【作者】 尹俊平；

【导师】 谭忠；

【作者基本信息】 厦门大学 ， 应用数学， 2008， 博士

【摘要】 本文主要考虑了Navier-Stokes-Poisson方程径向强解存在唯一性问题,以及Navier-Stokes-Poisson方程和完全的Navier-Stokes方程强解的存在性结果.Navier-Stokes-Poisson方程具有很强的物理背景,主要可以描述天体星云在有粘性和有重力时的运动状态.而Euler-Poisson方程刻画星云运动时并没有考虑粘性的影响;Navier-Stokes方程也能反映宇宙中流体的运动状况,但Navier-Stokes方程并没有考虑流体自身重力对流体运动的影响.因此,Navier-Stokes-Poisson方程是比Navier-Stokes方程和Euler-Poisson方程更精确地刻画星云运动的数学物理模型.对于Euler-Poisson方程早在十九世纪已有大量研究,并且得到了很多有意义的结果,其中包括存在性,唯一性,稳定性,静态解的存在性,多解问题,有固核和无固核的径向解的存在性;而Navier-Stokes方程方程更是目前流体研究的热门之一,许多优秀的数学家都致力于这方面的研究,并且取得了许多结果,包括三维情形弱解的全局存在性,本质上有固核的径向弱解的全局存在性,以及一维情形强解的全局存在唯—性,解的大时间行为和一些稳定性结果,还有光滑解有限时刻爆破行为.这些研究在本文的参考文献中都可以查阅.但是对于Navier-Stokes-Poisson方程主要只有下面的少量结果:（1）[66]中关于有限能量弱解的全局存在性;（2）[81-82]中自由边界问题径向弱解的存在性;（3）[95]中给出了当粘性项消失时,确实Navier-Stokes-Poisson方程会收敛到Euler-Poisson方程.与Navier-Stokes方程及Euler-Poisson方程比较而言,Navier-Stokes-Poisson方程结果是很少的,既然Navier-Stokes-Poisson方程是更能精确描述星云运动规律的数学物理模型,自然我们想来探讨对于Navier-Stokes方程和Euler-Poisson方程成立的结果是否对Navier-Stokes-Poisson方程仍然成立.出于这种目的,本文对Navier-Stokes-Poisson方程进行了大量的研究,并取得了一些很有意义的结果.就研究方法而言,Navier-Stokes方程和Euler-Poisson方程主要是用紧性方法去处理弱解和强解的存在性问题,当然紧性方法也可以用于Navier-Stokes-Poisson方程解的存在性证明,并且我们用传统的紧性方法也进行了一些研究,主要得到了:（1）三维Navier-Stokes-Poisson方程有固核的径向强解的存在性;（2）一维Navier-Stokes-Poisson方程强解的全局存在性;（3）三维Navier-Stokes-Poisson方程强解的局部存在性等结果.紧性方法的主要思路是:（i）处理原系统的逼近系统的一致估计;（ii）利用逼近解所在空间的紧性对逼近解抽取收敛子列收敛到某—个极限函数;（iii）证明逼近解收敛到的极限函数为原系统的解.其中紧性方法难点之—是（iii）,为了克服或避免繁琐的紧性来得到收敛过程,在处理强解的方法上我们有所创新,我们引入了所谓的迭代方法,且用迭代方法我们得到关于Navier-Stokes-Poisson方程及完全的Navier-Stokes方程新的结果:（1）三维情形Navier-Stokes-Poisson方程强解的局部存在性及大时间爆破判据;（2）三维情形带能量方程的Navier-Stokes-Poisson方程的强解局部存在唯一性;（3）三维情形粘性项依赖于密度时完全的Navier-Stokes方程强解的局部存在性.我们引入的迭代方法的主要思路为:（i）考虑线性化系统强解的存在性;（ii）通过线性化系统构造迭代系统,并对迭代逼近系统的强解做一致估计;（iii）处理迭代逼近解序列的收敛性,具体地说,以完全的Navier-Stokes-Poisson方程为例,设{p^k},{u^k},{e^k},{Φ^k}是我们考虑迭代逼近系统的逼近解序列,利用（ii）中的一致估计,通过处理迭代逼近系统相互作差所得的系统,可以得到下面的估计:通过这—估计,自然可以得到迭代解序列的强收敛性.与紧性方法相比,我们引入的处理强解存在性的新方法的主要优点在于我们处理逼近解的收敛性时,直接从迭代逼近系统的估计可以得到,从而避免了通过繁琐的紧性讨论来得到解的收敛性的过程.并且本文用迭代方法处理了带有热传导的能量方程的Navier-Stokes-Poisson方程强解问题,得到了较—般系统更具有意义和更新颖的结果.本文各章内容安排如下:第一章:引言.主要介绍物理背景,研究历史.第二章:基础知识.介绍本文的主要符号和所用的基本引理.第三章:用紧性方法处理三维情形Navier-Stokes-Poisson方程径向强解的存在唯一性.主要结果是:当初值满足0≤p₀∈H¹,u₀∈H₀¹∩H²且满足初值建立如下兼容性条件时,得到了Navier-Stokes-Poisson方程径向强解的存在唯一性.第四章:用紧性方法处理一维情形Navier-Stokes-Poisson方程全局强解的存在唯一性.主要结果是:当初值满足p₀∈H¹（0,1）,u₀∈H₀¹（0,1）,无需兼容性条件时,得到了1-维Navier-Stokes-Poisson方程全局强解的存在性;以及当初值满足p₀∈H¹,u₀∈H₀¹∩H²,且对初值建立如下兼容性条件时,1-维Navier-Stokes-Poisson方程正则性更高的强解存在唯一性.值得一提的是,这部分内容中,我们利用Lagrangian流的特点,把问题转化为Lagrangian流以后,证明了密度的全局有界性,正是由于该结果,才导致后面我们定理的结果.第五章:用紧性方法处理三维情形Navier-Stokes-Poisson方程局部强解的存在性,唯一性,稳定性.主要结果是:当初值满足p₀∈W^1,6,u₀∈H₀¹∩H²且对初值建立如下兼容性条件:时,得到了Navier-Stokes-Poisson方程的局部强解的存在性、唯—性、稳定性.第六章:用迭代方法处理三维情形Navier-Stokes-Poisson方程局部强解存在唯一性及大时间爆破判据.主要结果是:当初值满足p₀∈W^1,q,u₀∈H₀¹∩H²（3<q<+∞）且△Φ₀=4πg（p₀-（?））,对初值建立如下兼容性条件:时,Navier-Stokes-Poisson方程局部强解的存在唯一性及大时间爆破判据,并且揭示了兼容性条件与强解存在性之间的关系.第七章:用迭代方法处理带能量方程的Navier-Stokes-Poisson方程局部强解的存在唯—性.主要结果是:当初值满足p₀≥0,p₀∈W^1,q,（e₀,u₀）∈H₀¹∩H²且对初值建立如下兼容性条件:时,得到了完全的Navier-Stokes-Poisson方程的强解局部存在唯一性.第八章:用迭代方法处理完全的Navier-Stokes方程局部强解的存在唯一性.主要结果是:当初值满足0≤p₀∈W^1,q,（e₀,u₀）∈H₀¹∩H²并对初值建立如下兼容性条件:时,得到了完全的Navier-Stokes方程粘性系数依赖于密度时的强解存在唯一性.更多还原

【Abstract】 In this paper, we consider the existence and the uniqueness of radially symmetric strong solutions to the Navier-Stokes-Poisson equations, and the existence of the strong solutions to the Navier-Stokes-Poisson equations and the full Navier-Stokes equations.The Navier-Stokes-Poisson equations are derived from physical problems, which mainly describe the motion of gasous stars with viscosity and self-gravitation. However, the Euler-Poisson equations depict the motion of celestial bodies without viscosity, and the Navier-Stokes equations may reflect the motion of the fluids too, which do not consider the infection of the self-gravition to fluids. The Navier-Stokes-Poisson equations are the more rigorous and better models that reflect the motion of gasous stars than the Navier-Stokes equations and the Euler-Poisson equations.In the 19th century, as to the Euler-Poisson equations, many mathematicians have had lots of research and gotten various results, including existence, uniqueness, stability, the existence for stationary solutions, and the existence of the symmetric solutions with solid core and without solid core. Moreover, presently, the Navier-Stokes equations are the very hot direction in fluids. Many excellent mathematicians devote to this field, and have gotten a great deal of results, including the global existence of weak solutions in 3-dimension, the global existence of the symmetric weak and strong solutions with solid core in 3-dimension, the global existence of the strong solutions in 1-dimension, the local existence of the strong solutions in 3-dimension, the large time behavior of weak or strong solutions, some stabilities relying on initial condition or viscous terms, the large time blow-up behavior for the smooth solutions, and so on. For details, please refer the references. But, as to the the Navier-Stokes-Poisson equations, there are a few results, mainly including （1） in [66], the global existence of the limited energy weak solutions in 3-dimension; （2） in [81-82], the global existence of the symmetric weak solution to the free-boundary systems; （3） in [95], when the viscosity of the Navier-Stokes-Poisson equations disappears, the Navier-Stokes-Poisson equations would converge to the Euler-Poisson equations. Compared to the Navier-Stokes and Euler-Poisson equations, the Navier-Stokes-Poisson equations have a lack of results. Since the Navier-Stokes-Poisson equations can more properly describe the motions of gasous stars, naturally, it is very important to consider the well posedness of the weak and strong solutions to the Navier-Stokes-Poisson equations. Hence we do some research and have gotten many significant results.As to methods, usually, compact method can handle the existence of the weak or strong solutions to the Navier-Stokes and Euler-Poisson equations. Obviously, it can be used to deal with the Navier-Stokes-Poisson equations. Moreover, using compact method, we concluded the following results: （1） the global existence and uniqueness of the radially symmetric strong solution to the Navier-Stokes-Poisson equations in three dimension; （2） the global existence, uniqueness of strong solution to the Navier-Stokes-Poisson equations in one dimension; （3） the local existence, uniqueness, stability of the strong solution to the Navier-Stokes-Poisson equations in three dimension. The main ideas of compact method are that: （i） we get the uniform estimates of approximate system; （ii） using the compactness for the spaces of approximate solutions, we can obtain that a subsequence of approximate solution converges to a limit function; （iii） we are required to prove that the solution of original problem is this limitation. The mainly difficult problem of compact method is step （iii）. To avoid and overcome fussy discussion to compactness, we innovate methods to deal with the existence of strong solutions. We introduce the iterative method. And using our iterative method, we have gotten the following results: （1） the local existence of the strong solutions to the Navier-Stokes-Poisson equations and a blow-up criterion in three dimensions; （2） the local existence of strong solutions to the full Navier-Stokes-Poisson equations in three dimensions;（3） the local existence of strong solutions to the density-dependent full Navier-Stokes equations in three dimensions. The main ideas of iterative method are that: （i） prove the existence of the strong solution to linearized systems; （ii） from the existence of the strong solution to linearized systems, construct iterative approximate systems, and estimate uniformly the strong solution of iterative approximate system; （iii） deal with the convergence to the sequence of the iterative approximate strong solutions, precisely, for example, as to the full Navier-Stokes-Poisson equations, if we suppose that {p^k}, {u^k}, {e^k}, {Φ^k} are the sequences of the strong solutions to the iterative approximate systems, using the uniform estimates in step （ii）, we deal with the difference systems of the iterative approximate systems,thenThrough this estimates, naturally, we can get the strong convergence of the iterative approximate solutions. The virtue of iterative method is that we use the estimates of the iterative approximate systems to deal with the convergence of iterative approximate solution,and then avoid the fussy discussion of compact methods. Furthermore, in this paper, we get the local existence of strong solutions to the Navier-Stokes-Poisson equations with heat conductivity, and some more meaningful and novel results.This paper is organized as follows:ChapterⅠ:Introduction. It includes physical backgrounds and researched history.ChapterⅡ:Basic knowledge. It includes basic notions and basic Lemmas.ChapterⅢ:The uniqueness and existence of the radially symmetric strong solutions to the Navier-Stokes-Poisson equations in three dimension. This result was dealed with by compact method. For detail, the main results are: if 0≤p₀∈H¹,u₀∈H₀¹∩H² and satisfy the compatibility conditionwe get the existence and the uniqueness of the symmetric strong solutions to Navier-Stokes-Poisson equations in three dimensions.ChapterⅣ:The global existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations in one dimension. This result was gotten by compact method. For detail, the main results are: supposed that p₀∈H¹（0,1）,u₀∈H₀¹（0,1）,we get the global strong solutions to the Navier-Stokes-Poisson equations in one dimension. Furthermore, if p₀∈H¹,u₀∈H₀¹∩H² and the compatibility conditionthere exists the global strong solution with higher regularity for 1-D NFavier-Stokes-Poisson equations. It is important that through Lagrangian techniques（we translate the initial system into Lagrangian fluids, then combine the character of Lagrangian fluids）, we get the global uniform boundary of density. Many excellent estimates can be concluded by this boundary;ChapterⅤ:The well-posedness problems of the strong solutions to the Navier-Stokes-Poisson equations in three dimensions. This result can be proved by compact method. For detail, the main results are: if p₀∈W^1,6,u₀∈H₀¹∩H² and satisfy the compatibility conditionwe get the exitence, uniqueness, stability of local strong solutions to the Navier-Stokes-Poisson equations in three dimensions;ChapterⅥ:The local existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations and a blow-up criterion in three dimensions. This result was dealed with by iterative method. For detail, the main results are: if p₀∈W^1,q,u₀∈H₀¹∩H²（3<q< +∞）,ΔΦ₀=4πg（p₀-（?））,and satisfy the compatibility conditionwe get the local existence and uniqueness of the strong solutions to the Navier-Stokes-Poisson equations and the blow-up criterion with large time, describe the relation between the existence of the strong solutions and the compatibility conditions;ChapterⅦ:The uniqueness and existence of the strong solutions to the full Navier-Stokes-Poisson equations in three dimensions. This result was gotten by iterative method. For detail, the main results are: if p₀>0,p₀∈W^1,q,（e₀,u₀）∈H₀¹∩H² and satisfy the compatibility condition there exists the local strong solutions to the full Navier-Stokes-Poisson equations in threedimensions;ChapterⅧ:The uniqueness and existence of the strong solutions to the full Navier-Stokesequations with density-dependent in three dimensions. This result was gotten by iterativemethod. For detail, the main results are: if 0≤p₀∈W^1,q,（e₀,u₀）∈H₀¹∩H² and satisfythe compatibility condition,the existence and uniqueness of local strong solution to the density-dependent full Navier-Stokes equations in three dimensions can be concluded.更多还原