Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows and Newton's sine-squared law

We formulate a mathematical problem on hypersonic-limit of three-dimensional steady uniform non-isentropic compressible Euler flows of polytropic gases passing a straight cone with arbitrary cross-section and attacking angle, which is to study Radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2-sphere. The construction of a measure solution with density containing Dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations (ODE). For a circular cone with zero attacking angle, we then proved the Newton's sine-squared law by obtaining such a measure solution. This provides a mathematical foundation for the Newton's theory of pressure distribution on three-dimensional bodies in hypersonic flows.