Euler equations with spherical symmetry and an outing absorbing boundary

In this paper, we consider the Euler equations with spherical symmetry in ℝ³ outside a region of the center. By assuming an outing absorbing boundary and within the leading term with respect to the amplitude of wave strength, we prove that there exists a uniform bound for the approximate solutions constructed by Glimm's scheme. This implies that the main difficulty left for the full nonlinear system when x ≥ 1 is to estimate the infinite reflections of waves inside the region {(x, t)|1 ≤ x ≤ 1 + λ*t, t ≥ 0}, where λ* is the supremum of the absolute value of the characteristics under consideration.