  We present a rigorous approach and related techniques to construct global
solutions of the 2-D Riemann problem with four-shock interactions for the Euler
equations for potential flow. With the introduction of three critical angles:
the vacuum critical angle from the compatibility conditions, the detachment
angle, and the sonic angle, we clarify all configurations of the Riemann
solutions for the interactions of two-forward and two-backward shocks,
including the subsonic-subsonic reflection configuration that has not emerged
in previous results. To achieve this, we first identify the three critical
angles that determine the configurations, whose existence and uniqueness follow
from our rigorous proof of the strict monotonicity of the steady detachment and
sonic angles for 2-D steady potential flow with respect to the Mach number of
the upstream state. Then we reformulate the 2-D Riemann problem into the shock
reflection-diffraction problem with respect to a symmetric line, along with two
independent incident angles and two sonic boundaries varying with the choice of
incident angles. With these, the problem can be further reformulated as a free
boundary problem for a second-order quasilinear equation of mixed
elliptic-hyperbolic type. The difficulties arise from the degenerate
ellipticity of the nonlinear equation near the sonic boundaries, the
nonlinearity of the free boundary condition, the singularity of the solution
near the corners of the domain, and the geometric properties of the free
boundary. To the best of our knowledge, this is the first rigorous result for
the 2-D Riemann problem with four-shock interactions for the Euler equations.
The approach and techniques developed for the Riemann problem for four-wave
interactions should be useful for solving other 2-D Riemann problems for more
general Euler equations and related nonlinear hyperbolic systems of
conservation laws.
