【作者】 蔡晓静；

【导师】 酒全森； 辛周平；

【作者基本信息】 首都师范大学 ， 基础数学， 2008， 博士

【摘要】 本文由四部分组成:第一部分研究带有阻尼项α|u|^β-1u（α＞0）的不可压Navier-Stokes方程解的存在性和唯一性。利用Galerkin方法,我们得到了带有阻尼项不可压Navier-Stokes方程Cauchy问题当β≥1时,存在全局的弱解;当β≥7/2时,存在全局的强解,并且当7/2≤β≤5时强解是唯一的。第二部分研究的是带有阻尼项α|u|^β-1u（α＞0）的不可压Navier-Stokes方程Cauchy问题弱解的大时间性态。利用Fourier分解方法,我们得到了解的衰减率为-3/2,这与经典的不可压Navier-Stokes方程的衰减率是一样的。第三部分对三维有界及无界区域上的非齐次不可压Navier-Stokes方程解的全局存在性和全局稳定性进行了讨论。利用细致的能量估计,在解满足适当的条件下,对初始数据加上更正则的条件和相容性条件,如果给初值一个小扰动,我们证明了解是整体存在的（0,∞）,而且此解是全局稳定的。第四部分对二维Boussinesq方程进行了一些讨论。利用Schauder不动点定理,我们得到了当初始旋度光滑时,二维Boussinesq方程存在唯一的古典解。但当初始旋度属于L¹（R²）或者是Radon测度空间时,是否存在整体的解还未证明,进一步的研究正在进行中。更多还原

【Abstract】 This thesis is composed of four parts.In the first part,the existence and uniqueness of the incompressible Navier-Stokes equations with damping are studied.By the Galerkin method,we show that the Cauchy problem of the Navier-Stokes equations with dampingα｜u|^β-1u （α＞0） has global weak solutions for anyβ≥1,global strong solutions for anyβ≥7/2 and that the strong solution is unique for any 7/2≤β≤5.In the second part,we investigate the large time behavior of weak solutions for the Cauchy problem of the Navier-Stokes equations with dampingα｜u｜^β-1u（α＞0,β≥3）.By the Fourier splitting method,we show that the decay rate of the solutions is -3/2,which is the same as the classical incompressible Navier-Stokes equations.In the third part,we mainly study the global existence and stability of the solutions for the nonhomogeneous incompressible Navier-Stokes equations in three-dimensional bounded or unbounded domains.By delicate energy estimates,under some more regular conditions and some compatibility condition on the initial data,we obtain that if the initial data are small perturbation on those of a known strong solution,then there exists a global solution,which defined on（0,∞） for the nonhomogeneous incompressible Navier-Stokes equations and is a perturbation of the known one.The strong solution of the nonhomogeneous incompressible Navier-Stokes equations is stable.In the fourth part,we give some remarks on the two dimensional Boussinesq equations.By the Schauder fixed points,we get that when the initial vorticity is smooth,there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.When the initial vorticity belongs to L¹（R²）（or the Radon measure space）,whether there exists global existence of the solutions is still open.It will be investigated in future.更多还原