KMS Of Academy of mathematics and systems sciences, CAS
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 | |
Source Publication | ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS |
ISSN | 0003-9527 |
Volume | 243Issue:3Pages:1699-1771 |
Abstract | 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 |
Indexed By | SCI |
Language | 英语 |
Funding Project | 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 Research Area | Mathematics ; Mechanics |
WOS Subject | Mathematics, Applied ; Mechanics |
WOS ID | WOS:000750982100001 |
Publisher | SPRINGER |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/59963 |
Collection | 应用数学研究所 |
Corresponding Author | Chen, Gui-Qiang G. |
Affiliation | 1.Univ Oxford, Math Inst, Oxford OX2 6GG, England 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China |
Recommended Citation 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. |
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