Methods and Applications of Analysis
Volume 10 (2003)
Global Entropy Solutions in Linfinity to the Euler Equations and Euler-Poisson Equations for Isothermal Fluids with Spherical Symmetry
Pages: 215 – 244
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.