## Methods and Applications of Analysis

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

###
Global Entropy Solutions in *L*^{infinity} 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 *L*^{infinity
} 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
*L*^{infinity}. 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