关于高维可压缩流体方程的一些研究
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摘要
本文主要对高维可压缩流体方程做一些研究.包括可压缩Euler方程两维黎曼问题和可压缩Navier-Stokes-Poisson方程弱解的L~1稳定性以及球对称弱解的整体存在性.
全文共分为四章,其中第一章为引言,介绍所考虑问题的相关物理和数学背景以及本论文的主要结果.
在第二章里,对于两维可压缩Euler方程,我们构造了古典分片光滑解中一组光滑片解,这种解我们称为“半双曲斑片解”(semi-hyperbolic patch).这不同于以往的平面疏散波,冲击波,简单波或者是拟亚音速区域.“斑片解”是这样一个区域,它的一族特征或者是连续的,或者通过冲击波可以追溯到在无穷远处的边界;而它的另一族特征开始于音速线上,结束于音速线上或者在跨音速冲击波上.这种“斑片解”不仅在两维黎曼问题的数值模拟中常见,而且也出现在飞机机翼跨音速扰流问题和试图解释空气动力学中von Neumann悖论的Guderley反射中.
粘性系数依赖于密度的等熵可压缩Navier-Stokes-Poisson(NSP)方程既描述电子器件中带电粒子的输运,又可以用来描述天体物理学中气态星体的运动.由于NSP方程的粘性系数是密度依赖的,当真空(即ρ=0)出现时方程是退化的,就没有了关于速度的先验估计.因此,对于此方程逼近解的构造至今还是一个尚未解决的问题.作为研究的出发点,在第三章里,我们首先考虑弱解的L~1稳定性,通过得到关于密度梯度的新熵估计和精细的能量估计,我们可以证明逼近解的收敛性.
在第四章里,对于粘性系数依赖于密度的等熵可压缩NSP方程,在R~3中包含球心的球状区域内,我们得到了大始值的,球对称弱解的整体存在性.这是通过在两个球之间的环状区域上,对逼近解取极限得到的.
In this thesis we study the multi-dimensional compressible fluid flow,including two-dimensional Riemann problem for the compressible Euler Equations and the L~1-stability of weak solutions together with the global existence of spherically symmetric weak solutions for the compressible Navier-Stokes-Poisson(NSP) equations.
There are four chapters in this thesis.In Chapter 1,we introduce some related physical and mathematical backgrounds about the equations and problems we studied and our main results obtained in this thesis.
In Chapter 2,we construct patches of solutions,in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves,to the two-dimensional Euler equations.The semi-hyperbolic patch of solutions is different from the planar rarefaction,shock,simple,or pseudo-subsonic waves.This type of solutions appears in the transonic flow over an airfoil and Guderley reflection,and is common in the numerical solutions of Riemann problems.
In Chapter 3,we consider the isentropic compressible NSP equations arising from transport of charged particles or motion of gaseous stars in astrophysics.We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of(density) vacuum.We prove the L~1-stability of weak solutions for arbitrarily large data on multi-dimensional bounded or periodic domain or whole space.
In Chapter 4,we prove the existence of global weak solutions to the compressible NSP equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. The solutions are obtained as limits of solutions in annular regions between two balls.
全文共分为四章,其中第一章为引言,介绍所考虑问题的相关物理和数学背景以及本论文的主要结果.
在第二章里,对于两维可压缩Euler方程,我们构造了古典分片光滑解中一组光滑片解,这种解我们称为“半双曲斑片解”(semi-hyperbolic patch).这不同于以往的平面疏散波,冲击波,简单波或者是拟亚音速区域.“斑片解”是这样一个区域,它的一族特征或者是连续的,或者通过冲击波可以追溯到在无穷远处的边界;而它的另一族特征开始于音速线上,结束于音速线上或者在跨音速冲击波上.这种“斑片解”不仅在两维黎曼问题的数值模拟中常见,而且也出现在飞机机翼跨音速扰流问题和试图解释空气动力学中von Neumann悖论的Guderley反射中.
粘性系数依赖于密度的等熵可压缩Navier-Stokes-Poisson(NSP)方程既描述电子器件中带电粒子的输运,又可以用来描述天体物理学中气态星体的运动.由于NSP方程的粘性系数是密度依赖的,当真空(即ρ=0)出现时方程是退化的,就没有了关于速度的先验估计.因此,对于此方程逼近解的构造至今还是一个尚未解决的问题.作为研究的出发点,在第三章里,我们首先考虑弱解的L~1稳定性,通过得到关于密度梯度的新熵估计和精细的能量估计,我们可以证明逼近解的收敛性.
在第四章里,对于粘性系数依赖于密度的等熵可压缩NSP方程,在R~3中包含球心的球状区域内,我们得到了大始值的,球对称弱解的整体存在性.这是通过在两个球之间的环状区域上,对逼近解取极限得到的.
In this thesis we study the multi-dimensional compressible fluid flow,including two-dimensional Riemann problem for the compressible Euler Equations and the L~1-stability of weak solutions together with the global existence of spherically symmetric weak solutions for the compressible Navier-Stokes-Poisson(NSP) equations.
There are four chapters in this thesis.In Chapter 1,we introduce some related physical and mathematical backgrounds about the equations and problems we studied and our main results obtained in this thesis.
In Chapter 2,we construct patches of solutions,in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves,to the two-dimensional Euler equations.The semi-hyperbolic patch of solutions is different from the planar rarefaction,shock,simple,or pseudo-subsonic waves.This type of solutions appears in the transonic flow over an airfoil and Guderley reflection,and is common in the numerical solutions of Riemann problems.
In Chapter 3,we consider the isentropic compressible NSP equations arising from transport of charged particles or motion of gaseous stars in astrophysics.We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of(density) vacuum.We prove the L~1-stability of weak solutions for arbitrarily large data on multi-dimensional bounded or periodic domain or whole space.
In Chapter 4,we prove the existence of global weak solutions to the compressible NSP equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. The solutions are obtained as limits of solutions in annular regions between two balls.
引文
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