  We are concerned with the stability of steady multi-wave configurations for
the full Euler equations of compressible fluid flow. In this paper, we focus on
the stability of steady four-wave configurations that are the solutions of the
Riemann problem in the flow direction, consisting of two shocks, one vortex
sheet, and one entropy wave, which is one of the core multi-wave configurations
for the two-dimensional Euler equations. It is proved that such steady
four-wave configurations in supersonic flow are stable in structure globally,
even under the BV perturbation of the incoming flow in the flow direction. In
order to achieve this, we first formulate the problem as the Cauchy problem
(initial value problem) in the flow direction, and then develop a modified
Glimm difference scheme and identify a Glimm-type functional to obtain the
required BV estimates by tracing the interactions not only between the strong
shocks and weak waves, but also between the strong vortex sheet/entropy wave
and weak waves. The key feature of the Euler equations is that the reflection
coefficient is always less than 1, when a weak wave of different family
interacts with the strong vortex sheet/entropy wave or the shock wave, which is
crucial to guarantee that the Glimm functional is decreasing. Then these
estimates are employed to establish the convergence of the approximate
solutions to a global entropy solution, close to the background solution of
steady four-wave configuration.
