Solution of the Dorodnitsyn problem

We solve the problem of rigorous substantiation of the maximum entropy on the surface of a smooth convex nose of a body streamlined by a uniform supersonic flow of an ideal gas in the asymmetric case. Even before 1968, M. D. Ladyzhensky proposed a proof of this fact, but only for rotation bodies at certain small angles of attack. There was no estimate of the range of such angles. In this regard, academician A. A. Dorodnitsyn in 1968 stated the absence of a rigorous justification of the maximum entropy on a smooth convex nose of the surface of a body, even for rotation bodies (thereby setting the task of finding such a justification). For asymmetric nose of a body and for axisymmetric bodies at the nonzero angle of attack, a rigorous justification is obtained by G. B. Sizykh in 2019. The corresponding proof was published in short form, so that the validity of some steps of the proof was difficult to understand. The proof of 2019 is given in expanded form, with a detailed justification of all steps.