双曲平衡律系统解的定性行为

发布时间：2018-01-29 01:34

本文关键词： 双曲平衡律双曲守恒律 Jin-Xin松弛系统可压缩Euler方程大时间稳定性和奇性形成　出处：《清华大学》2016年博士论文　论文类型：学位论文

【摘要】：双曲平衡律系统是气体动力学的主要研究对象,双曲守恒律系统是双曲平衡律系统中最简单最典型的例子.近150多年来,关于双曲平衡律系统,特别是关于双曲守恒律系统的研究硕果累累.其中,主要的研究结果是关于其Cauchy问题古典解的爆破理论以及弱解的稳定性.在这些成果的基础上,本论文证明了一维空间Jin-Xin松弛系统在初始值有小扰动情形下的激波和接触间断波耦合解的大时间渐近稳定性和一维空间可压缩Euler方程组在一般压力及大初始值情形下的古典解会在有限时间内爆破.关于Jin-Xin松弛系统弱解的大时间稳定性问题,本论文证明了对于一维空间Jin-xin松弛系统的Cauchy问题,在初始具有小扰动时,其整体解在时间趋于无穷大时收敛到Jin-Xin松弛系统的平衡态系统Riemann问题的激波和接触间断波耦合解.具体的证明过程主要依赖于经典的加权能量估计和关于热核的精细衰减估计.特别地,本论文移除了在单一接触间断波的结果中施加在线性退化特征场上的一个结构性条件.关于Euler方程组古典解的爆破问题,本论文证明了对于一维空间可压缩Euler方程组的Cauchy问题,在初始值不添加小性假设及压力只是密度和熵的一般函数时,其古典解会在有限时间内爆破.通过建立密度的一致上界估计及对于由解的导数构成的非耦合Riccati型常微分方程的精细估计,最终证明了解的导数会在有限时间内爆破.值得注意的是,密度的一致上界估计是得到大初始值情形下爆破发生的核心.
[Abstract]:Hyperbolic equilibrium law system is the main research object of gas dynamics. Hyperbolic conservation law system is the simplest and most typical example of hyperbolic equilibrium law system. Especially, the research on hyperbolic conservation law system is fruitful. The main research results are about the blasting theory of classical solution of Cauchy problem and the stability of weak solution. On the basis of these results. In this paper, we prove the large time asymptotic stability of coupled solutions of shock wave and contact discontinuous wave for one-dimensional space Jin-Xin relaxation system with small initial disturbance. In the case of general pressure and large initial value, the classical solution will blow up in finite time. For the large time stability problem of weak solution of Jin-Xin relaxation system. In this paper, we prove that for the Cauchy problem of one-dimensional space Jin-xin relaxation system, when the initial perturbation is small. The global solution converges to the coupling solution of shock wave and contact discontinuous wave for the equilibrium system Riemann problem of Jin-Xin relaxation system when the time tends to infinity. The concrete proof process mainly depends on the classical. Weighted energy estimates and fine decay estimates for hot nuclei. In particular. This paper removes a structural condition applied to the linear degenerate characteristic field in the results of a single contact discontinuous wave. On the blow-up problem of classical solutions for Euler equations. In this paper, we prove that for the Cauchy problem of compressible Euler equations in one-dimensional space, when the initial values do not add small assumptions and the pressure is only a general function of density and entropy. Its classical solution will blow up in finite time by establishing a uniform upper bound estimate of density and a fine estimate of the uncoupled Riccati type ordinary differential equation consisting of the derivatives of the solution. It is proved that the derivative of the solution will blow up in a finite time. It is worth noting that the uniformly upper bound estimate of density is the core of the explosion in the case of large initial values.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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