Methods and Applications of Analysis

Global Entropy Solutions in Linfinity to the Euler Equations and Euler-Poisson Equations for Isothermal Fluids with Spherical Symmetry

Gui-Qiang Chen and Tian-Hong Li

Abstract

We prove the existence of global entropy solutions in Linfinity to the multidimensional Euler equations and Euler-Poisson equations for compressible isothermal fluids with spherically symmetric initial data that allows vacuum and unbounded velocity outside a solid ball. The multidimensional existence problem can be reduced to the existence problem for the one-dimensional Euler equations and Euler-Poisson equations with geometrical source terms. Due to the presence of the geometrical source terms, new variables- weighted density and momentum-are first introduced to transform the nonlinear system into a new nonlinear hyperbolic system to reduce the geometric source effect. We then develop a shock capturing scheme of Lax-Friedrichs type to construct approximate solutions for the weighted density and momentum. Since the velocity may be unbounded, the Courant-Friedrichs-Lewy stability condition may fail for the standard fractional-step Lax-Friedrichs scheme; hence we introduce a cut-off technique to modify the approximate density functions and adjust the ratio of the space and time mesh sizes to construct our approximate solutions. Finally we establish the convergence and consistency of the approximate solutions using the method of compensated compactness and obtain global entropy solutions in Linfinity. The solutions we obtain allow unbounded velocity near vacuum, one of the essential difficulties here, which is different from the isentropic case.

Article information

Source
Methods Appl. Anal., Volume 10, Number 2 (2003), 215-244.

Dates
First available in Project Euclid: 17 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.maa/1119018754

Mathematical Reviews number (MathSciNet)
MR2074749

Zentralblatt MATH identifier
1059.35094

Citation

Chen, Gui-Qiang; Li, Tian-Hong. Global Entropy Solutions in L infinity to the Euler Equations and Euler-Poisson Equations for Isothermal Fluids with Spherical Symmetry. Methods Appl. Anal. 10 (2003), no. 2, 215--244. https://projecteuclid.org/euclid.maa/1119018754


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