Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

doi:10.1007/s00205-021-01742-4

CSpace > 应用数学研究所

	Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density
	Chen, Gui-Qiang G.1; Wang, Yong2
	2022-03-01
发表期刊	ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
ISSN	0003-9527
卷号	243 期号:3 页码:1699-1771
摘要	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 so that the total initial-energy is unbounded. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. For the large initial data of positive far-field density, various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at a 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 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 the 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, though the density may blow up near the origin at certain time and the wave propagation is not at finite speed.
DOI	10.1007/s00205-021-01742-4
收录类别	SCI
语种	英语
资助项目	UK Engineering and Physical Sciences Research Council[EP/L015811/1] ; UK Engineering and Physical Sciences Research Council[EP/V008854/1] ; Royal Society-Wolfson Research Merit Award (UK) ; National Natural Sciences Foundation of China[12022114] ; National Natural Sciences Foundation of China[11771429] ; National Natural Sciences Foundation of China[11671237] ; National Natural Sciences Foundation of China[11688101]
WOS研究方向	Mathematics ; Mechanics
WOS类目	Mathematics, Applied ; Mechanics
WOS记录号	WOS:000750982100001
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/59963
专题	应用数学研究所
通讯作者	Chen, Gui-Qiang G.
作者单位	1.Univ Oxford, Math Inst, Oxford OX2 6GG, England 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Chen, Gui-Qiang G.,Wang, Yong. Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density[J]. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS,2022,243(3):1699-1771.
APA	Chen, Gui-Qiang G.,&Wang, Yong.(2022).Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density.ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS,243(3),1699-1771.
MLA	Chen, Gui-Qiang G.,et al."Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density".ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 243.3(2022):1699-1771.