可压等熵Navier-Stokes方程组的适定性和奇性形成

发布时间：2018-03-13 08:37

本文选题：可压流体　切入点：等熵Navier-stokes方程组　出处：《上海交通大学》2015年博士论文　论文类型：学位论文

【摘要】：初值含真空的光滑大解的适定性和奇性理论是流体力学方程组数学理论的一个重要分支.本文主要在流体粘性系数为密度的幂律的情形下,对高维的可压等熵Navier-Stokes方程组的Cauchy问题进行了一系列研究.由于该系统在真空区域高度的退化性,光滑解的构造一直以来都是大家所关心的公开问题.本文在这个方向上做出了有一定价值的贡献.在第一章绪论中,我们简要介绍相关的数学模型及物理背景,回顾研究现状,阐述本文考虑问题的特点,数学上的主要困难及本文的创新之处,并且给出了必要的预备知识.本论文主要结果如下：一.含真空的光滑解的局部存在性.基于对该系统在真空区域中数学结构的分析,我们合理地给出了流体速度在真空区域上随时间变化的演化机制.在此基础上,通过充分利用该方程组中双曲算子的对称结构和退化椭圆算子的弱光滑性,以及引进人工粘性,我们得到了一系列与初始密度的下界无关的先验估计,从而成功的建立了初值含真空的高维光滑大解的局部(时间)存在性理论.在这里,我们特别指出,在真空区域上,粘性系数关于密度的不同幂次对应了控制流体速度随时间演化的方程组的不同数学结构.二.粘性消失极限.对于高维的可压等熵流的含真空(在某些开集或者无穷远处)的光滑大解,我们建立了从Navier-Stokes方程组至Euler方程组的粘性消失极限,并且在Sobolev空间Hs(s'为属于[2,3)的任意常数)中给出了相应的收敛速率.事实上,我们可以证明可压等熵Navier-Stokes方程组的解的生命跨度具有与粘性系数无关的一致正下界.并且,值得注意的是,我们还可以对所得的解,在H3空间中建立一个与粘性系数无关的一致能量估计,这样就使得粘性消失极限的建立成为可能.三.解的奇性形成.基于流体速度在真空区域上随时间变化的演化机制的分析,我们给出了两类必然会导致正则解在有限时间内爆破的初始条件.事实上,我们将会证明,对于某些初值,如果真空出现在某些局部开集上,那么我们得到的解肯定会在有限时间内发生爆破,不论初值是多么的光滑,并且在任意意义下充分小.那么这就使得,即使在初值充分小的情形下,建立一般的整体光滑解存在的理论变得不可能.同时我们指出了,由于我们得到的光滑解满足积分意义下的动量守恒定律,那么就不会得到满足流体速度的L∞范数会随着时间趋向无穷而衰减到零的整体解.四.Beale-Kato-Majda型爆破准则.对应于前面给出的奇异性结果,我们在粘性系数线性依赖于密度的情形下给出了相应的Beale-Kato-Majda型爆破准则.更准确地说,我们证明了△ρ/ρ和D(u)=(%絬+%絬T)/2的L∞范数控制了光滑解可能的爆破.这就意味着,如果解在初始时刻是光滑的,但是在以后的某一个时刻失去了光滑性,那么这个奇异性一定是由于解在这个临界时间上失去了Vρ/ρ或者D(u)的L∞范数的上界引起的.
[Abstract]:Well posedness and singularity theory of smooth solution of initial vacuum is an important branch in the theory of fluid mechanics equations in mathematics. This paper mainly in the viscous coefficients for power-law density in the case, conducted a series of studies on high dimensional compressible isentropic Navier-Stokes equations of the Cauchy problem. Because of the system in the vacuum region highly degenerate, smooth solution structure has always been open issues of concern to everyone. This paper makes a valuable contribution in this direction. In the first chapter, we briefly introduce the mathematical model and the physical background related to the review of the research status, this paper consider the characteristics of the problem. In this paper, the mathematics of the main difficulties and innovation, and give the necessary prior knowledge. The main results of this thesis are as follows: 1. The local existence of smooth solutions of the vacuum based on the system. In the vacuum region mathematical structure analysis, we reasonably show the evolution mechanism of the fluid velocity varies with time in the vacuum area. On this basis, by making full use of the structure of symmetric hyperbolic operator equations and degenerate elliptic operators and weak smoothness, and the introduction of artificial viscosity, we obtained a priori estimates independent the lower bound of a series of initial density, thus successfully established local initial vacuum high dimensional smooth solution (time) existence theory. Here, we pointed out that in the vacuum region, the viscosity coefficient on the density of different power corresponding to the equations of different mathematical structures with time evolution control fluid speed. Two. The vanishing viscosity limit. For high dimensional vacuum pressure containing isentropic flow (in some open or infinity) smooth solution, we established from Navier-Stokes to E equations The Uler equations of the vanishing viscosity limit, and in the Sobolev space Hs (S') belongs to [2,3) the arbitrary constants are given in the corresponding convergence rate. In fact, we can prove that compressible isentropic Navier-Stokes equations of the life span has nothing to do with the viscous coefficient of uniform positive lower bound. Moreover, worth noting is that we can on the solution, to establish a consistent energy independent and viscous coefficient estimation in H3 space, which makes possible the establishment of viscous limit disappeared. Three. Singular solution. The formation of evolution mechanism of fluid velocity varies with time in the vacuum region based on our proposed the two class will inevitably lead to the regular solution of initial conditions of blow up in finite time. In fact, we will prove that for some initial value, if the vacuum appeared in some local open set, then we get the solution must be in Blow up in finite time, regardless of the initial value is so smooth, and in any sense. So it has a sufficiently small, even in the case of the initial value is small enough, the establishment of a general theory of the existence of global smooth solution is not possible. At the same time, we point out, because we get the smooth solution to meet the law of conservation of momentum integral in the sense, then it does not satisfy the L norm of the fluid velocity with time tends to infinity and decay to zero solutions. Four.Beale-Kato-Majda type blasting criterion. The singularity corresponds to the result given above, we depend on the density of the case are given Beale-Kato-Majda type blasting corresponding criterion in viscous coefficient linear. Precisely, we prove that the delta Rho / Rho and D (U) = (T +%% Xie Xie) L norm /2 control of the smooth solution possible blasting. This means that if the solution at the initial moment is smooth However, a certain time in the future loss of smoothness, then the singularity is due to certain solutions in this critical time lost V P / P or D (U) caused by the L norm of the upper bound.

【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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