论文标题
可压缩欧拉方程的全局溶液，具有球形对称性和远场密度的大初始数据
Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
论文作者
论文摘要
我们关注的是多维压缩欧拉方程的球形对称解的全球存在理论，其远场密度的初始数据很大。溶液的主要特征是波的加强是在向内向原点径向向内移动时的增强。各种例子表明，特定时间在原点附近爆炸的欧拉方程的球形对称解。一个基本的未解决的问题是，当总的初始能量无界且波传播最初不以有限的速度启动时，全局溶液的密度是否会形成浓度，以成为附近原点的度量。在本文中，我们建立了一个全球存在理论，用于可压缩欧拉方程的球形对称解，具有大量的远场密度和相对有限能量的初始数据。这是通过通过调整一类简并密度依赖性粘度项来开发新方法的，从而使Navier-Stokes方程的全球弱解决方案消失的粘度限制具有密度依赖性粘度项的严格证明，以对相应的全球方程进行EULER方程的相应全球对法，具有大型的远光度和偏见范围的初始数据。我们的主要观察结果之一是，依赖于浅层水的粘度术语的改编的简并密度依赖性粘度项不仅包括浅水（Saint Venant）流动的Navier-Stokes方程的粘度项，而且更重要的是，更重要的是，适合于实现我们本文的主要目标。这些结果表明，即使在无限制的总初始能源不受限制的情况下，本文中构建的Navier-Stokes近似值的消失粘度极限也不会形成浓度。
We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. Various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded and the wave propagation is not at a finite speed starting initially. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit for the Navier-Stokes approximations constructed in this paper even when the total initial-energy is unbounded.
