Non-uniqueness Of Admissible Weak Solutions To 3-D Compress- Ible Steady Euler Equations For The Supersonic Incoming Flow Past A 3-D Wedge

The uniqueness of weak solutions to the multiple Euler system is always widely and highly concerned by researchers. The first result of non-uniqueness for the multiple Euler equations was due to Scheffer in 1993. He constructed a nontrivial L2 weak so-lution to the incompressible Euler equations with compact support in time and space. Then in 2015, under an entropy condition, Camillo DeLellis and Szekelyhidi Lasz-lo applied the method of differential inclusion to the isentropic compressible Euler equations in two space dimensions, where they also used Baire Theorem to construct infinitely many bounded solutions to a Cauchy problem with some special Riemann da-ta. Multiple shocks play an important role in multiple conservation laws, while shocks occur in real life as a simultaneous phenomenon as well, like the airplanes flying with supersonic speed. We consider another physical phenomenon, when a steady superson-ic flow comes from minus infinity and hits a sharp symmetric wedge, as indicated in "Supersonic flow and shock waves", it then follows from the Rankine-Hugoniot con-ditions and the physical entropy condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a su-personic shock or a transonic shock, respectively. The appearance of the supersonic shock or the transonic shock is related to the downstream pressure. In this paper, for the supersonic flow past a 3-D wedge, we shall show that there are infinitely many admissible bounded weak solutions to 3-D steady compressible Euler equations when the concrete oblique shock structures are neglected. And the crucial point in the proof is the construction of a sub-solution, with which we can apply the methods of Camillo DeLellis and Szekelyhidi Laszlo to construct infinitely many bounded admissible weak solutions.