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We present a rigorous approach and related techniques to construct globalsolutions of the 2-D Riemann problem with four-shock interactions for the Eulerequations for potential flow. With the introduction of three critical angles:the vacuum critical angle from the compatibility conditions, the detachmentangle, and the sonic angle, we clarify all configurations of the Riemannsolutions for the interactions of two-forward and two-backward shocks,including the subsonic-subsonic reflection configuration that has not emergedin previous results. To achieve this, we first identify the three criticalangles that determine the configurations, whose existence and uniqueness followfrom our rigorous proof of the strict monotonicity of the steady detachment andsonic angles for 2-D steady potential flow with respect to the Mach number ofthe upstream state. Then we reformulate the 2-D Riemann problem into the shockreflection-diffraction problem with respect to a symmetric line, along with twoindependent incident angles and two sonic boundaries varying with the choice ofincident angles. With these, the problem can be further reformulated as a freeboundary problem for a second-order quasilinear equation of mixedelliptic-hyperbolic type. The difficulties arise from the degenerateellipticity of the nonlinear equation near the sonic boundaries, thenonlinearity of the free boundary condition, the singularity of the solutionnear the corners of the domain, and the geometric properties of the freeboundary. To the best of our knowledge, this is the first rigorous result forthe 2-D Riemann problem with four-shock interactions for the Euler equations.The approach and techniques developed for the Riemann problem for four-waveinteractions should be useful for solving other 2-D Riemann problems for moregeneral Euler equations and related nonlinear hyperbolic systems ofconservation laws.

Global Solutions of the Two-Dimensional Riemann Problem with Four-Shock Interactions for the Euler Equations for Potential Flow