Existence of a global attractor for the compressible Euler equation in a bounded interval

In this paper, we are concerned with the one-dimensional initial boundary value problem for isentropic gas dynamics. Through the contribution of great researchers such as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay theory of solutions was established. They treated with the Cauchy problem and the corresponding initial data have the small total variation. On the other hand, the decay for initial data with large oscillation has been open for half a century. In addition, due to the reflection of shock waves at the boundaries, little is known for the decay of the boundary value problem on a bounded interval. Our goal is to 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.