Admissibility criteria for convex integration fan solutions and contact discontinuities in the Euler equation

For piecewise constant fan subsolutions to the isentropic Euler equations, the entropy and action rates of any associated convex integration solution, relative to a common reference solution, depend only on the fan and are computed explicitly. Moreover, these explicit expressions may be decomposed into a kinetic energy mismatch and an internal energy mismatch, from which we characterize agreement of Dafermos’ entropy rate criterion with action rate criteria. The known disagreement for the Chiodaroli–Kreml two-shock data is recovered. For the planar contact reference, we prove that both criteria strictly prefer every convex integration solution associated with a strictly dissipative admissible fan to the classical contact discontinuity. Certified computations locate the Krupa–Sz'{e}kelyhidi solutions in this region, where the criteria agree, while Horimoto’s analytical construction provides such fans for every strictly increasing pressure law.