  In this paper, we are concerned with the Cauchy problem for isentropic gas
dynamics. Through the contribution of many researchers such as Lax, P. D.,
Glimm, J., DiPerna, R. J. and Liu, T. P., the decay of solutions was
established. They treated with initial data with the small total variation. On
the other hand, the decay for large initial data has been open for half a
century. Our goal is to provide a new method to analyze this problem. We prove
the existence of a global attractor, which yields a decay of solutions for
large data. To construct approximate solutions, we introduce a modified Godunov
scheme.
