Conservation theorems and weak solutions in gas dynamics

The formulation of gas dynamics laws using conservation theorems in integral form or weak solutions is investigated. The development of the integral forms of the mass, momentum, and energy theorems is examined; the laws are presented in space-time divergence form. Haar's lemma, which states that for each weak solution of a partial differential equation a corresponding integral equation of conservation law type is valid for almost every family member, is utilized to verify the weak solutions of differential equations; the inversion of Haar's lemma is also validated. The relationship between the integral forms and weak solutions of the laws is analyzed; it is concluded that both formulations of conservation laws are equivalent. The divergence concept hypothesized by Mueller (1957) is studied and an example is provided.