程存在弱解。Scheffer和Shnirelman得到了时空中具有紧支集的Euler方程的弱解。Caffarelli、Kohn和Nirenberg改进了Scheffer的结果，F. H. Lin简化了J.Leray结果的证明。关于Euler和Navier-斯托克斯方程的弱解的行为的许多问题和定理在Bertozzi和Majda、Constantin或Lemarié-Rieusset的著作中有描述。在$\cite{TT10}$中，利用Fourier变换和Laplace变换，用解析迭代法求解了二维和三维Navier-Stokes方程和Euler方程的初值问题（Cauchy问题）。这些解决方案是以无限可微函数的形式收到的，这使我们能够在更深的层次上和更详细地分析问题的各个方面。这种光滑解也满足$\cite{CF06}$中Navier-斯托克斯方程问题所要求的条件。对于问题参数的几种组合，得到数值结果，并以图形$\cite{TT10}$，$\;\cite{TT11}$表示。本文详细证明了三维Navier-斯托克斯方程Cauchy问题的分析迭代法的收敛性。收敛性示出了广泛的问题的参数范围。得到了迭代过程收敛域边界在系统参数空间的估计公式。同时，我们也证明了三维Navier-斯托克斯方程Cauchy问题的解析迭代解法的正确性。
英文摘要：
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about behavior of weak solutions of the Euler and Navier-Stokes equations are described in the books of Bertozzi and Majda, Constantin or Lemarié-Rieusset. Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for 2D and 3D cases were obtained in the convergence series form by analytical iterative method using Fourier and Laplace transforms in paper $\cite{TT10}$. These solutions were received in a form of infinitely differentiable functions, and that allows us to analyze all aspects of the problem on a much deeper level and with more details. Also such smooth solutions satisfy the conditions required in $\cite{CF06}$ for the problem of Navier-Stokes equations. For several combinations of problem parameters numerical results were obtained and presented as graphs $\cite{TT10}$,$\;\cite{TT11}$. This paper describes detailed proof of convergence of the analitical iterative method for solution of the Cauchy problem for the 3D Navier - Stokes equations. The convergence is shown for wide ranges of the problem's parameters. Estimated formula for the border of convergence area of the iterative process in the space of system parameters is obtained. Also we have provided justification of the analytical iterative method solution for Cauchy problem for the 3D Navier-Stokes equations.
Global Well-Posedness of Classical Solutions to the Cauchy problem of Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data
Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Navier-Stokes Equations with Vacuum
Global Existence and Incompressible Limit of the Cauchy Problem for 2D Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）
The local existence and uniqueness of strong solutions for Cauchy problem of three-dimensional inhomogeneous incompressible Navier-Stokes-Vlasov equations
Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting Navier-Stokes and magnetohydrodynamic equations with vacuum at infinity
Global well-posedness and exponential decay to the Cauchy problem of nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum in $\mathbb{R}^2$
研究了具有依赖于密度的粘性和初始密度允许为零的非齐次Navier-斯托克斯方程在$\mathbb{R}^2$中强解的整体适定性.利用对数插值不等式和精细的能量估计，证明了当$\|\nablaμ(ρ_0)\|_{L^q}$适当小时，该方程强解的整体存在性，改进了Huang和Wang [SIAM J. Math. Anal. 46，1771- 1788（2014）]到整个空间情况。此外，我们还导出了解的指数衰减率。特别地，尽管存在真空，但不需要要求额外的初始相容性条件。
Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum
Strong solutions to the Cauchy problem of the two-dimensional compressible Navier-Stokes-Smoluchowski equations with vacuum
研究了以真空为远场密度的二维流体-颗粒相互作用模型Cauchy问题的局部强解的存在性。请注意，Ding等人在\cite{SBH}中使用的用于强解的相应3D局部适定性的技术不能处理2D情况，因为速度$u$的$L^p$ -范数（$p>2$）在这里不能仅根据$\sqrtρu$和$\nabla u$来控制。在本文中，我们将使用[J. Li，Z. Liang，On classical solutions to the Cauchy problem of the two-dimensional barotropic compressable Navier-Stokes equations with vacuum，J. Math. Pures Appl.（2014）640- 671]对于Navier-Stokes方程，以获得强解的局部存在性，前提是混合物中的初始密度和粒子密度在无穷远处不会非常缓慢地衰减。特别地，初始密度可以具有紧凑的支撑。本文扩展了方等。的结果\cite{DRZ}和Ding et al.的结果\cite{SBH}，其中当空间维数分别为$N=1$和$N=3$时，得到了存在性.
On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院）
